Weak martingale solution of stochastic critical Oldroyd-B type models perturbed by pure jump noise

Abstract

We investigate the existence of a weak martingale solution for a two-dimensional critical viscoelastic flow of the Oldroyd type driven by pure jump Lévy noise. Due to the viscoelastic nature, noise in the equation modeling stress tensor is considered in the Marcus canonical form. Owing to the lack of dissipation and taking into account of the structure of the non-linear terms, the proof requires higher order estimates.

Keywords:

• Martingale solution
• Oldroyd-B model
• Marcus canonical form

2010 Mathematics Subject Classification:

• 60H30
• 76A10
• 37L40
• 76M35

1. Introduction

1.1. Description of the deterministic model

It is well-known that the full classical Oldroyd type models for viscoelastic fluids (see Oldroyd [1]) in ${\mathbb{R}}^2$ can be written as (1.1a) $$\frac{\partial \mathbf{v}}{\partial t}+(\mathbf{v}·\nabla )\mathbf{v}-\nu \Delta \mathbf{v}+\nabla p={\nu }_{1}Div \mathbf{\tau }\hspace{1em}\text{in}\hspace{1em}{\mathbb{R}}^2\times (0,T),$$(1.1a) (1.1b) $$\frac{\partial \mathbf{\tau }}{\partial t}+(\mathbf{v}·\nabla )\mathbf{\tau }-\kappa \Delta \tau +a\mathbf{\tau }+\mathbf{Q}(\mathbf{\tau },\nabla \mathbf{v})={\nu }_2\mathcal{D}(\mathbf{v})\hspace{1em}\text{in}\hspace{1em}{\mathbb{R}}^2\times (0,T),$$(1.1b) (1.1c) $$\nabla ·\mathbf{v}=0\mathrm{\hspace{1em}}\mathrm{\hspace{1em}}\hspace{1em} \text{in}\hspace{1em}{\mathbb{R}}^2\times (0,T),$$(1.1c) (1.1d) $$\mathbf{v}(·,0)={\mathbf{v}}_0, \mathbf{\tau }(·,0)={\mathbf{\tau }}_0\mathrm{\hspace{1em}}\mathrm{\hspace{1em}}\hspace{1em} \text{in}\hspace{1em}{\mathbb{R}}^2,$$(1.1d) where $\mathbf{v}$ is the velocity vector field which is assumed to be divergence free, $\mathit{\tau }$ is the non-Newtonian part of the stress tensor (i.e., $\mathit{\tau }(x,t)$ is a (2, 2) symmetric matrix), p is the pressure of the fluid, which is an unknown scalar, i.e., part of the solution. The parameters ν (the viscosity of the fluid), a (the reciprocal of the relaxation time), ν₁ and ν₂ (determined by the dynamical viscosity of the fluid, the retardation time and a) are assumed to be non-negative. $\mathcal{D}(\mathbf{v})$ is called the deformation tensor and is the symmetric part of the velocity gradient $$\mathcal{D}(\mathbf{v})=\frac{1}{2}(\nabla \mathbf{v}+{\nabla }^t\mathbf{v}).$$ $\mathbf{Q}$ is a quadratic form in $(\mathit{\tau },\nabla \mathbf{v})$ and for Oldroyd fluids one usually chooses $$\mathbf{Q}(\mathit{\tau },\nabla \mathbf{v})=\mathit{\tau }\mathcal{W}(\mathbf{v})-\mathcal{W}(\mathbf{v})\mathit{\tau }-b\left(\mathcal{D}(\mathbf{v})\mathit{\tau }+\mathit{\tau }\mathcal{D}(\mathbf{v})\right),$$ where $b\in [-1,1]$ is a constant and $\mathcal{W}(\mathbf{v})=\frac{1}{2}(\nabla \mathbf{v}-{\nabla }^t\mathbf{v})$ is the vorticity tensor, and is the skew-symmetric part of velocity gradient. We impose an idealized condition (or far-field condition) on the fluid, i.e., $$\mathbf{v}(x,t)\to 0\hspace{1em}\text{and}\hspace{1em}\mathit{\tau }(x,t)\to 0\hspace{1em}\text{as}\hspace{1em}\left|x\right|\to \infty .$$

The case κ = 0 in (1.1)(1.1a) $$\frac{\partial \mathbf{v}}{\partial t}+(\mathbf{v}·\nabla )\mathbf{v}-\nu \Delta \mathbf{v}+\nabla p={\nu }_{1}Div \mathbf{\tau }\hspace{1em}\text{in}\hspace{1em}{\mathbb{R}}^2\times (0,T),$$(1.1a) is a classical issue and analysis of the model under this situation is quiet challenging. For more review of such problems, we refer to [2, 3] and the references cited therein. Authors in [4] have studied the two dimensional model when there is a lack of dissipation in the velocity equation only, i.e., when $\nu =0,\kappa >0,$ and when $\mathbf{Q}$ has a special structure (that is, in the corotational case). As an improvization of [4], the authors in [5] have exploited the method of transferring dissipation and have obtained the existence of a unique global solution in three dimensional space provided the initial data is small in the H${}^s$ Sobolev-norm with $s>\frac{5}{2}.$ It is worth noting that even when $\mathbf{Q}=\mathbf{0},$ the coupling is critical with respect to the smoothing effect provided by the partial parabolic regularization (see, for instance, [4] for details). When the diffusion is present in both the equations i.e., $\nu ,\kappa >0,$ the problem is subcritical in dimension two, and global well-posedness of strong solutions can be obtained in [6] due to the smoothing effect provided by the fully parabolic system. One can clearly observe that (1.1a)(1.1a) $$\frac{\partial \mathbf{v}}{\partial t}+(\mathbf{v}·\nabla )\mathbf{v}-\nu \Delta \mathbf{v}+\nabla p={\nu }_{1}Div \mathbf{\tau }\hspace{1em}\text{in}\hspace{1em}{\mathbb{R}}^2\times (0,T),$$(1.1a) is similar to Navier-Stokes equations (NSE) if ${\nu }_1=0.$ We have fairly good understanding of two dimensional NSE in both deterministic and stochastic set-ups. Although in three dimensional NSE settings, there are still open questions on global solvability and only partial results are available in this direction. In this regard, we restrict ourselves to the two dimensional problem only.

Let us now briefly describe the connection between some deterministic systems with critical coupling which are close to our model. In particular, a critical coupling for the Boussinesq system has been studied in Hmidi et al. [7, 8], Manna and Panda [9], for the MHD system in Caflisch et al. [10], for the liquid crystal model in Lin et al. [11], to name a few. Many other interesting results are devoted to the Oldroyd-B model and related models and are available in [12–16] and the references therein.

Stochastic analysis of the above critical coupled systems influenced by random forcing are very limited and quite recent (e.g. see Yamazaki [17] for Boussinesq system with zero dissipation, Manna et al. [18] for the non-resistive MHD system, Brzeźniak et al. [19] for the liquid crystal model). Existence and asymptotic behavior of a linear visoelastic fluid equation perturbed by additive or multiplicative Wiener processes has been studied, in Barbu et al. [20] and Razafimandimby [21].

In this context, it is worthy to mention that some recent works have been devoted to understand the fluid behaviors in the micro-macro regimes. In order to build a micro-macro model, one needs to go down to the microscopic scale and make use of the kinetic theory to obtain a mathematical model for the evolution of the microstructures of the fluid (e.g. configurations of the polymer chains in the case of polymeric fluid). For a quick survey and more details, we refer to our earlier paper [2, 3] and references therein.

1.2. The stochastic model

In the present work, we are interested in the mathematical analysis of a stochastic version of (1.1a)–(1.1d). Our work is motivated by the importance of external perturbation on the dynamics of the velocity field for fluids with memory. The viscoelastic property demands that the material must return to its original shape after any deforming external force has been removed (i.e., it will show an elastic response) even though it may take time to do so. Hence the equation modeling stress tensor (i.e. the equation for $\mathbf{\tau }$) is invariant under coordinate transformation. Therefore the question of how to incorporate a suitable perturbation modeling the stress tensor without destroying its invariance property is a delicate one.

Hence for a complete knowledge of the effect of fluctuating forcing field on the behavior of the viscoelastic fluids, one needs to take into account the dynamics of $\mathbf{v}$ and $\mathbf{\tau }.$ To initiate this kind of investigation, we propose a mathematical study of the following system of equations which basically describes an approximation of the system governing the viscoelastic fluids under the influence of fluctuating external forces.

1.2.1. Physical motivation for the choice of noise

In continuum mechanics, it is well-known that in a solid material, the elastic component of the stress can be attributed to the deformation of the bonds between the tiniest particles of the material. As a subject of interest, in a fluid, the elastic stress can be assigned to the change in the mean spacing of the particles. This phenomena in the fluid affects the particles’ interaction or collision rate and hence the transfer of momentum across the fluid takes place. Therefore, it is related to the microscopic thermal random component of the particles’ motion, which motivates us to establish more realistic models by incorporating stochastic processes. The randomness leads to a variate of new phenomena and may have highly non-trivial impact on the behavior of the solution. As per the existing literature, the viscous component of the stress arises from the macroscopic mean velocity of the particles. It can be attributed to friction or particle diffusion between adjacent parcels of the medium that have different mean velocities, which inspires us to incorporate noise (both continuous and jump) for both the equations (see [22]) for further details.

1.3. Model problem

In this work we consider the following two dimensional stochastic viscoelastic equation ($\nu =0,\kappa >0$ and $\mathbf{Q}=\mathbf{0}$) with pure jump noise (in Itô-Lévy sense for the velocity equation and Marcus canonical sense for the stress tensor equation) (1.2a) $$d\mathbf{v}(t)+[(\mathbf{v}(t)·\nabla )\mathbf{v}(t)+\nabla p]dt={\nu }_{1}Div \mathit{\tau }(t)dt+{\int }_{Z}F(\mathbf{v}(t-),z){\tilde{N}}_1(dt,dz),$$(1.2a) (1.2b) $$d\mathit{\tau }(t)+[(\mathbf{v}(t)·\nabla )\mathit{\tau }(t)+a\mathit{\tau }(t)-\kappa \Delta \mathit{\tau }(t)]dt={\nu }_2\mathcal{D}(\mathbf{v}(t))dt+\sum _{i=1}^k(h_i\otimes \mathit{\tau }(t))\diamond d{L}_i(t),$$(1.2b) (1.2c) $$\nabla ·\mathbf{v}=0, \mathbf{v}(·,0)={\mathbf{v}}_0, \mathit{\tau }(·,0)={\mathit{\tau }}_0.$$(1.2c) Each $h_i;i=1,\dots ,k$ is a bounded function in ${\mathbb{R}}^{2\times 2},L(t):=(L_1(t),\cdots ,L_k(t))$ is a ${\mathbb{R}}^k$-valued pure jump Lévy process with the intensity measure λ₂ such that $\mathit{supp}{\lambda }_2\subset B,$ where $B:=\mathbb{B}(0,1)\backslash \left\{0\right\}\subset {\mathbb{R}}^k; l\in {\mathbb{R}}^k$ i.e., $$L(t)={\int }_0^t{\int }_B\mathrm{}l {\tilde{N}}_2(ds,dl)$$ where ${\tilde{N}}_2$ represents time homogeneous compensated Poisson random measure. Here ${\tilde{N}}_1$ represents time homogeneous compensated Poisson random measure on a certain measurable space $(Z,\mathcal{Z}).$ Precise definition of $\diamond$ will be stated later. The tensor product ⊗ denotes usual matrix multiplication. Properties of F have been stated in Assumption 2.3.

1.3.1. Why Marcus type noise?

The choice of the noise modeling the stress tensor for the Stratonovich type noise (for continuous noise) and Lévy noise (for jump noise) in the Marcus canonical form are due to its special property of invariance under diffeomorphism. We refer to our earlier paper [2] where the Equation (1.1a)(1.1a) $$\frac{\partial \mathbf{v}}{\partial t}+(\mathbf{v}·\nabla )\mathbf{v}-\nu \Delta \mathbf{v}+\nabla p={\nu }_{1}Div \mathbf{\tau }\hspace{1em}\text{in}\hspace{1em}{\mathbb{R}}^2\times (0,T),$$(1.1a) for velocity field is perturbed by a very general noise of multiplicative Lévy type, whereas Equation (1.1b)(1.1b) $$\frac{\partial \mathbf{\tau }}{\partial t}+(\mathbf{v}·\nabla )\mathbf{\tau }-\kappa \Delta \tau +a\mathbf{\tau }+\mathbf{Q}(\mathbf{\tau },\nabla \mathbf{v})={\nu }_2\mathcal{D}(\mathbf{v})\hspace{1em}\text{in}\hspace{1em}{\mathbb{R}}^2\times (0,T),$$(1.1b) for the stress tensor is perturbed by a Wiener noise in the Stratonovich form. However, interested readers may look into the recent manuscript [23] of the first author and his collaborator, where for the first time a certain kind of SPDE driven by Lévy noise in the Marcus canonical form has been studied. Such invariance property is known in the literature for Stratonovich integrals (see e.g. Brzeźniak and Elworthy [24, Theorem 3.3 and Proposition 4.4]). However, one needs an analogue of the Stratonovich integral in the case of stochastic integral with respect to compensated Poisson random measure. The work of Marcus [25], developed later by Applebaum and Kunita, (see e.g. Section 6.10 of [26] and [27]; see also [28]) provides a framework to resolve this technical issue. In other words, one needs a result of the following type (which is framed in SDE set-up with notations as above). Authors of the current paper have studied the critically coupled system in [2] (when $\nu >0, \kappa =0, \mathbf{Q}\ne \mathbf{0}$) under the influence of a Wiener process in Stratonovich form and in [3] in presence of Marcus noise (when $\nu >0, \kappa >0, \mathbf{Q}=\mathbf{0}$) in ${\mathbb{R}}^2.$

1.3.2. Novelty of our work

We construct a martingale solution of the stochastic critical Oldroyd-B system (1.2)(1.2a) $$d\mathbf{v}(t)+[(\mathbf{v}(t)·\nabla )\mathbf{v}(t)+\nabla p]dt={\nu }_{1}Div \mathit{\tau }(t)dt+{\int }_{Z}F(\mathbf{v}(t-),z){\tilde{N}}_1(dt,dz),$$(1.2a) with zero dissipation (in the velocity equation) and a convection diffusion (in the equation for $\mathit{\tau }$) in dimension two. The proof is based on a variant of the Faedo-Galerkin method. Furthermore, we use tightness criteria (see Appendix A.2 of the current manuscript and the references cited therein) in certain spaces of cádlág functions and Jakubowski’s generalization of the Skorokhod-Theorem to nonmetric spaces; we refer to [29, 30], Proposition 9 in [31], Proposition 3.7 in Step IV of the current manuscript.

Since the original system (3.1)(3.1a) $$d\mathbf{v}(t)+[B(\mathbf{v}(t))-{\nu }_{1}Div \mathit{\tau }(t)]dt={\int }_{Z}F(\mathbf{v}(t-),z){\tilde{N}}_1(dt,dz),\hspace{1em}t\ge 0$$(3.1a) lacks dissipation, it behaves more like an incompressible Euler equation. Hence mere ${\mathrm{L}}^2$-estimate on the velocity is not enough to talk about well-posedness of such problems and one needs to consider more regular initial data to obtain an enstrophy estimate. Besides, the rigorous analysis involved in this paper owing to the critical coupling of the random system in presence of pure jump noise is quiet technical and delicate compared to the deterministic ones, see [4, 5].

Proposition 1.1.

Let L(t) be a ${\mathbb{R}}^k$-valued pure jump Lévy process with the intensity measure λ such that $\mathit{supp}\lambda \subset B:=\mathbb{B}(0,1)\subset {\mathbb{R}}^k; l\in {\mathbb{R}}^k$ i.e., $$L(t)={\int }_0^t{\int }_B\mathrm{}l \tilde{N}(ds,dl)$$ where $\tilde{N}$ represents time homogeneous compensated Poisson random measure. Let $\mathbf{u},{\mathbf{w}}_1,\dots ,{\mathbf{w}}_k:{\mathbb{R}}^2\to {\mathbb{R}}^2$ be complete C${}^1$-vector fields. Define $\mathbf{w}:{\mathbb{R}}^2\to \mathcal{L}({\mathbb{R}}^k,{\mathbb{R}}^2)$ such that $\mathbf{w}(y)(h):={\sum }_{j=1}^k{\mathbf{w}}_j(y)h_j,h\in {\mathbb{R}}^k,y\in {\mathbb{R}}^2$. Consider the following “Marcus” stochastic differential equation (1.3) $$dY(t)=\mathbf{u}(Y(t)) dt+\mathbf{w}(Y(t))\diamond dL(t)=\mathbf{u}(Y(t)) dt+\sum _{j=1}^k{\mathbf{w}}_j(Y(t))\diamond d{L}_j(t),$$(1.3) which is defined in the integral form as follows $$\begin{array}{c}Y(t)=Y_0+{\int }_0^t\mathbf{u}(Y(s)) ds+{\int }_0^t{\int }_B[\Phi (1,l,Y(s-))-Y(s-)]\tilde{N}(ds,dl)\\ \hspace{1em}+{\int }_0^t{\int }_B[\Phi (1,l,Y(s))-Y(s)-\sum _{j=1}^k{l}_j{\mathbf{w}}_j(Y(s))]\lambda (dl)ds,\end{array}$$ where $z(t):=\Phi (t,l,z_0)$ solves $$\frac{dz}{dt}=\sum _{j=1}^k{l}_j{\mathbf{w}}_j(z),\hspace{1em}\mathit{\text{with}}\mathrm{ }\mathit{\text{initial}}\mathrm{ }\mathit{\text{condition}}\mathrm{ }z(0)=z_0.$$ Suppose, Y, an ${\mathbb{R}}^2$-valued process, is a solution to (1.3)(1.3) $$dY(t)=\mathbf{u}(Y(t)) dt+\mathbf{w}(Y(t))\diamond dL(t)=\mathbf{u}(Y(t)) dt+\sum _{j=1}^k{\mathbf{w}}_j(Y(t))\diamond d{L}_j(t),$$(1.3) . Let $\phi :{\mathbb{R}}^2\to {\mathbb{R}}^2$ be a C${}^1$ diffeomorphism. Define for each $j=0,1,\dots ,k$, the “Push-forward” of the vector fields $\mathbf{u}$ and ${\mathbf{w}}_j$ by $\phi \prime$ as $$\begin{array}{c}\widehat{\mathbf{u}}:{\mathbb{R}}^2\ni y\mapsto \phi \prime ({\phi }^{-1}(y))(\mathbf{u}({\phi }^{-1}(y)))\in {\mathbb{R}}^2,\\ {\widehat{\mathbf{w}}}_j:{\mathbb{R}}^2\ni y\mapsto \phi \prime ({\phi }^{-1}(y))({\mathbf{w}}_j({\phi }^{-1}(y)))\in {\mathbb{R}}^2.\end{array}$$ Let $\widehat{\mathbf{w}}:{\mathbb{R}}^2\to \mathcal{L}({\mathbb{R}}^k,{\mathbb{R}}^2)$ be such that $\widehat{\mathbf{w}}(y)(h):={\sum }_{j=1}^k{\widehat{\mathbf{w}}}_j(y)h_j,$ $h\in {\mathbb{R}}^k.$

Then Y is a solution to (1.3)(1.3) $$dY(t)=\mathbf{u}(Y(t)) dt+\mathbf{w}(Y(t))\diamond dL(t)=\mathbf{u}(Y(t)) dt+\sum _{j=1}^k{\mathbf{w}}_j(Y(t))\diamond d{L}_j(t),$$(1.3) if and only if $Z(t):=\phi (Y(t))$ is a solution to $$dZ(t)=\widehat{\mathbf{u}}(Z(t)) dt+\widehat{\mathbf{w}}(Z(t))\diamond dL(t),\hspace{1em}{Z}_0=\phi (Y_0).$$

Proof of the above result can be borrowed from [32, Theorem B.1], where Brzeźniak and the first named author of this article have obtained the result for the pure jump case.

Let us briefly describe the content of the paper. In Section 2, we present detailed description of the functional spaces, the Marcus map and assumptions of this paper. In Section 3, we discuss statement of the main result and its proof.

Appendix A is split into two parts. The very first subsection has been presented by series of technical lemmas which are useful during the course of analysis made throughout the paper. The second subsection has been devoted to certain useful compactness and tightness criterion for cádlag functions and predictable processes.

2. Functional settings, assumptions and the Marcus map

We denote $C_c^{\infty }({\mathbb{R}}^2;{\mathbb{R}}^2)$ as the space of all ${\mathbb{R}}^2-$valued compactly supported $C^{\infty }$ functions in ${\mathbb{R}}^2,$ $$\mathcal{V}:=\{\mathbf{v}\in C_c^{\infty }({\mathbb{R}}^2;{\mathbb{R}}^2):\mathit{Div} \mathbf{v}=0\}, \tilde{\mathcal{V}}:=\{\mathit{\tau }\in C_c^{\infty }({\mathbb{R}}^2;{\mathbb{R}}^4):{\mathit{\tau }}^T=\mathit{\tau }\}.$$ We define $H,V,\tilde{H}$ and $\tilde{V}$ be the closure of $\mathcal{V}$ in ${\mathrm{L}}^2({\mathbb{R}}^2;{\mathbb{R}}^2),H^1({\mathbb{R}}^2;{\mathbb{R}}^2),{\mathrm{L}}^2({\mathbb{R}}^2;{\mathbb{R}}^4)$ and $H^1({\mathbb{R}}^2;{\mathbb{R}}^4)$ respectively. For $s>0,$ let V_s and $\widetilde{V_s}$ be the closure of $\mathcal{V}$ and $\tilde{\mathcal{V}}$ in $H^s({\mathbb{R}}^2;{\mathbb{R}}^2)$ and $H^s({\mathbb{R}}^2;{\mathbb{R}}^4)$ respectively. It is clear from the definitions that $V_1=V$ and ${\tilde{V}}_1=\tilde{V}.$

Notation 2.1.

Throughout this paper, we denote $X\prime$ as the dual of X where X is a Hilbert space and C as a generic constant.

We note the following:

1. H is a Hilbert space with scalar product ${(\mathbf{u},\mathbf{v})}_H:={(\mathbf{u},\mathbf{v})}_{{\mathrm{L}}^2}, \forall \mathbf{u},\mathbf{v}\in H$ and V is a Hilbert space with scalar product ${(\mathbf{u},\mathbf{v})}_V:={(\mathbf{u},\mathbf{v})}_{{\mathrm{L}}^2}+{(\nabla \mathbf{u},\nabla \mathbf{v})}_{{\mathrm{L}}^2}, \forall \mathbf{u},\mathbf{v}\in V.$

2. $\tilde{H}$ is a Hilbert space with scalar product ${(\mathbf{u},\mathbf{v})}_{\tilde{H}}:={(\mathbf{u},\mathbf{v})}_{{\mathrm{L}}^2} \forall \mathbf{u},\mathbf{v}\in \tilde{H}$ and $\tilde{V}$ is a Hilbert space with scalar product ${(\mathbf{u},\mathbf{v})}_{\tilde{V}}:={(\mathbf{u},\mathbf{v})}_{{\mathrm{L}}^2}+{(\nabla \mathbf{u},\nabla \mathbf{v})}_{{\mathrm{L}}^2},\hspace{1em}\forall \mathbf{u},\mathbf{v}\in \tilde{V}.$

2.1. Trilinear operators b and $\tilde{b}$

We define the trilinear operators $b:{\mathrm{L}}^p({\mathbb{R}}^2;{\mathbb{R}}^2)\times W^{1,q}({\mathbb{R}}^2;{\mathbb{R}}^2)\times {\mathrm{L}}^r({\mathbb{R}}^2;{\mathbb{R}}^2)\to \mathbb{R}$ and $\tilde{b}:{\mathrm{L}}^p({\mathbb{R}}^2;{\mathbb{R}}^2)\times W^{1,q}({\mathbb{R}}^2;{\mathbb{R}}^4)\times {\mathrm{L}}^r({\mathbb{R}}^2;{\mathbb{R}}^4)\to \mathbb{R}$ by $$\begin{array}{c}b(\mathbf{u},\mathbf{v},\mathbf{w}):=\sum _{i,j=1}^2{\int }_{{\mathbb{R}}^2}{\mathbf{u}}^{(i)}{\partial }_{x_i}{\mathbf{v}}^{(j)}{\mathbf{w}}^{(j)}dx,\hspace{1em}\mathbf{u}\in {\mathrm{L}}^p({\mathbb{R}}^2;{\mathbb{R}}^2), \mathbf{v}\in W^{1,q}({\mathbb{R}}^2;{\mathbb{R}}^2), \mathbf{w}\in {\mathrm{L}}^r({\mathbb{R}}^2;{\mathbb{R}}^2)\\ \tilde{b}(\mathbf{v},\mathit{\tau },\theta ):=\sum _{i,j,k=1}^2{\int }_{{\mathbb{R}}^2}{\mathbf{v}}^{(i)}{\partial }_{x_i}{\mathit{\tau }}^{(jk)}{\theta }^{(jk)}dx,\hspace{1em}\mathbf{v}\in {\mathrm{L}}^p({\mathbb{R}}^2;{\mathbb{R}}^2), \mathit{\tau }\in W^{1,q}({\mathbb{R}}^2;{\mathbb{R}}^4), \theta \in {\mathrm{L}}^r({\mathbb{R}}^2;{\mathbb{R}}^4)\end{array}$$ with $\frac{1}{p}+\frac{1}{q}+\frac{1}{r}=1$ and $p,q,r\in [1,\infty ].$ Let us define bilinear maps $B:V\times V\to V\prime$ and $\tilde{B}:V\times \tilde{V}\to \tilde{V}\prime$ by $B(\mathbf{u},\mathbf{v})=b(\mathbf{u},\mathbf{v},·)$ and $\tilde{B}(\mathbf{v},\mathit{\tau })=\tilde{b}(\mathbf{v},\mathit{\tau },·).$ Then we can show that $B:V\times V\to V\prime$ and $\tilde{B}:V\times \tilde{V}\to \tilde{V}\prime$ are linear and continuous.

2.2. Some additional operators and notations

If $s>2,$ then by the Sobolev embedding theorem $$H^{s-1}({\mathbb{R}}^2;{\mathbb{R}}^2)\subset C_b({\mathbb{R}}^2;{\mathbb{R}}^2)\subset {\mathrm{L}}^{\infty }({\mathbb{R}}^2;{\mathbb{R}}^2)$$ where $C_b({\mathbb{R}}^2;{\mathbb{R}}^2)$ denotes the space of continuous and bounded ${\mathbb{R}}^2$ valued functions defined on ${\mathbb{R}}^2.$ We have the following conclusions: (2.1) $$V_s\underset{j_s}{\hookrightarrow }V\underset{j}{\hookrightarrow }H\cong H\prime \underset{j\prime }{\hookrightarrow }V\prime \underset{j{\prime }_s}{\hookrightarrow }{V}_s^{\prime },\mathrm{\hspace{1em}}\mathrm{\hspace{1em}}\text{and}\mathrm{\hspace{1em}}\mathrm{\hspace{1em}}\widetilde{V_s}\underset{{\tilde{j}}_s}{\hookrightarrow }\tilde{V}\underset{\tilde{j}}{\hookrightarrow }\tilde{H}\cong \tilde{H}\prime \underset{\tilde{j}\prime }{\hookrightarrow }\tilde{V}\prime \underset{{\tilde{j}}_s^{\prime }}{\hookrightarrow }\widetilde{V_s}\prime .$$(2.1) The following embeddings $j,j_s,\tilde{j},{\tilde{j}}_s$ are continuous and dense. By Lemma C.1 in Appendix C of [23], there exists a Hilbert space $\mathcal{U}$ such that $\mathcal{U}\hookrightarrow V_s$ and (2.2) $$\text{the}\mathrm{ }\text{natural}\mathrm{ }\text{embedding}\mathrm{ }{l}_s:\mathcal{U}\hookrightarrow V_s\mathrm{ }\text{is}\mathrm{ }\text{dense}\mathrm{ }\text{and}\mathrm{ }\text{compact}.$$(2.2) Combining (2.1)(2.1) $$V_s\underset{j_s}{\hookrightarrow }V\underset{j}{\hookrightarrow }H\cong H\prime \underset{j\prime }{\hookrightarrow }V\prime \underset{j{\prime }_s}{\hookrightarrow }{V}_s^{\prime },\mathrm{\hspace{1em}}\mathrm{\hspace{1em}}\text{and}\mathrm{\hspace{1em}}\mathrm{\hspace{1em}}\widetilde{V_s}\underset{{\tilde{j}}_s}{\hookrightarrow }\tilde{V}\underset{\tilde{j}}{\hookrightarrow }\tilde{H}\cong \tilde{H}\prime \underset{\tilde{j}\prime }{\hookrightarrow }\tilde{V}\prime \underset{{\tilde{j}}_s^{\prime }}{\hookrightarrow }\widetilde{V_s}\prime .$$(2.1) and (2.2)(2.2) $$\text{the}\mathrm{ }\text{natural}\mathrm{ }\text{embedding}\mathrm{ }{l}_s:\mathcal{U}\hookrightarrow V_s\mathrm{ }\text{is}\mathrm{ }\text{dense}\mathrm{ }\text{and}\mathrm{ }\text{compact}.$$(2.2) we have $$\mathcal{U}\underset{l_s}{\hookrightarrow }{V}_s\underset{j_s}{\hookrightarrow }V\underset{j}{\hookrightarrow }H\cong H\prime \underset{j\prime }{\hookrightarrow }V\prime \underset{j{\prime }_s}{\hookrightarrow }{V}_s^{\prime }\underset{l{\prime }_s}{\hookrightarrow }\mathcal{U}\prime .$$ Let us consider the adjoint of the following maps $j,j_s,l_s$ denoted by $j^{*},j_s^{*},l_s^{*}$ respectively and are defined as: $$j^{*}:H\to V, j_s^{*}:V\to V_s, l_s^{*}:V_s\to \mathcal{U}.$$ Since all the embeddings $j,j_s,l_s$ are dense so $j^{*},j_s^{*},l_s^{*}$ are one-one. Let us define the map $i:\mathcal{U}\hookrightarrow H\hspace{1em}\text{by}\hspace{1em}i:=j°j_s°l_s.$ Note that i is compact and range of i is dense in H. We consider the adjoint of i defined as $i^{*}:H\to \mathcal{U},$ which is one-one. In the similar manner, one can have $\tilde{\mathcal{U}}\underset{{\tilde{l}}_s}{\hookrightarrow }\widetilde{V_s}\underset{{\tilde{j}}_s}{\hookrightarrow }\tilde{V}\underset{\tilde{j}}{\hookrightarrow }\tilde{H}\cong \tilde{H}\prime \underset{\tilde{j}\prime }{\hookrightarrow }\tilde{V}\prime \underset{{\tilde{j}}_s^{\prime }}{\hookrightarrow }\widetilde{V_s}\prime \underset{{\tilde{l}}_s^{\prime }}{\hookrightarrow }\tilde{\mathcal{U}}\prime$ and $\tilde{i}:\tilde{\mathcal{U}}\hookrightarrow \tilde{H}\hspace{1em}\text{by}\hspace{1em}\tilde{i}:=\tilde{j}°{\tilde{j}}_s°{\tilde{l}}_s.$ Let us define $(A,D(A))$ by $$D(A):=j^{*}(H)\subset V, A\mathbf{v}:={(j^{*})}^{-1}\mathbf{v},\hspace{1em}\forall \mathbf{v}\in D(A).$$ In the similar manner, we can define $(A_s,D(A_s)),(\mathfrak{A},D(\mathfrak{A})),({\mathfrak{A}}_s,D({\mathfrak{A}}_s)),(\tilde{A},D(\tilde{A})),({\tilde{A}}_s,D({\tilde{A}}_s)),(\tilde{\mathfrak{A}},D(\tilde{\mathfrak{A}}))$ and $({\tilde{\mathfrak{A}}}_s,D({\tilde{\mathfrak{A}}}_s)),$ corresponding to maps ${(j_s^{*})}^{-1},$ ${(i^{*})}^{-1},{(i_s^{*})}^{-1},{({(\tilde{j})}^{*})}^{-1},{({({\tilde{j}}_s)}^{*})}^{-1},{({(\tilde{i})}^{*})}^{-1}$ and ${({({\tilde{i}}_s)}^{*})}^{-1}$ respectively. We note that $\mathfrak{A}=A°A_s°{\mathfrak{A}}_s$ and $\tilde{\mathfrak{A}}=\tilde{A}°{\tilde{A}}_s°{\tilde{\mathfrak{A}}}_s.$ Note that for each $\mathbf{v}\in V,$ using the continuity of $\mathbf{u}\mapsto {(\nabla \mathbf{v},\nabla \mathbf{u})}_H,$ one can conclude existence of an element $\mathcal{A}\mathbf{v}\in V\prime$ such that $${}_{V\prime }{〈\mathcal{A}\mathbf{v},\mathbf{u}〉}_V={(\nabla \mathbf{v},\nabla \mathbf{u})}_H,\hspace{1em}\forall \mathbf{v},\mathbf{u}\in V.$$ Similarly we can define $\tilde{\mathcal{A}}:\tilde{V}\to \tilde{V}\prime$ by ${}_{\tilde{V}\prime }{〈\tilde{\mathcal{A}}\mathit{\tau },\mathbf{w}〉}_{\tilde{V}}={(\nabla \mathit{\tau },\nabla \mathbf{w})}_H,\hspace{1em}\forall \mathit{\tau },\mathbf{w}\in \tilde{V}.$

Remark 1.

In view of [23], one can observe that both $\mathfrak{A},\tilde{\mathfrak{A}}$ are self adjoint and ${\mathfrak{A}}^{-1},{(\tilde{\mathfrak{A}})}^{-1}$ are compact.

Lemma 2.2.

For $\mathbf{v}\in D(A),\mathbf{u}\in V,\mathit{\tau }\in D(\tilde{A}),\mathbf{w}\in \tilde{V},$ the following results hold:

1. ${((A-I)\mathbf{v},\mathbf{u})}_H={(\nabla \mathbf{v},\nabla \mathbf{u})}_H{=}_{V\prime }{〈\mathcal{A}\mathbf{v},\mathbf{u}〉}_V, \mathit{\text{and}} \hspace{1em}|\mathcal{A}\mathbf{v}{|}_{V\prime }\le |A\mathbf{v}-\mathbf{v}{|}_H.$

2. ${((\tilde{A}-I)\mathit{\tau },\mathbf{w})}_{\tilde{H}}={(\nabla \mathit{\tau },\nabla \mathbf{w})}_{\tilde{H}}{=}_{\tilde{V}\prime }{〈\tilde{\mathcal{A}}\mathit{\tau },\mathbf{w}〉}_{\tilde{V}}, \mathit{\text{and}} \hspace{1em}|\tilde{\mathcal{A}}\mathit{\tau }{|}_{\tilde{V}\prime }\le |\tilde{A}\mathit{\tau }-\mathit{\tau }{|}_{\tilde{H}}.$

For proof see [23].

Let $(S,\mathcal{S})$ be a measurable space and let $M_{\widehat{\mathbb{N}}}(S)$ be the set of all $\widehat{\mathbb{N}}$ valued measures on $(S,\mathcal{S}),$ where $\widehat{\mathbb{N}}=\mathbb{N}\cup \{0,\infty \}.$ On the set $M_{\widehat{\mathbb{N}}}(S),$ we consider the $\sigma -$field ${\mathcal{M}}_{\widehat{\mathbb{N}}}(S)$ defined as the smallest $\sigma -$field for all $A\in \mathcal{S}:$ the map $$i_A:M_{\widehat{\mathbb{N}}}(S)\ni {\lambda }_j\mapsto {\lambda }_j(A)\in \widehat{\mathbb{N}}$$ is measurable.

Let $(Z,\mathcal{Z})$ be a measurable space. Let N₁ on $(Z,\mathcal{Z})$ over a filtered probability space $(\Omega ,\mathcal{F},\mathbb{F},\mathbb{P})$ be the time homogeneous Poisson random measure (for definition, see e.g. [33]), which is known to be a measurable function $$N_1:(\Omega ,\mathcal{F})\to \left(M_{\widehat{\mathbb{N}}}({\mathbb{R}}_{+}\times Z),{\mathcal{M}}_{\widehat{\mathbb{N}}}({\mathbb{R}}_{+}\times Z)\right).$$ The formula $${\lambda }_1(A):=\mathbb{E}\left[N_1((0,1]\times A)\right],\hspace{1em}A\in \mathcal{Z}$$ defines a measure on $(Z,\mathcal{Z}),$ called an intensity measure of N₁. Moreover, for all $T<\infty$ and all $A\in \mathcal{Z}$ such that $\mathbb{E}\left[N_1((0,T]\times A)\right]<\infty ,$ the $\mathbb{R}$-valued process ${\left\{N_1((0,t]\times A)-t{\lambda }_1(A)\right\}}_{t\in (0,T]}$ is an integrable martingale on $(\Omega ,\mathcal{F},\mathbb{F},\mathbb{P}).$ The random measure $\mathit{Leb}\otimes {\lambda }_1$ on $\mathcal{B}({\mathbb{R}}_{+})\otimes \mathcal{Z}$ is called the compensator of N₁ and ${\tilde{N}}_1:=N_1-\mathit{Leb}\otimes {\lambda }_1$ is called the compensated time homogeneous Poisson random measure.

Similarly, one can define the time homogeneous Poisson random measure N₂.

Assumption 2.3.

Below we state the assumptions made throughout this paper.

(A.1) $\mathfrak{L}=(\Omega ,\mathcal{F},\mathbb{F},\mathbb{P})$ is a filtered probability space, where $\mathbb{F}={({\mathcal{F}}_t)}_{t\ge 0}$ is the filtration, satisfying the usual conditions, i.e.,

1. $\mathbb{P}$ is complete on $(\Omega ,\mathcal{F}),$

2. for each $t\ge 0,\mathrm{ }{\mathcal{F}}_t$ contains all $(\mathbb{F},\mathbb{P})$-null sets,

3. the filtration ${\mathcal{F}}_t$ is right-continuous.

(A.2) N₁ is a time homogeneous Poisson random measure on a measurable space $(Z,\mathcal{Z})$ over the above probability space with intensity measure λ₁ (and compensator $\mathit{Leb}\otimes {\lambda }_1$).

(A.3) ${(L(t))}_{t\ge 0}$ is a ${\mathbb{R}}^k$-valued, $({\mathcal{F}}_t)$-adapted Lévy process of pure jump type defined on the above probability space with drift 0 and the corresponding time homogenous Poisson random measure N₂.

(A.4) The intensity measure λ₂ of N₂ (with compensator $\mathit{Leb}\otimes {\lambda }_2$) is such that $\mathit{supp}{\lambda }_2\subset B$ and ${\lambda }_2(B)<\infty ,$ where $B:=\mathbb{B}(0,1)\backslash \left\{0\right\}\subset {\mathbb{R}}^k.$

(A.5) N₁ and N₂ are mutually independent.

(A.6) $h_i\in W^{1,\infty },$ for each $i=1,2,\dots k.$

(A.7) $F:H\times Z\to H$ is a measurable function such that there exists a positive constant L such that (2.3) $${\int }_Z|F({\mathbf{v}}_1,z)-F({\mathbf{v}}_2,z){|}_H^2 {\lambda }_1(dz)\le L|{\mathbf{v}}_1-{\mathbf{v}}_2{|}_H^2,{\mathbf{v}}_1,{\mathbf{v}}_2\in H,$$(2.3) and for each $p\ge 1$ there exists a positive constant C_p such that (2.4) $${\int }_Z|F(\mathbf{v},z){|}_H^p {\lambda }_1(dz)\le C_p(1+|\mathbf{v}{|}_H^p),\hspace{1em}\mathbf{v}\in H.$$(2.4)

(A.8) $F:V\times Z\to V$ is a measurable function such that there exists a positive constant C_p, depending upon $p\ge 2,$ such that $${\int }_Z|C\mathit{url} F(\mathbf{v},z){|}_H^p{\lambda }_1(dz)\le C_p(1+|\nabla \mathbf{v}{|}_H^p), \mathrm{ }\mathbf{v}\in V.$$

Assumption (A.8) is required for the gradient and curl estimates in Lemma 3.4 and Corollary 3.5.

2.3. The Marcus map

For general discussion about this topic, we refer to the Appendix A. For $h_i\in W^{1,\infty },$ we consider the bounded linear map $g_i:\tilde{H}\to \tilde{H}\hspace{1em}\text{by}\hspace{1em}{g}_i(\mathit{\tau })=h_i\otimes \mathit{\tau }.$ It is easy to observe that $g_i:\tilde{V}\to \tilde{V}$ is also a continuous linear map. We define a generalized Marcus mapping $\Phi :{\mathbb{R}}_{+}\times {\mathbb{R}}^k\times \tilde{H}\to \tilde{H}$ such that for each fixed $l\in {\mathbb{R}}^k, {\mathit{\tau }}_0\in \tilde{H},$ the function $t\mapsto \Phi (t,l,{\mathit{\tau }}_0)$ is the continuously differentiable solution of the ordinary differential equation $$\frac{dy(t)}{dt}=\sum _{i=1}^k{l}_i{g}_i(y(t))\hspace{1em}t\ge 0,\hspace{1em}y(0)={\mathit{\tau }}_0.$$ Therefore, we can write $$\Phi (t,l,{\mathit{\tau }}_0)=\Phi (0,l,{\mathit{\tau }}_0)+\sum _{i=1}^k{\int }_0^t{l}_i{g}_i(\Phi (s,l,{\mathit{\tau }}_0))ds,\hspace{1em}t\ge 0.$$

Notation.

Let us fix t = 1 and denote $\Phi (l,·)=\Phi (1,l,·).$

Equation (1.2)(1.2a) $$d\mathbf{v}(t)+[(\mathbf{v}(t)·\nabla )\mathbf{v}(t)+\nabla p]dt={\nu }_{1}Div \mathit{\tau }(t)dt+{\int }_{Z}F(\mathbf{v}(t-),z){\tilde{N}}_1(dt,dz),$$(1.2a) with notation $\diamond$ is defined in the integral form as following: (2.5) $$\begin{array}{c}\mathit{\tau }(t)={\mathit{\tau }}_0-{\int }_0^t[(\mathbf{v}(s)·\nabla )\mathit{\tau }(s)+a\mathit{\tau }(s)+\kappa \Delta \mathit{\tau }(s)-{\nu }_2\mathcal{D}(\mathbf{v}(s))]ds+{\int }_0^t{\int }_B[\Phi (l,\mathit{\tau }(s-))-\mathit{\tau }(s-)]{\tilde{N}}_2(ds,dl)\\ \hspace{1em}+{\int }_0^t{\int }_B\{\Phi (l,\mathit{\tau }(s))-\mathit{\tau }(s)-\sum _{i=1}^k{l}_i{g}_i(\mathit{\tau }(s))\}{\lambda }_2(dl)ds,\end{array}$$(2.5) where ${\mathit{\tau }}_0$ is a ${\mathcal{F}}_0$-measurable random variable.

For $z\in \tilde{H},$ we denote $G(l,z):=\Phi (l,z)-z, K(l,z):=\Phi (l,z)-z-{\sum }_{i=1}^k{l}_i{g}_i(z),\text{and} b(z):={\int }_{B}K(l,z) {\lambda }_2(dl).$ With these above notations, (2.5)(2.5) $$\begin{array}{c}\mathit{\tau }(t)={\mathit{\tau }}_0-{\int }_0^t[(\mathbf{v}(s)·\nabla )\mathit{\tau }(s)+a\mathit{\tau }(s)+\kappa \Delta \mathit{\tau }(s)-{\nu }_2\mathcal{D}(\mathbf{v}(s))]ds+{\int }_0^t{\int }_B[\Phi (l,\mathit{\tau }(s-))-\mathit{\tau }(s-)]{\tilde{N}}_2(ds,dl)\\ \hspace{1em}+{\int }_0^t{\int }_B\{\Phi (l,\mathit{\tau }(s))-\mathit{\tau }(s)-\sum _{i=1}^k{l}_i{g}_i(\mathit{\tau }(s))\}{\lambda }_2(dl)ds,\end{array}$$(2.5) can be written as: $$\begin{array}{c}\mathit{\tau }(t)={\mathit{\tau }}_0-{\int }_0^t[(\mathbf{v}(s)·\nabla )\mathit{\tau }(s)+a\mathit{\tau }(s)+\kappa \Delta \mathit{\tau }(s)-{\nu }_2\mathcal{D}(\mathbf{v}(s))]ds\\ \hspace{1em}+{\int }_0^t{\int }_{B}G(l,\mathit{\tau }(s-)){\tilde{N}}_2(ds,dl)+{\int }_0^{t}b(\mathit{\tau }(s))ds.\end{array}$$ Now, define a linear operator $$\mathcal{R}:\tilde{H}\ni \mathit{\tau }\mapsto \sum _{i=1}^k{l}_i{g}_i(\mathit{\tau })\in \tilde{H}.$$ Then $\Vert \mathcal{R}{\Vert }_{\mathcal{L}(\tilde{H})}\le |l{|}_{{\mathbb{R}}^k}\Vert g{\Vert }_{\mathcal{L}(\tilde{H})}.$

Thus, if we denote $y(t):=\Phi (t,l,x),$ then y satisfies $\frac{dy}{dt}=\mathcal{R}y,y(0)=x.$

Hence $y(t)=e^{t\mathcal{R}}x={\sum }_{j=0}^{\infty }\frac{t^j}{j!}{\mathcal{R}}^{j}x.$

We then have the following useful result. We postpone the proof to Appendix A.

Lemma 2.4.

Let $\psi :\tilde{H}\to \mathbb{R}$ be defined by $\psi (\mathit{\tau })=|\mathit{\tau }{|}_{\tilde{H}}^p,\mathrm{ }p\ge 1.$ If $\mathfrak{N}:={\int }_0^1\Vert e^{s\mathcal{R}}{\Vert }^{p}ds,$ then

1. $|\psi (\Phi (l,\mathit{\tau }))-\psi (\mathit{\tau })|\le \mathfrak{N} p \left|l{|}_{{\mathbb{R}}^k}\Vert g{\Vert }_{\mathcal{L}(\tilde{H})}\right|\mathit{\tau }{|}_{\tilde{H}}^p.$

2. $|\psi (\Phi (l,\mathit{\tau }))-\psi (\mathit{\tau })-\psi \prime (\mathit{\tau })\mathcal{R}\mathit{\tau }|\le \mathfrak{N} p^2 \left|l{|}_{{\mathbb{R}}^k}^2\Vert g{\Vert }_{\mathcal{L}(\tilde{H})}^2\right|\mathit{\tau }{|}_{\tilde{H}}^p.$

3. Main result and key ideas of the proof

We now consider the following stochastic critical Oldroyd-B system (with zero dissipation in the velocity equation and a convection diffusion equation for $\mathit{\tau }$) in dimension two (3.1a) $$d\mathbf{v}(t)+[B(\mathbf{v}(t))-{\nu }_{1}Div \mathit{\tau }(t)]dt={\int }_{Z}F(\mathbf{v}(t-),z){\tilde{N}}_1(dt,dz),\hspace{1em}t\ge 0$$(3.1a) (3.1b) $$d\mathit{\tau }(t)+[\kappa \tilde{\mathcal{A}}\mathit{\tau }(t)+\tilde{B}(\mathbf{v}(t),\mathit{\tau }(t))+a\mathit{\tau }(t)-{\nu }_2\mathcal{D}(\mathbf{v}(t))]dt={\int }_{B}G(l,\mathit{\tau }(t-)){\tilde{N}}_2(dt,dl)+b(\mathit{\tau }(t))dt,\hspace{1em}t\ge 0$$(3.1b) (3.1c) $$\mathbf{v}(0)={\mathbf{v}}_0; \mathit{\tau }(0)={\mathit{\tau }}_0.$$(3.1c) We first introduce the following functional spaces endowed with the respective topologies: $$D([0,T];\mathcal{U}\prime ):=\text{the}\mathrm{ }\text{space}\mathrm{ }\text{of}\mathrm{ }\mathrm{c}\stackrel{`}{\mathrm{a}}\text{dl}\stackrel{`}{\mathrm{a}}\mathrm{g}\mathrm{ }\text{functions} \mathbf{v}:[0,T]\to \mathcal{U}\prime$$ with the topology ${\mathcal{T}}_1$ induced by the Skorokhod metric ${\delta }_{T,\mathcal{U}\prime };$ $${\mathrm{L}}_w^2(0,T;V):=\text{the space} {\mathrm{L}}^2(0,T;V) \text{with the weak topology} {\mathcal{T}}_2;$$ $${\mathrm{L}}^2(0,T;H_{\mathit{loc}}):=\text{the space of measurable functions} \mathbf{v}:[0,T]\to H \text{such that for all} R>0,$$ $$p_{T,R}(\mathbf{v}):=\Vert \mathbf{v}{\Vert }_{{\mathrm{L}}^2(0,T;H_{B_R})}:={\left({\int }_0^T{\int }_{B_R}|\mathbf{v}(x,t){|}_H^2 dx dt\right)}^{1/2}<\infty ,$$ with the topology ${\mathcal{T}}_3$ generated by the seminorms ${(p_{T,R})}_{R>0},$ where B_R is a closed ball of radius R; $$H_w:=\text{the}\mathrm{ }\text{Hilbert}\mathrm{ }\text{space} H \text{endowed}\mathrm{ }\text{with}\mathrm{ }\text{the}\mathrm{ }\text{weak}\mathrm{ }\text{topology};$$ $$D([0,T];H_w):=\text{the}\mathrm{ }\text{space}\mathrm{ }\text{of}\mathrm{ }\text{all}\mathrm{ }\text{weakly}\mathrm{ }\mathrm{c}\stackrel{`}{\mathrm{a}}\text{dl}\stackrel{`}{\mathrm{a}}g\mathrm{ }\text{functions} \mathbf{v}:[0,T]\to \mathrm{ }H$$ with the weakest topology ${\mathcal{T}}_4$ such that for all $h\in H$ the mappings $$D([0,T];H_w)\ni \mathbf{v}\mapsto {(\mathbf{v}(·),h)}_H\in D([0,T];\mathbb{R})$$ are continuous. In particular, ${\mathbf{v}}_n\to \mathbf{v}$ in $D([0,T];H_w)$ iff for all $h\in H$ $${({\mathbf{v}}_n(·),h)}_H\to {(\mathbf{v}(·),h)}_H\mathrm{ }\text{in}\mathrm{ }D([0,T];\mathbb{R}).$$ Similarly, one can define the spaces $D([0,T];\tilde{\mathcal{U}}\prime ),\mathrm{ }{\mathrm{L}}_w^2(0,T;\tilde{V}),\mathrm{ }{\mathrm{L}}^2(0,T;{\tilde{H}}_{\mathit{loc}})$ and $D([0,T];{\tilde{H}}_w).$

Now let us consider the spaces ${\mathcal{Z}}_i; i=1,2$ defined by (3.2) $$\{\begin{array}{c}{\mathcal{Z}}_1:=D([0,T];\mathcal{U}\prime )\cap D([0,T];H_w)\cap {\mathrm{L}}_w^2(0,T;V)\cap {\mathrm{L}}^2(0,T;H_{\mathit{loc}}),\hfill \\ {\mathcal{Z}}_2:=D([0,T];\tilde{\mathcal{U}}\prime )\cap D([0,T];{\tilde{H}}_w)\cap {\mathrm{L}}_w^2(0,T;\tilde{V})\cap {\mathrm{L}}^2(0,T;{\tilde{H}}_{\mathit{loc}}).\hfill \end{array}$$(3.2) and let ${\mathcal{K}}_i$ be the supremum¹ of the corresponding topologies in the spaces ${\mathcal{Z}}_i,i=1,2.$ Let $\mathcal{Z}:={\mathcal{Z}}_1\times {\mathcal{Z}}_2$ and $\mathcal{T}$ be the product topology on $\mathcal{Z}.$

We now recall the definition of a weak martingale solution.

Definition 3.1.

A weak martingale solution of (3.1)(3.1a) $$d\mathbf{v}(t)+[B(\mathbf{v}(t))-{\nu }_{1}Div \mathit{\tau }(t)]dt={\int }_{Z}F(\mathbf{v}(t-),z){\tilde{N}}_1(dt,dz),\hspace{1em}t\ge 0$$(3.1a) is a system $(\widehat{\mathfrak{L}},\widehat{\mathbf{v}},\widehat{\mathit{\tau }},{\widehat{N}}_1,{\widehat{N}}_2),$ where

1. $\widehat{\mathfrak{L}}=(\widehat{\Omega },\widehat{\mathcal{F}},\widehat{\mathbb{F}},\widehat{\mathbb{P}})$ is a filtered probability space with a filtration $\widehat{\mathbb{F}}={({\widehat{\mathcal{F}}}_t)}_{t\ge 0},$

2. ${\widehat{N}}_1$ is a time homogeneous Poisson random measure on $(Z,\mathcal{Z})$ over $\widehat{\mathfrak{L}}$ with the intensity measure λ₁ and ${\widehat{N}}_2$ is a time homogeneous Poisson random measure on $(B,\mathcal{B}(B))$ with the intensity measure λ₂,

3. ${\mathbf{v}}_0\in {\mathrm{L}}^2(\widehat{\Omega };V)$ and ${\mathit{\tau }}_0\in {\mathrm{L}}^2(\widehat{\Omega };\tilde{V}),$

4. $\widehat{\mathbf{v}}:[0,T]\times \widehat{\Omega }\to H$ is an $\widehat{\mathbb{F}}$-progressively measurable process with $\widehat{\mathbb{P}}$-a.e. paths $$\widehat{\mathbf{v}}(·,\omega )\in D([0,T];H_w)\cap {\mathrm{L}}^2(0,T;V)$$ such that for all $t\in [0,T]$ and all $\varphi \in V$ $$\begin{array}{c}{(\widehat{\mathbf{v}}(t),\varphi )}_H+{\int }_0^t〈B(\widehat{\mathbf{v}}(s)),\varphi 〉 ds\\ ={({\mathbf{v}}_0,\varphi )}_H+{\nu }_1{\int }_0^t〈Div \widehat{\mathit{\tau }}(s),\varphi 〉 ds+{\int }_0^t{\int }_Z(F(\widehat{\mathbf{v}}(s),z),\varphi {)}_H{\tilde{\widehat{N}}}_1(ds,dz),\mathrm{\hspace{1em}}\mathrm{\hspace{1em}}\widehat{\mathbb{P}}-\mathrm{a}.\mathrm{s}.\end{array}$$ and satisfies the inequality $$\widehat{\mathbb{E}}\left(\underset{0\le t\le T}{\sup }|\widehat{\mathbf{v}}(t){|}_H^2+\underset{0\le t\le T}{\sup }|\nabla \times \widehat{\mathbf{v}}(t){|}_H^2\right)<\infty ,$$ where $\widehat{\mathbb{E}}$ denotes the expectation with respect to $\widehat{\mathbb{P}}.$

5. $\widehat{\mathit{\tau }}:[0,T]\times \widehat{\Omega }\to \tilde{H}$ is an $\widehat{\mathbb{F}}$-progressively measurable process with $\widehat{\mathbb{P}}$-a.e. paths $$\widehat{\mathit{\tau }}(·,\omega )\in D([0,T];{\tilde{H}}_w)\cap {\mathrm{L}}^2(0,T;\tilde{V})$$ such that for all $t\in [0,T]$ and all $\psi \in \tilde{V},$ $$\begin{array}{c}{(\widehat{\mathit{\tau }}(t),\psi )}_{\tilde{H}}+\kappa {\int }_0^t〈\tilde{\mathcal{A}}\widehat{\mathit{\tau }}(s),\psi 〉 ds+{\int }_0^t〈\tilde{B}(\widehat{\mathbf{v}}(s),\widehat{\mathit{\tau }}(s)),\psi 〉 ds+a{\int }_0^t〈\widehat{\mathit{\tau }}(s),\psi 〉 ds\\ ={(\widehat{\mathit{\tau }}(0),\psi )}_{\tilde{H}}+{\nu }_2{\int }_0^t〈\mathcal{D}(\widehat{\mathbf{v}}(s)),\psi 〉 ds+{\int }_0^t〈b(\widehat{\mathit{\tau }}(s)),\psi 〉 ds\\ \hspace{1em}+{\int }_0^t{\int }_B(G(l,\widehat{\mathit{\tau }}(s-)),\psi {)}_{\tilde{H}} {\tilde{\widehat{N}}}_2(ds,dl),\mathrm{\hspace{1em}}\mathrm{\hspace{1em}}\widehat{\mathbb{P}}-\mathrm{a}.\mathrm{s}.,\end{array}$$ and satisfies the inequality $$\widehat{\mathbb{E}}\left(\underset{0\le t\le T}{\sup }|\widehat{\mathit{\tau }}(t){|}_{\tilde{H}}^2+2\kappa {\int }_0^T|\widehat{\mathit{\tau }}(t){|}_{\tilde{V}}^2 dt\right)<\infty .$$

We now present our main result.

Theorem 3.1.

For the sake of brevity, we take ${\nu }_1={\nu }_2=\kappa =1.$ Let Assumption 2.3 hold and let ${\mathbf{v}}_0\in {\mathrm{L}}^2(\widehat{\Omega };V)$ and ${\mathit{\tau }}_0\in {\mathrm{L}}^2(\widehat{\Omega };\tilde{V})$. Then there exists a weak martingale solution $(\widehat{\mathfrak{L}},\widehat{\mathbf{v}},\widehat{\mathit{\tau }},{\widehat{N}}_1,{\widehat{N}}_2)$ to the system (3.1)(3.1a) $$d\mathbf{v}(t)+[B(\mathbf{v}(t))-{\nu }_{1}Div \mathit{\tau }(t)]dt={\int }_{Z}F(\mathbf{v}(t-),z){\tilde{N}}_1(dt,dz),\hspace{1em}t\ge 0$$(3.1a) satisfying $$\widehat{\mathbb{E}}[\underset{0\le s\le T}{\sup }(|\widehat{\mathbf{v}}(s){|}_H^2+|\widehat{\mathit{\tau }}(s){|}_{\tilde{H}}^2)]+2\widehat{\mathbb{E}}[{\int }_0^T|\widehat{\mathit{\tau }}(s){|}_{\tilde{V}}^{2}ds]<\infty ,\hspace{1em}\widehat{\mathbb{E}}\left(\underset{0\le s\le T}{\sup }|\nabla \times \widehat{\mathbf{v}}(s){|}_H^2\right)<\infty ,$$

3.1. Main ideas of the proof

Due to lack of dissipation in the velocity equation (3.1)(3.1a) $$d\mathbf{v}(t)+[B(\mathbf{v}(t))-{\nu }_{1}Div \mathit{\tau }(t)]dt={\int }_{Z}F(\mathbf{v}(t-),z){\tilde{N}}_1(dt,dz),\hspace{1em}t\ge 0$$(3.1a) , we first incorporate an artificial dissipative term. In other words, for every $1>{\nu }_0\ge \nu >0,$ ν₀ being fixed, we consider the following approximating Oldroyd-B system: (3.3a) $$d{\mathbf{v}}^{\nu }(t)+[\nu \mathcal{A}{\mathbf{v}}^{\nu }(t)+B({\mathbf{v}}^{\nu }(t))-{\nu }_{1}Div {\mathit{\tau }}^{\nu }(t)]dt={\int }_{Z}F({\mathbf{v}}^{\nu }(t-),z){\tilde{N}}_1(dt,dz),\hspace{1em}t\ge 0$$(3.3a) (3.3b) $$d{\mathit{\tau }}^{\nu }(t)+[\kappa \tilde{\mathcal{A}}{\mathit{\tau }}^{\nu }(t)+\tilde{B}({\mathbf{v}}^{\nu }(t),{\mathit{\tau }}^{\nu }(t))+a{\mathit{\tau }}^{\nu }(t)-{\nu }_2\mathcal{D}({\mathbf{v}}^{\nu }(t))]dt={\int }_{B}G(l,{\mathit{\tau }}^{\nu }(t-)){\tilde{N}}_2(dt,dl)+b({\mathit{\tau }}^{\nu }(t))dt,\hspace{1em}t\ge 0$$(3.3b) (3.3c) $${\mathbf{v}}^{\nu }(0)={\mathbf{v}}_0; {\mathit{\tau }}^{\nu }(0)={\tau }_0.$$(3.3c) We now consider an Faedo-Galerkin approximation of the above system and let the Galerkin solution be $({\mathbf{v}}_n^{\nu },{\mathit{\tau }}_n^{\nu }).$ Our strategy is the following: we prove that the system $(\widehat{\mathfrak{L}},\widehat{\mathbf{v}},\widehat{\mathit{\tau }},{\widehat{N}}_1,{\widehat{N}}_2)$ is a weak martingale solution to (3.1)(3.1a) $$d\mathbf{v}(t)+[B(\mathbf{v}(t))-{\nu }_{1}Div \mathit{\tau }(t)]dt={\int }_{Z}F(\mathbf{v}(t-),z){\tilde{N}}_1(dt,dz),\hspace{1em}t\ge 0$$(3.1a) , which is obtained as a limit (as $n\to \infty$ or $\nu \to 0$) of a random variable $({\widehat{\mathbf{v}}}_n^{\nu },{\widehat{\mathit{\tau }}}_n^{\nu },{\widehat{N}}_1^n,{\widehat{N}}_2^n)$ on the new probability space $\widehat{\mathfrak{L}}$ and this random variable has the same laws as the random variable $({\mathbf{v}}_n^{\nu },{\mathit{\tau }}_n^{\nu },N_1^n,N_2^n)$ on the original probability space $\mathfrak{L}.$ The proof is split into a few steps.

Firstly, our aim is to have uniform energy bounds for the Galerkin solution $({\mathbf{v}}_n^{\nu },{\mathit{\tau }}_n^{\nu }),$ which are also valid bounds for the Galerkin solution of the original system (3.1)(3.1a) $$d\mathbf{v}(t)+[B(\mathbf{v}(t))-{\nu }_{1}Div \mathit{\tau }(t)]dt={\int }_{Z}F(\mathbf{v}(t-),z){\tilde{N}}_1(dt,dz),\hspace{1em}t\ge 0$$(3.1a) . Therefore the bounds need to be free from the choice of ν (and n). Toward this, we obtain a basic energy estimate for the Galerkin solution $({\mathbf{v}}_n^{\nu },{\mathit{\tau }}_n^{\nu }),$ which reads, for $p\ge 2,$ $$\mathbb{E}\left[\underset{0\le s\le T}{\sup }\right(\left|{\mathbf{v}}_n^{\nu }\right(s){|}_H^p+|{\mathit{\tau }}_n^{\nu }\left(s\right){|}_{\tilde{H}}^p\left)\right]+2\mathbb{E}\left[{\int }_0^T|{\mathit{\tau }}_n^{\nu }\right(s\left){|}_{\tilde{V}}^2\right|{\mathit{\tau }}_n^{\nu }\left(s\right){|}_{\tilde{H}}^{p-2}ds]\le C$$ where C depends on the parameter of the problem but is independent of n and $\nu .$ Since the original system (3.1) lacks dissipation, it behaves more like an incompressible Euler equation. Hence mere ${\mathrm{L}}^2$-estimate on the velocity is not enough to talk about well-posedness of such problems and one needs to consider more regular initial data to obtain an enstrophy estimate. Toward this goal our aim is to get the ${\mathrm{L}}^2$-estimate on the finite dimensional projection on the vorticity, i.e. for $\mathit{Curl} {\mathbf{v}}_n^{\nu }:={\mathbf{w}}_n^{\nu }.$ Drawing motivation from [4], we define the singular integral operator R, that operates on matrices in the following way: $$R(·):=-{(-\Delta )}^{-1}C\mathit{url} \mathit{Div}.$$ Now applying $\mathit{Curl}(:=\nabla \wedge )$ and R to (3.3), we get for $t\in [0,T],$ $$\begin{array}{c}d{\mathbf{w}}_n^{\nu }(t)+[({\mathbf{v}}_n^{\nu }(t)·\nabla ){\mathbf{w}}_n^{\nu }(t)-\nu \Delta {\mathbf{w}}_n^{\nu }(t)]dt=\nabla \wedge \mathit{Div} {\mathit{\tau }}_n^{\nu }(t)dt+{\int }_Z\nabla \wedge F_n({\mathbf{v}}_n^{\nu }(t-),z){\tilde{N}}_1(dt,dz)\\ dR({\mathit{\tau }}_n^{\nu })(t)+[-R\Delta {\mathit{\tau }}_n^{\nu }(t)+R({\mathbf{v}}_n^{\nu }(t)·\nabla ){\mathit{\tau }}_n^{\nu }(t)+aR({\mathit{\tau }}_n^{\nu })(t)]dt=R(\mathcal{D}({\mathbf{v}}_n^{\nu })(t))dt+{\int }_{B}R{G}_n(l,{\mathit{\tau }}_n^{\nu }(t-)){\tilde{N}}_2(dt,dl)\\ +R{b}_n({\mathit{\tau }}_n^{\nu }(t))dt.\end{array}$$ Define $${\Gamma }_n^{\nu }:={\mathbf{w}}_n^{\nu }-{\mathcal{C}}_{\nu }R({\mathit{\tau }}_n^{\nu }),$$ where ${\mathcal{C}}_{\nu }:=\frac{1}{1-\nu }>0,$ for $\nu \in (0,{\nu }_0],1>{\nu }_0$ being fixed.

Using the commutator property $R({\mathbf{v}}_n^{\nu }·\nabla ){\mathit{\tau }}_n^{\nu }=({\mathbf{v}}_n^{\nu }·\nabla )R({\mathit{\tau }}_n^{\nu })-[{\mathbf{v}}_n^{\nu }·\nabla ,R]{\mathit{\tau }}_n^{\nu }$ and a crucial simplification due to the identity $R\mathcal{D}({\mathbf{v}}_n^{\nu })={\mathbf{w}}_n^{\nu },$ we see that ${\Gamma }_n^{\nu }$ satisfies the following equation for $t\in [0,T],$ $$\begin{array}{c}d{\Gamma }_n^{\nu }(t)+[({\mathbf{v}}_n^{\nu }(t)·\nabla ){\Gamma }_n^{\nu }(t)+{\mathcal{C}}_{\nu }{\Gamma }_n^{\nu }(t)-\nu \Delta {\Gamma }_n^{\nu }(t)+{\mathcal{C}}_{\nu }[{\mathbf{v}}_n^{\nu }(t)·\nabla ,R]{\mathit{\tau }}_n^{\nu }(t)+({\mathcal{C}}_{\nu }{}^2-a{\mathcal{C}}_{\nu })R({\mathit{\tau }}_n^{\nu }(t))]dt\\ ={\int }_Z\nabla \wedge F_n({\mathbf{v}}_n^{\nu }(t-),z){\tilde{N}}_1(dt,dz)-{\mathcal{C}}_{\nu }{\int }_{B}R{G}_n(l,{\mathit{\tau }}_n^{\nu }(t-)){\tilde{N}}_2(dt,dl)-{\mathcal{C}}_{\nu }R{b}_n({\mathit{\tau }}_n^{\nu }(t))dt.\end{array}$$ Next, exploiting already obtained good bounds on ${\mathit{\tau }}_n^{\nu }$ and properties of R, we find an uniform energy bound in ${\mathbb{L}}^2$ for ${\Gamma }_n^{\nu },$ $$\mathbb{E}\left[\underset{t\in [0,T]}{\sup }\right|{\Gamma }_n^{\nu }(t){|}_{{\mathbb{L}}^2}^2]\le C,$$ which further yields $\mathbb{E}[\underset{t\in [0,T]}{\sup }|{\mathbf{w}}_n^{\nu }(t){|}_{{\mathbb{L}}^2}^2]\le C.$ Further, with the help of an associated elliptic problem, we prove $$\mathbb{E}\left[\underset{t\in [0,T]}{\sup }\right|{\mathbf{v}}_n^{\nu }(t){|}_V^2]\le C.$$ As a next step, due to the above uniform bounds (which are independent of ν), tightness of the sequence of measures ${\{\mathcal{L}({\mathbf{v}}_n^{\nu },{\mathit{\tau }}_n^{\nu })\}}_{n\in \mathbb{N}}$ is tight on the space $(\mathcal{Z},\mathcal{T})$ (see (3.20)(3.20) $$\{\begin{array}{c}{\mathcal{Z}}_1:=D([0,T];\mathcal{U}\prime )\cap D([0,T];H_w)\cap {\mathrm{L}}_w^2(0,T;V)\cap {\mathrm{L}}^2(0,T;H_{\mathit{loc}}),\hfill \\ {\mathcal{Z}}_2:=D([0,T];\tilde{\mathcal{U}}\prime )\cap D([0,T];{\tilde{H}}_w)\cap {\mathrm{L}}_w^2(0,T;\tilde{V})\cap {\mathrm{L}}^2(0,T;{\tilde{H}}_{\mathit{loc}}).\hfill \end{array}$$(3.20) ). Next by choosing $\nu =\frac{1}{n}$ and employing Skorokhod-Jakubowski theorem, we have a new probability space $\widehat{\mathfrak{L}}$ and random variables $(\widehat{\mathbf{v}},\widehat{\mathit{\tau }},{\widehat{N}}_1,{\widehat{N}}_2),({\widehat{\mathbf{v}}}_n^{\nu },{\widehat{\mathit{\tau }}}_n^{\nu },{\widehat{N}}_1^n,{\widehat{N}}_2^n),n\in \mathbb{N}$ on this new probability space, such that $\widehat{\mathbb{P}}$-a.s. $$\left({\widehat{\mathbf{v}}}_n^{\nu },{\widehat{\mathit{\tau }}}_n^{\nu }\right)\to \left(\widehat{\mathbf{v}},\widehat{\mathit{\tau }}\right) \text{in} \mathcal{Z}.$$ As a last step, we show that on the new probability space $\widehat{\mathfrak{L}},$ the limiting sequence $(\widehat{\mathbf{v}},\widehat{\mathit{\tau }},{\widehat{N}}_1,{\widehat{N}}_2),$ i.e. the system $(\widehat{\mathfrak{L}},\widehat{\mathbf{v}},\widehat{\mathit{\tau }},{\widehat{N}}_1,{\widehat{N}}_2)$ is a weak martingale solution of the original problem (3.1)(3.1a) $$d\mathbf{v}(t)+[B(\mathbf{v}(t))-{\nu }_{1}Div \mathit{\tau }(t)]dt={\int }_{Z}F(\mathbf{v}(t-),z){\tilde{N}}_1(dt,dz),\hspace{1em}t\ge 0$$(3.1a) .

3.2. Proof of Theorem 3.1

We present the proof of Theorem 3.1 below.

Proof.

We split the proof in few steps.

Step I: (Faedo-Galerkin Approximation and Energy Estimate). By Remark 1, there exist orthonormal bases $\left\{e_i\right\},\left\{{\tilde{e}}_i\right\}$ of $H,\tilde{H}$ respectively consist of eigenvectors of operators $\mathfrak{A},\tilde{\mathfrak{A}}.$ Let $\left\{{\lambda }_i\right\},\left\{{\tilde{\lambda }}_i\right\}$ be the corresponding eigenvalues of $\mathfrak{A}$ and $\tilde{\mathfrak{A}}$ respectively. Therefore we have $\mathfrak{A}{e}_i={\lambda }_i{e}_i$ and $\tilde{\mathfrak{A}}{\tilde{e}}_i={\tilde{\lambda }}_i{\tilde{e}}_i, \forall i\in \mathbb{N}.$ Let us denote $$H_n=\mathit{span}\{e_1,\dots ,{\mathrm{e}}_n\}, {\tilde{H}}_n=\mathit{span}\{{\tilde{e}}_1,\dots ,{\tilde{e}}_n\}.$$ Let $P_n,{\tilde{P}}_n$ be defined as $P_n:\mathcal{U}\prime \to \mathcal{U},$ ${\tilde{P}}_n:\tilde{\mathcal{U}}\prime \to \tilde{\mathcal{U}}$ by $$P_n{\mathbf{v}}^{\nu }:=\sum _{i=1}^n{}_{\mathcal{U}\prime }{〈{\mathbf{v}}^{\nu },e_i〉}_{\mathcal{U}}{e}_i,\hspace{1em}{\tilde{P}}_n{\mathit{\tau }}^{\nu }=\sum _{i=1}^n{}_{\tilde{\mathcal{U}}\prime }{〈{\mathit{\tau }}^{\nu },{\tilde{e}}_i〉}_{\tilde{\mathcal{U}}}{\tilde{e}}_i, \hspace{1em}\forall {\mathbf{v}}^{\nu }\in \mathcal{U}, {\mathit{\tau }}^{\nu }\in \tilde{\mathcal{U}}\prime .$$ For ${\mathbf{v}}^{\nu }\in H_n,{\mathit{\tau }}^{\nu }\in {\tilde{H}}_n$ and $z\in Z,$ let us consider the following maps:

1. $B_n:H_n\to H_n\hspace{1em}\text{by}\hspace{1em}{B}_n({\mathbf{v}}^{\nu }):=P_{n}B({\mathbf{v}}^{\nu }),$

2. ${\tilde{B}}_n:H_n\times {\tilde{H}}_n\to {\tilde{H}}_n\hspace{1em}\text{by}\hspace{1em}{\tilde{B}}_n({\mathbf{v}}^{\nu },{\mathit{\tau }}^{\nu }):={\tilde{P}}_n\tilde{B}({\mathbf{v}}^{\nu },{\mathit{\tau }}^{\nu }),$

3. $F_n:H_n\times Z\to H_n\hspace{1em}\text{by}\hspace{1em}{F}_n({\mathbf{v}}^{\nu },z):=P_{n}F({\mathbf{v}}^{\nu },z),$

4. $g_{in}:{\tilde{H}}_n\to {\tilde{H}}_n\hspace{1em}\text{by}\hspace{1em}{g}_{in}({\mathit{\tau }}^{\nu }):={\tilde{P}}_n(h_i\otimes {\mathit{\tau }}^{\nu }).$

Let us denote ${\mathbf{v}}_n^{\nu }(0)=P_n{\mathbf{v}}_0^{\nu }$ and ${\mathit{\tau }}_n^{\nu }(0)={\tilde{P}}_n{\mathit{\tau }}_0^{\nu }.$

Let ${\Phi }_n(t,l,{\mathbf{v}}_n^{\nu }(0))$ be a flow on ${\tilde{H}}_n$ assosciated to the vector field ${\sum }_{i=1}^k{l}_i{g}_{in},$ i.e., $$\begin{array}{c}\frac{d{\Phi }_n}{dt}(t,l,{\mathit{\tau }}_n^{\nu }(0))=\sum _{i=1}^k{l}_i{g}_{in}{\Phi }_n(t,l,{\mathit{\tau }}_n^{\nu }(0)), t\ge 0,\\ {\Phi }_n(t,l,{\mathit{\tau }}_n^{\nu }(0))={\mathit{\tau }}_n^{\nu }(0)\in {\tilde{H}}_n.\end{array}$$ For ${\mathit{\tau }}_n^{\nu }\in {\tilde{H}}_n,$ we define $$\begin{array}{c}{G}_n(l,{\mathit{\tau }}_n^{\nu }):={\Phi }_n(l,{\mathit{\tau }}_n^{\nu })-{\mathit{\tau }}_n^{\nu },\\ K_n(l,{\mathit{\tau }}_n^{\nu }):={\Phi }_n(l,{\mathit{\tau }}_n^{\nu })-{\mathit{\tau }}_n^{\nu }-\sum _{i=1}^k{l}_i{g}_{in}({\mathit{\tau }}_n^{\nu })\\ b_n({\mathit{\tau }}_n^{\nu }):={\int }_B[{\Phi }_n(l,{\mathit{\tau }}_n^{\nu })-{\mathit{\tau }}_n^{\nu }-\sum _{i=1}^k{l}_i{g}_{in}({\mathit{\tau }}_n^{\nu })]{\lambda }_2(dl)={\int }_B{K}_n(l,{\mathit{\tau }}_n^{\nu }){\lambda }_2(dl).\end{array}$$ Consider the following mappings $$\begin{array}{c}B{\prime }_n({\mathbf{v}}^{\nu }):=P_{n}B({\chi }_n({\mathbf{v}}^{\nu }),{\mathbf{v}}^{\nu }),\hspace{1em}{\mathbf{v}}^{\nu }\in H_n,\\ \tilde{B}{\prime }_n({\mathbf{v}}^{\nu },{\mathit{\tau }}^{\nu }):=P_n\tilde{B}({\chi }_n({\mathbf{v}}^{\nu }),{\mathit{\tau }}^{\nu }),\hspace{1em}{\mathbf{v}}^{\nu }\in H_n,{\mathit{\tau }}^{\mathbf{\nu }}\in {\tilde{H}}_n,\end{array}$$ where ${\chi }_n:H\to H$ is defined by ${\chi }_n({\mathbf{v}}^{\nu })={\theta }_n(|{\mathbf{v}}^{\nu }{|}_{V\prime }){\mathbf{v}}^{\nu },$ where ${\theta }_n:\mathbb{R}\to [0,1]$ of class ${\mathcal{C}}^{\infty }$ such that $${\theta }_n(r)=1\mathit{\text{\hspace{1em}if\hspace{1em}r}}\le n\hspace{1em}\mathit{and}\hspace{1em}{\theta }_n(r)=0\mathit{\text{\hspace{1em}if\hspace{1em}r}}\ge n+1.$$ Since $H_n\subset H$ and ${\tilde{H}}_n\subset \tilde{H},$ the mappings $B{\prime }_n$ and $\tilde{B}{\prime }_n$ are well defined. Moreover, $B{\prime }_n:H_n\to H_n$ and $\tilde{B}{\prime }_n:H_n\times {\tilde{H}}_n\to {\tilde{H}}_n$ are globally Lipschitz continuous.

We now consider the following $H_n\times {\tilde{H}}_n$-valued modified approximated system: (3.4a) $$d{\mathbf{v}}_n^{\nu }(t)+[\nu P_n\mathcal{A}{\mathbf{v}}_n^{\nu }(t)+B{\prime }_n({\mathbf{v}}_n^{\nu }(t))-\mathit{Div} {\mathit{\tau }}_n^{\nu }(t)]dt={\int }_Z{F}_n({\mathbf{v}}_n^{\nu }(t-),z){\tilde{N}}_1(dt,dz),\hspace{1em}t\ge 0$$(3.4a) (3.4b) $$\begin{array}{c}d{\mathit{\tau }}_n^{\nu }(t)+[{\tilde{P}}_n\tilde{\mathcal{A}}{\mathit{\tau }}_n^{\nu }(t)+\tilde{B}{\prime }_n({\mathbf{v}}_n^{\nu }(t),{\mathit{\tau }}_n^{\nu }(t))+a{\mathit{\tau }}_n^{\nu }(t)-{\nu }_2\mathcal{D}({\mathbf{v}}_n^{\nu }(t))]dt\\ \hspace{1em}={\int }_B{G}_n(l,{\mathit{\tau }}_n^{\nu }(t-)){\tilde{N}}_2(dt,dl)+b_n({\mathit{\tau }}_n^{\nu }(t))dt.\\ \end{array}$$(3.4b) Since all relevant maps are globally Lipschitz (see Lemma A.1–A.3 in Appendix A), we have the following standard result, see e.g. [34] for a reference.

Lemma 3.2.

For each $n\in \mathbb{N},$ there exists a unique global, $\mathbb{F}$-progressively measurable, $H_n\times {\tilde{H}}_n$-valued càdlàg process $({\mathbf{v}}_n^{\nu },{\mathit{\tau }}_n^{\nu })$ satisfying the modified Galerkin approximated system (3.4)(3.4a) $$d{\mathbf{v}}_n^{\nu }(t)+[\nu P_n\mathcal{A}{\mathbf{v}}_n^{\nu }(t)+B{\prime }_n({\mathbf{v}}_n^{\nu }(t))-\mathit{Div} {\mathit{\tau }}_n^{\nu }(t)]dt={\int }_Z{F}_n({\mathbf{v}}_n^{\nu }(t-),z){\tilde{N}}_1(dt,dz),\hspace{1em}t\ge 0$$(3.4a) .

It follows easily, see for instance [34], that for each $n\in \mathbb{N},$ the approximated system (3.4)(3.4a) $$d{\mathbf{v}}_n^{\nu }(t)+[\nu P_n\mathcal{A}{\mathbf{v}}_n^{\nu }(t)+B{\prime }_n({\mathbf{v}}_n^{\nu }(t))-\mathit{Div} {\mathit{\tau }}_n^{\nu }(t)]dt={\int }_Z{F}_n({\mathbf{v}}_n^{\nu }(t-),z){\tilde{N}}_1(dt,dz),\hspace{1em}t\ge 0$$(3.4a) has a unique local maximal solution. In two dimensional setting, by a combination of the proof of [34, Theorem 3.1] with Lemma 3.3 below, we infer that the approximated system (3.4)(3.4a) $$d{\mathbf{v}}_n^{\nu }(t)+[\nu P_n\mathcal{A}{\mathbf{v}}_n^{\nu }(t)+B{\prime }_n({\mathbf{v}}_n^{\nu }(t))-\mathit{Div} {\mathit{\tau }}_n^{\nu }(t)]dt={\int }_Z{F}_n({\mathbf{v}}_n^{\nu }(t-),z){\tilde{N}}_1(dt,dz),\hspace{1em}t\ge 0$$(3.4a) has a unique global solution.

Step II: (Estimates for the velocity field and stress tensor)

Lemma 3.3.

Let $p\ge 2.$ Let ${\mathbf{v}}_0\in {\mathrm{L}}^p(\Omega ;H)$ and ${\mathit{\tau }}_0\in {\mathrm{L}}^p(\Omega ;\tilde{H}).$ Then for every T > 0, there exists $C>0,$ depending on ${\mathbf{v}}_0,{\mathit{\tau }}_0$ (but is independent of n and ν), such that $$\underset{n\in \mathbb{N}}{\sup } \mathbb{E}[\underset{0\le s\le T}{\sup }(|{\mathbf{v}}_n^{\nu }(s){|}_H^p+|{\mathit{\tau }}_n^{\nu }(s){|}_{\tilde{H}}^p)]+\nu p \underset{n\in \mathbb{N}}{\sup }\mathbb{E}[{\int }_0^T|\nabla {\mathbf{v}}_n^{\nu }(s){|}_H^2|{\mathbf{v}}_n^{\nu }(s){|}_H^{p-2}ds]\mathrm{ }$$ (3.5) $$\hspace{1em}+p \underset{n\in \mathbb{N}}{\sup } \mathbb{E}[{\int }_0^T|\nabla {\mathit{\tau }}_n^{\nu }(s){|}_{\tilde{H}}^2|{\mathit{\tau }}_n^{\nu }(s){|}_{\tilde{H}}^{p-2}ds]\le C.$$(3.5)

Proof.

Estimate (3.5)(3.5) $$\hspace{1em}+p \underset{n\in \mathbb{N}}{\sup } \mathbb{E}[{\int }_0^T|\nabla {\mathit{\tau }}_n^{\nu }(s){|}_{\tilde{H}}^2|{\mathit{\tau }}_n^{\nu }(s){|}_{\tilde{H}}^{p-2}ds]\le C.$$(3.5) can be proved by applying Itô’s Lemma to the function $f(x)=\frac{1}{p}|x{|}_H^p$ (for the process ${\mathbf{v}}_n^{\nu }(t)$) and $\frac{1}{p}|x{|}_{\tilde{H}}^p$ (for the process ${\mathit{\tau }}_n^{\nu }(t)$) to the system (3.4)(3.4a) $$d{\mathbf{v}}_n^{\nu }(t)+[\nu P_n\mathcal{A}{\mathbf{v}}_n^{\nu }(t)+B{\prime }_n({\mathbf{v}}_n^{\nu }(t))-\mathit{Div} {\mathit{\tau }}_n^{\nu }(t)]dt={\int }_Z{F}_n({\mathbf{v}}_n^{\nu }(t-),z){\tilde{N}}_1(dt,dz),\hspace{1em}t\ge 0$$(3.4a) . □

Step III: (Estimate of the vorticity)

Lemma 3.4.

Let ${\mathbf{w}}_0\in {\mathrm{L}}^2(\Omega ;H)$ and ${\mathit{\tau }}_0\in {\mathrm{L}}^2(\Omega ;\tilde{V}).$ Then for every T > 0 there exists a constant $C>0,$ depending on ${\mathbf{v}}_0,{\mathit{\tau }}_0$ (but is independent of n and ν), such that (3.6) $$\mathbb{E}\left[\underset{0\le s\le T}{\sup }\right|{\mathbf{w}}_n^{\nu }(s){|}_H^2]\le C,$$(3.6) where ${\mathbf{w}}^{\nu }:=\text{Curl} {\mathbf{v}}^{\nu }$ and ${\mathbf{w}}_n^{\nu }$ is the finite dimensional approximation of ${\mathbf{w}}^{\nu }.$

Proof.

Let us recall that R denote the singular integral operator that operates on matrices in the following way: $$R(·):=-{(-\Delta )}^{-1}C\mathit{url} \mathit{Div}.$$ We note that R is a degree 0 singular integral operator having the following properties (see [4]):

1. R is a bounded linear transformation in ${\mathrm{L}}^2({\mathbb{R}}^{2\times 2}).$

2. For divergence free ${\mathbf{v}}^{\nu },$ we have (3.7) $$\left|\right[{\mathbf{v}}^{\nu }·\nabla ,R]{\mathit{\tau }}^{\nu }{|}_{{\mathbb{L}}^2}\le C|\nabla {\mathbf{v}}^{\nu }{|}_H|{\mathit{\tau }}^{\nu }{|}_{\tilde{V}}.$$(3.7)

Now applying $\mathit{Curl}(:=\nabla \wedge )$ and R to (3.4)(3.4a) $$d{\mathbf{v}}_n^{\nu }(t)+[\nu P_n\mathcal{A}{\mathbf{v}}_n^{\nu }(t)+B{\prime }_n({\mathbf{v}}_n^{\nu }(t))-\mathit{Div} {\mathit{\tau }}_n^{\nu }(t)]dt={\int }_Z{F}_n({\mathbf{v}}_n^{\nu }(t-),z){\tilde{N}}_1(dt,dz),\hspace{1em}t\ge 0$$(3.4a) , we get for $t\in [0,T],$ $$\begin{array}{c}d{\mathbf{w}}_n^{\nu }(t)+[({\mathbf{v}}_n^{\nu }(t)·\nabla ){\mathbf{w}}_n^{\nu }(t)-\nu \Delta {\mathbf{w}}_n^{\nu }(t)]dt=\nabla \wedge \mathit{Div} {\mathit{\tau }}_n^{\nu }(t)dt+{\int }_Z\nabla \wedge F_n({\mathbf{v}}_n^{\nu }(t-),z){\tilde{N}}_1(dt,dz)\\ dR({\mathit{\tau }}_n^{\nu })(t)+[-R\Delta {\mathit{\tau }}_n^{\nu }(t)+R({\mathbf{v}}_n^{\nu }(t)·\nabla ){\mathit{\tau }}_n^{\nu }(t)+aR({\mathit{\tau }}_n^{\nu })(t)]dt=R(\mathcal{D}({\mathbf{v}}_n^{\nu })(t))dt+{\int }_{B}R{G}_n(l,{\mathit{\tau }}_n^{\nu }(t-)){\tilde{N}}_2(dt,dl)\\ +R{b}_n({\mathit{\tau }}_n^{\nu }(t))dt.\end{array}$$ Define ${\mathcal{C}}_{\nu }:=\frac{1}{1-\nu }>0, \text{for} \nu \in (0,{\nu }_0], 1>{\nu }_0 \text{being fixed}.$

Let ${\Gamma }_n^{\nu }:={\mathbf{w}}_n^{\nu }-{\mathcal{C}}_{\nu }R({\mathit{\tau }}_n^{\nu }).$ Then using properties of R, we see that ${\Gamma }_n^{\nu }$ satisfies (3.8) $$\begin{array}{c}d{\Gamma }_n^{\nu }(t)+[({\mathbf{v}}_n^{\nu }(t)·\nabla ){\Gamma }_n^{\nu }(t)+{\mathcal{C}}_{\nu }{\Gamma }_n^{\nu }(t)-\nu \Delta {\Gamma }_n^{\nu }(t)+{\mathcal{C}}_{\nu }[{\mathbf{v}}_n^{\nu }(t)·\nabla ,R]{\mathit{\tau }}_n^{\nu }(t)+({\mathcal{C}}_{\nu }{}^2-a{\mathcal{C}}_{\nu })R({\mathit{\tau }}_n^{\nu }(t))]dt\\ ={\int }_Z\nabla \wedge F_n({\mathbf{v}}_n^{\nu }(t-),z){\tilde{N}}_1(dt,dz)-{\mathcal{C}}_{\nu }{\int }_{B}R{G}_n(l,{\mathit{\tau }}_n^{\nu }(t-)){\tilde{N}}_2(dt,dl)-{\mathcal{C}}_{\nu }R{b}_n({\mathit{\tau }}_n^{\nu }(t))dt\end{array}$$(3.8) Exploiting the properties of R and Young’s inequality, we now estimate the following two terms which lead to (3.9) $$\begin{array}{c}|({\mathcal{C}}_{\nu }{}^2-a{\mathcal{C}}_{\nu }){〈R({\mathit{\tau }}_n^{\nu }(t)),{\Gamma }_n^{\nu }(t)〉}_{{\mathbb{L}}^2}|\le \frac{{\mathcal{C}}_{\nu }}{4}|{\Gamma }_n^{\nu }(t){|}_{{\mathbb{L}}^2}^2+{\mathcal{C}}_{\nu }|{\mathcal{C}}_{\nu }-a{|}^2|R({\mathit{\tau }}_n^{\nu }(t)){|}_{{\mathbb{L}}^2}^2\\ \left|{\mathcal{C}}_{\nu }{〈[{\mathbf{v}}_n^{\nu }(t)·\nabla ,R]{\mathit{\tau }}_n^{\nu }(t),{\Gamma }_n^{\nu }(t)〉}_{{\mathbb{L}}^2}\right|\le \frac{{\mathcal{C}}_{\nu }}{4}|{\Gamma }_n^{\nu }(t){|}_{{\mathbb{L}}^2}^2+{\mathcal{C}}_{\nu }|[{\mathbf{v}}_n^{\nu }(t)·\nabla ,R]{\mathit{\tau }}_n^{\nu }(t){|}_{{\mathbb{L}}^2}^2\mathrm{ }\end{array}$$(3.9) (3.10) $$\le \frac{{\mathcal{C}}_{\nu }}{4}|{\Gamma }_n^{\nu }(t){|}_{{\mathbb{L}}^2}^2+C {\mathcal{C}}_{\nu }|\nabla {\mathbf{v}}_n^{\nu }(t){|}_H^2|{\mathit{\tau }}_n^{\nu }(t){|}_{\tilde{V}}^2.$$(3.10) For $N\in \mathbb{N}$ fixed, let us define a stopping time by $${\mathfrak{S}}_N^n=\underset{t\ge 0}{\inf }\left\{t:\underset{t\in [0,T]}{\sup }|{\Gamma }_n^{\nu }(t){|}_{{\mathbb{L}}^2}>N\right\}\wedge \underset{t\ge 0}{\inf }\left\{t:\underset{t\in [0,T]}{\sup }|{\mathbf{w}}_n^{\nu }(t){|}_H>N\right\}.$$ Therefore applying Itô’s formula to the function $\frac{1}{2}|{\Gamma }_n^{\nu }(t){|}_{{\mathbb{L}}^2}^2$ for the Equation (3.8)(3.8) $$\begin{array}{c}d{\Gamma }_n^{\nu }(t)+[({\mathbf{v}}_n^{\nu }(t)·\nabla ){\Gamma }_n^{\nu }(t)+{\mathcal{C}}_{\nu }{\Gamma }_n^{\nu }(t)-\nu \Delta {\Gamma }_n^{\nu }(t)+{\mathcal{C}}_{\nu }[{\mathbf{v}}_n^{\nu }(t)·\nabla ,R]{\mathit{\tau }}_n^{\nu }(t)+({\mathcal{C}}_{\nu }{}^2-a{\mathcal{C}}_{\nu })R({\mathit{\tau }}_n^{\nu }(t))]dt\\ ={\int }_Z\nabla \wedge F_n({\mathbf{v}}_n^{\nu }(t-),z){\tilde{N}}_1(dt,dz)-{\mathcal{C}}_{\nu }{\int }_{B}R{G}_n(l,{\mathit{\tau }}_n^{\nu }(t-)){\tilde{N}}_2(dt,dl)-{\mathcal{C}}_{\nu }R{b}_n({\mathit{\tau }}_n^{\nu }(t))dt\end{array}$$(3.8) , using (3.9)(3.9) $$\begin{array}{c}|({\mathcal{C}}_{\nu }{}^2-a{\mathcal{C}}_{\nu }){〈R({\mathit{\tau }}_n^{\nu }(t)),{\Gamma }_n^{\nu }(t)〉}_{{\mathbb{L}}^2}|\le \frac{{\mathcal{C}}_{\nu }}{4}|{\Gamma }_n^{\nu }(t){|}_{{\mathbb{L}}^2}^2+{\mathcal{C}}_{\nu }|{\mathcal{C}}_{\nu }-a{|}^2|R({\mathit{\tau }}_n^{\nu }(t)){|}_{{\mathbb{L}}^2}^2\\ \left|{\mathcal{C}}_{\nu }{〈[{\mathbf{v}}_n^{\nu }(t)·\nabla ,R]{\mathit{\tau }}_n^{\nu }(t),{\Gamma }_n^{\nu }(t)〉}_{{\mathbb{L}}^2}\right|\le \frac{{\mathcal{C}}_{\nu }}{4}|{\Gamma }_n^{\nu }(t){|}_{{\mathbb{L}}^2}^2+{\mathcal{C}}_{\nu }|[{\mathbf{v}}_n^{\nu }(t)·\nabla ,R]{\mathit{\tau }}_n^{\nu }(t){|}_{{\mathbb{L}}^2}^2\mathrm{ }\end{array}$$(3.9) , (3.10)(3.10) $$\le \frac{{\mathcal{C}}_{\nu }}{4}|{\Gamma }_n^{\nu }(t){|}_{{\mathbb{L}}^2}^2+C {\mathcal{C}}_{\nu }|\nabla {\mathbf{v}}_n^{\nu }(t){|}_H^2|{\mathit{\tau }}_n^{\nu }(t){|}_{\tilde{V}}^2.$$(3.10) then integrating in $[0,T]$ and then taking supremum and expectation, we get (3.11) $$\begin{array}{c}\frac{1}{2}\mathbb{E}\left[\underset{t\in [0,T\wedge {\mathfrak{S}}_N^n]}{\sup }\right|{\Gamma }_n^{\nu }(t){|}_{{\mathbb{L}}^2}^2]+\nu \mathbb{E}[{\int }_0^{T\wedge {\mathfrak{S}}_N^n}|\nabla {\Gamma }_n^{\nu }(t){|}_{{\mathbb{L}}^2}^{2}dt]+\frac{{\mathcal{C}}_{\nu }}{2}\mathbb{E}[{\int }_0^{T\wedge {\mathfrak{S}}_N^n}|{\Gamma }_n^{\nu }(t){|}_{{\mathbb{L}}^2}^{2}dt]\\ \le \frac{1}{2}\mathbb{E}\left[\right|{\Gamma }^{\nu }(0){|}_{{\mathbb{L}}^2}^2]+|{\mathcal{C}}_{\nu }\left|\right|{\mathcal{C}}_{\nu }-a{|}^2\mathbb{E}[{\int }_0^{T\wedge {\mathfrak{S}}_N^n}|R({\mathit{\tau }}_n^{\nu }(t)){|}_{{\mathbb{L}}^2}^{2}dt]+{\mathcal{C}}_{\nu }C \mathbb{E}[{\int }_0^{T\wedge {\mathfrak{S}}_N^n}|\nabla {\mathbf{v}}_n^{\nu }(t){|}_H^2|{\mathit{\tau }}_n^{\nu }(t){|}_{\tilde{V}}^{2}dt]\\ \hspace{1em}+\sum _{i=1}^2\mathbb{E}[\underset{t\in [0,T\wedge {\mathfrak{S}}_N^n]}{\sup }|{\int }_0^t{M}_i(s)ds|]+\sum _{i=1}^2\mathbb{E}[{\int }_0^{T\wedge {\mathfrak{S}}_N^n}|L_i(t)|dt],\end{array}$$(3.11) where (3.12a) $$M_1(t):={\int }_Z\{|{\Gamma }_n^{\nu }(t-)+\nabla \wedge F_n({\mathbf{v}}_n^{\nu }(t-),z){|}_{{\mathbb{L}}^2}^2-|{\Gamma }_n^{\nu }(t-){|}_{{\mathbb{L}}^2}^2\}{\tilde{N}}_1(dt,dz),$$(3.12a) (3.12b) $$L_1(t):={\int }_Z\{|{\Gamma }_n^{\nu }(t)+\nabla \wedge F_n({\mathbf{v}}_n^{\nu }(t),z){|}_{{\mathbb{L}}^2}^2-|{\Gamma }_n^{\nu }(t){|}_{{\mathbb{L}}^2}^2-2{〈\nabla \wedge F_n({\mathbf{v}}_n^{\nu }(t),z),{\Gamma }_n^{\nu }(t)〉}_{{\mathbb{L}}^2}\}{\lambda }_1(dz)dt,$$(3.12b) (3.12c) $$M_2(t):={\int }_B\{|{\mathcal{C}}_{\nu }R{\Phi }_n(l,{\mathit{\tau }}_n^{\nu }(t-)){|}_{{\mathbb{L}}^2}^2-|{\Gamma }_n^{\nu }(t-){|}_{{\mathbb{L}}^2}^2\}{\tilde{N}}_2(dt,dl),$$(3.12c) (3.12d) $$L_2(t):={\int }_B\{|{\mathcal{C}}_{\nu }R{\Phi }_n(l,{\mathit{\tau }}_n^{\nu }(t)){|}_{{\mathbb{L}}^2}^2-|{\Gamma }_n^{\nu }(t){|}_{{\mathbb{L}}^2}^2-2\sum _{i=1}^k{l}_i{〈{\mathcal{C}}_{\nu }R{g}_i({\mathit{\tau }}_n^{\nu }(t)),{\Gamma }_n^{\nu }(t)〉}_{{\mathbb{L}}^2}\}{\lambda }_2(dl)dt.$$(3.12d) Using Burkholder-Davis-Gundy inequality, (A.8) in Assumption 2.3, and Young’s inequality, we get (3.13) $$\mathbb{E}[\underset{t\in [0,T\wedge {\mathfrak{S}}_N^n]}{\sup }|{\int }_0^t{M}_1(s)ds|]\le CT+\frac{1}{8}\mathbb{E}\left[\underset{t\in [0,T\wedge {\mathfrak{S}}_N^n]}{\sup }\right|{\Gamma }_n^{\nu }(t){|}_{{\mathbb{L}}^2}^2]+C(T) \mathbb{E}[{\int }_0^{T\wedge {\mathfrak{S}}_N^n}|{\Gamma }_n^{\nu }(t){|}_{{\mathbb{L}}^2}^{2}dt].$$(3.13) In order to estimate M₂ term, we apply the boundedness property of R (see (3.7)(3.7) $$\left|\right[{\mathbf{v}}^{\nu }·\nabla ,R]{\mathit{\tau }}^{\nu }{|}_{{\mathbb{L}}^2}\le C|\nabla {\mathbf{v}}^{\nu }{|}_H|{\mathit{\tau }}^{\nu }{|}_{\tilde{V}}.$$(3.7) ) and part 1 of Lemma 2.4, and obtain (3.14) $$|R{\Phi }_n({\mathit{\tau }}_n^{\nu }(t)){|}_{{\mathbb{L}}^2}^4\le C|{\Phi }_n(l,{\mathit{\tau }}_n^{\nu }(t){|}_{{\mathbb{L}}^2}^4\le C(|{\mathit{\tau }}_n^{\nu }(t){|}_{\tilde{H}}^4+|l{|}_{{\mathbb{R}}^k}^2|{\mathit{\tau }}_n^{\nu }(t){|}_{\tilde{H}}^4).$$(3.14) Thus using (3.14)(3.14) $$|R{\Phi }_n({\mathit{\tau }}_n^{\nu }(t)){|}_{{\mathbb{L}}^2}^4\le C|{\Phi }_n(l,{\mathit{\tau }}_n^{\nu }(t){|}_{{\mathbb{L}}^2}^4\le C(|{\mathit{\tau }}_n^{\nu }(t){|}_{\tilde{H}}^4+|l{|}_{{\mathbb{R}}^k}^2|{\mathit{\tau }}_n^{\nu }(t){|}_{\tilde{H}}^4).$$(3.14) , (A.4) in Assumption 2.3 and bounds on ${\mathit{\tau }}_n$ we have, (3.15) $$\begin{array}{c}\mathbb{E}[\underset{t\in [0,T\wedge {\mathfrak{S}}_N^n]}{\sup }|{\int }_0^t{M}_2(s)ds|]\le C\mathbb{E}\left[{({{\int }_0^{T\wedge {\mathfrak{S}}_N^n}{\int }_B\{|{\mathcal{C}}_{\nu }R{\Phi }_n(l,{\mathit{\tau }}_n^{\nu }(t)){|}_{{\mathbb{L}}^2}^2-|{\Gamma }_n^{\nu }(t){|}_{{\mathbb{L}}^2}^2\}}^2{\lambda }_2(dl)dt)}^{1/2}\right]\\ \le {\mathcal{C}}_{\nu }C\mathbb{E}\left[{\left({\int }_0^{T\wedge {\mathfrak{S}}_N^n}|{\mathit{\tau }}_n^{\nu }\right(t\left){|}_{\tilde{H}}^{4}dt\right)}^{1/2}\right]+C\mathbb{E}\left[{\left({\int }_0^{T\wedge {\mathfrak{S}}_N^n}|{\Gamma }_n^{\nu }\right(t\left){|}_{{\mathbb{L}}^2}^{4}dt\right)}^{1/2}\right]\\ \le C{\mathcal{C}}_{\nu }T+\frac{1}{8}\mathbb{E}\left[\underset{t\in [0,T\wedge {\mathfrak{S}}_N^n]}{\sup }\right|{\Gamma }_n^{\nu }\left(t\right){|}_{{\mathbb{L}}^2}^2]+C(T) \mathbb{E}[{\int }_0^{T\wedge {\mathfrak{S}}_N^n}|{\Gamma }_n^{\nu }\left(t\right){|}_{{\mathbb{L}}^2}^{2}dt].\end{array}$$(3.15) Likewise, using (A.8) in Assumption 2.3, Lemma 3.3 and $T\wedge {\mathfrak{S}}_N^n\to T$ as $N\to \infty$ we have (3.16) $$\begin{array}{c}\mathbb{E}[{\int }_0^{T\wedge {\mathfrak{S}}_N^n}|L_1(t)|dt]\le \mathbb{E}[{\int }_0^{T\wedge {\mathfrak{S}}_N^n}{\int }_Z|\nabla \wedge F_n({\mathbf{v}}_n^{\nu }(t),z){|}_{{\mathbb{L}}^2}^2{\lambda }_1(dz)dt]\\ \le C\mathbb{E}[{\int }_0^{T\wedge {\mathfrak{S}}_N^n}(1+|\nabla {\mathbf{v}}_n^{\nu }(t){|}_H^2)dt]\le C(T),\end{array}$$(3.16) and employing Lemma 2.4, (A.4) in Assumption 2.3, boundedness property of R and bounds on ${\mathit{\tau }}_n^{\nu },$ we achieve (3.17) $$\begin{array}{c}\mathbb{E}[{\int }_0^{T\wedge {\mathfrak{S}}_N^n}|L_2(t)|dt]\le \mathbb{E}[{\int }_0^{T\wedge {\mathfrak{S}}_N^n}{\int }_B|{\mathcal{C}}_{\nu }R{\Phi }_n(l,{\mathit{\tau }}_n^{\nu }(t)){|}_{{\mathbb{L}}^2}^2+2{\mathcal{C}}_{\nu }\sum _{i=1}^k\left|l_i{〈R{g}_i({\mathit{\tau }}_n^{\nu }(t)),{\Gamma }_n^{\nu }(t)〉}_{{\mathbb{L}}^2}\right|{\lambda }_2(dl)dt]\\ \le C(T)+C ({\mathcal{C}}_{\nu }+1)\mathbb{E}[{\int }_0^{T\wedge {\mathfrak{S}}_N^n}|{\Gamma }_n^{\nu }(t){|}_{{\mathbb{L}}^2}^{2}dt].\end{array}$$(3.17) Finally we compute the third term on the left hand side of the inequality (3.11)(3.11) $$\begin{array}{c}\frac{1}{2}\mathbb{E}\left[\underset{t\in [0,T\wedge {\mathfrak{S}}_N^n]}{\sup }\right|{\Gamma }_n^{\nu }(t){|}_{{\mathbb{L}}^2}^2]+\nu \mathbb{E}[{\int }_0^{T\wedge {\mathfrak{S}}_N^n}|\nabla {\Gamma }_n^{\nu }(t){|}_{{\mathbb{L}}^2}^{2}dt]+\frac{{\mathcal{C}}_{\nu }}{2}\mathbb{E}[{\int }_0^{T\wedge {\mathfrak{S}}_N^n}|{\Gamma }_n^{\nu }(t){|}_{{\mathbb{L}}^2}^{2}dt]\\ \le \frac{1}{2}\mathbb{E}\left[\right|{\Gamma }^{\nu }(0){|}_{{\mathbb{L}}^2}^2]+|{\mathcal{C}}_{\nu }\left|\right|{\mathcal{C}}_{\nu }-a{|}^2\mathbb{E}[{\int }_0^{T\wedge {\mathfrak{S}}_N^n}|R({\mathit{\tau }}_n^{\nu }(t)){|}_{{\mathbb{L}}^2}^{2}dt]+{\mathcal{C}}_{\nu }C \mathbb{E}[{\int }_0^{T\wedge {\mathfrak{S}}_N^n}|\nabla {\mathbf{v}}_n^{\nu }(t){|}_H^2|{\mathit{\tau }}_n^{\nu }(t){|}_{\tilde{V}}^{2}dt]\\ \hspace{1em}+\sum _{i=1}^2\mathbb{E}[\underset{t\in [0,T\wedge {\mathfrak{S}}_N^n]}{\sup }|{\int }_0^t{M}_i(s)ds|]+\sum _{i=1}^2\mathbb{E}[{\int }_0^{T\wedge {\mathfrak{S}}_N^n}|L_i(t)|dt],\end{array}$$(3.11) to obtain, (3.18) $$\mathbb{E}[{\int }_0^{T\wedge {\mathfrak{S}}_N^n}|\nabla {\mathbf{v}}_n^{\nu }(t){|}_{{\mathbb{L}}^2}^2|{\mathit{\tau }}_n^{\nu }(t){|}_{\tilde{V}}^{2}dt]\le T \mathbb{E}[\underset{t\in [0,T\wedge {\mathfrak{S}}_N^n]}{\sup }|\nabla {\mathbf{v}}_n^{\nu }(t){|}_{\tilde{H}}^4]+\mathbb{E}[{\int }_0^{T\wedge {\mathfrak{S}}_N^n}|{\mathit{\tau }}_n^{\nu }(t){|}_{\tilde{V}}^{4}dt]$$(3.18) which is finite because of Lemma 3.3 and (3.5)(3.5) $$\hspace{1em}+p \underset{n\in \mathbb{N}}{\sup } \mathbb{E}[{\int }_0^T|\nabla {\mathit{\tau }}_n^{\nu }(s){|}_{\tilde{H}}^2|{\mathit{\tau }}_n^{\nu }(s){|}_{\tilde{H}}^{p-2}ds]\le C.$$(3.5) .

Now substituting (3.13)(3.13) $$\mathbb{E}[\underset{t\in [0,T\wedge {\mathfrak{S}}_N^n]}{\sup }|{\int }_0^t{M}_1(s)ds|]\le CT+\frac{1}{8}\mathbb{E}\left[\underset{t\in [0,T\wedge {\mathfrak{S}}_N^n]}{\sup }\right|{\Gamma }_n^{\nu }(t){|}_{{\mathbb{L}}^2}^2]+C(T) \mathbb{E}[{\int }_0^{T\wedge {\mathfrak{S}}_N^n}|{\Gamma }_n^{\nu }(t){|}_{{\mathbb{L}}^2}^{2}dt].$$(3.13) , (3.15)(3.15) $$\begin{array}{c}\mathbb{E}[\underset{t\in [0,T\wedge {\mathfrak{S}}_N^n]}{\sup }|{\int }_0^t{M}_2(s)ds|]\le C\mathbb{E}\left[{({{\int }_0^{T\wedge {\mathfrak{S}}_N^n}{\int }_B\{|{\mathcal{C}}_{\nu }R{\Phi }_n(l,{\mathit{\tau }}_n^{\nu }(t)){|}_{{\mathbb{L}}^2}^2-|{\Gamma }_n^{\nu }(t){|}_{{\mathbb{L}}^2}^2\}}^2{\lambda }_2(dl)dt)}^{1/2}\right]\\ \le {\mathcal{C}}_{\nu }C\mathbb{E}\left[{\left({\int }_0^{T\wedge {\mathfrak{S}}_N^n}|{\mathit{\tau }}_n^{\nu }\right(t\left){|}_{\tilde{H}}^{4}dt\right)}^{1/2}\right]+C\mathbb{E}\left[{\left({\int }_0^{T\wedge {\mathfrak{S}}_N^n}|{\Gamma }_n^{\nu }\right(t\left){|}_{{\mathbb{L}}^2}^{4}dt\right)}^{1/2}\right]\\ \le C{\mathcal{C}}_{\nu }T+\frac{1}{8}\mathbb{E}\left[\underset{t\in [0,T\wedge {\mathfrak{S}}_N^n]}{\sup }\right|{\Gamma }_n^{\nu }\left(t\right){|}_{{\mathbb{L}}^2}^2]+C(T) \mathbb{E}[{\int }_0^{T\wedge {\mathfrak{S}}_N^n}|{\Gamma }_n^{\nu }\left(t\right){|}_{{\mathbb{L}}^2}^{2}dt].\end{array}$$(3.15) –(3.18)(3.18) $$\mathbb{E}[{\int }_0^{T\wedge {\mathfrak{S}}_N^n}|\nabla {\mathbf{v}}_n^{\nu }(t){|}_{{\mathbb{L}}^2}^2|{\mathit{\tau }}_n^{\nu }(t){|}_{\tilde{V}}^{2}dt]\le T \mathbb{E}[\underset{t\in [0,T\wedge {\mathfrak{S}}_N^n]}{\sup }|\nabla {\mathbf{v}}_n^{\nu }(t){|}_{\tilde{H}}^4]+\mathbb{E}[{\int }_0^{T\wedge {\mathfrak{S}}_N^n}|{\mathit{\tau }}_n^{\nu }(t){|}_{\tilde{V}}^{4}dt]$$(3.18) in (3.11)(3.11) $$\begin{array}{c}\frac{1}{2}\mathbb{E}\left[\underset{t\in [0,T\wedge {\mathfrak{S}}_N^n]}{\sup }\right|{\Gamma }_n^{\nu }(t){|}_{{\mathbb{L}}^2}^2]+\nu \mathbb{E}[{\int }_0^{T\wedge {\mathfrak{S}}_N^n}|\nabla {\Gamma }_n^{\nu }(t){|}_{{\mathbb{L}}^2}^{2}dt]+\frac{{\mathcal{C}}_{\nu }}{2}\mathbb{E}[{\int }_0^{T\wedge {\mathfrak{S}}_N^n}|{\Gamma }_n^{\nu }(t){|}_{{\mathbb{L}}^2}^{2}dt]\\ \le \frac{1}{2}\mathbb{E}\left[\right|{\Gamma }^{\nu }(0){|}_{{\mathbb{L}}^2}^2]+|{\mathcal{C}}_{\nu }\left|\right|{\mathcal{C}}_{\nu }-a{|}^2\mathbb{E}[{\int }_0^{T\wedge {\mathfrak{S}}_N^n}|R({\mathit{\tau }}_n^{\nu }(t)){|}_{{\mathbb{L}}^2}^{2}dt]+{\mathcal{C}}_{\nu }C \mathbb{E}[{\int }_0^{T\wedge {\mathfrak{S}}_N^n}|\nabla {\mathbf{v}}_n^{\nu }(t){|}_H^2|{\mathit{\tau }}_n^{\nu }(t){|}_{\tilde{V}}^{2}dt]\\ \hspace{1em}+\sum _{i=1}^2\mathbb{E}[\underset{t\in [0,T\wedge {\mathfrak{S}}_N^n]}{\sup }|{\int }_0^t{M}_i(s)ds|]+\sum _{i=1}^2\mathbb{E}[{\int }_0^{T\wedge {\mathfrak{S}}_N^n}|L_i(t)|dt],\end{array}$$(3.11) , we have $$\begin{array}{c}\frac{1}{2}\mathbb{E}\left[\underset{t\in [0,T\wedge {\mathfrak{S}}_N^n]}{\sup }\right|{\Gamma }_n^{\nu }(t){|}_{{\mathbb{L}}^2}^2]+\nu \mathbb{E}[{\int }_0^{T\wedge {\mathfrak{S}}_N^n}|\nabla {\Gamma }_n^{\nu }(t){|}_{{\mathbb{L}}^2}^{2}dt]+\frac{{\mathcal{C}}_{\nu }}{2}\mathbb{E}[{\int }_0^{T\wedge {\mathfrak{S}}_N^n}|{\Gamma }_n^{\nu }(t){|}_{{\mathbb{L}}^2}^{2}dt]\\ \le \frac{1}{2}\mathbb{E}\left[\right|{\Gamma }^{\nu }(0){|}_{{\mathbb{L}}^2}^2]+C(T)+C(T)({\mathcal{C}}_{\nu }+{\mathcal{C}}_{\nu }|{\mathcal{C}}_{\nu }-a{|}^2)+C(T)(1+{\mathcal{C}}_{\nu })\mathbb{E}[{\int }_0^{T\wedge {\mathfrak{S}}_N^n}|{\Gamma }_n^{\nu }(t){|}_{{\mathbb{L}}^2}^{2}dt].\end{array}$$ Using Gronwall’s inequality and noting that as $N\to \infty ,$ ${\mathfrak{S}}_N^n\wedge T\to T,$ we finally achieve $$\mathbb{E}\left[\underset{t\in [0,T]}{\sup }\right|{\Gamma }_n^{\nu }(t){|}_{{\mathbb{L}}^2}^2]\le C(T,{\mathcal{C}}_{\nu }).$$ Finally, using the definition of ${\mathbf{w}}_n^{\nu }:={\Gamma }_n^{\nu }+{\mathcal{C}}_{\nu }R{\mathit{\tau }}_n,$ we have $$\mathbb{E}\left[\underset{t\in [0,T]}{\sup }\right|{\mathbf{w}}_n^{\nu }(t){|}_{{\mathbb{L}}^2}^2]\le 2\{\mathbb{E}[\underset{t\in [0,T]}{\sup }(|{\Gamma }_n^{\nu }(t){|}_{{\mathbb{L}}^2}^2+{\mathcal{C}}_{\nu }{}^2|R({\mathit{\tau }}_n^{\nu }(t)){|}_{{\mathbb{L}}^2}^2)]\}\le C(T,{\mathcal{C}}_{\nu }).$$ Note that, as $1>{\nu }_0\ge \nu >0,$ for a fixed ν₀, we have by the definition of ${\mathcal{C}}_{\nu }$ that $1<{\mathcal{C}}_{\nu }\le {\mathcal{C}}_{{\nu }_0}.$ Hence the right hand side of the above inequality can be further made less than some constant, which is independent of n and ν. □

Corollary 3.5.

Under the above mathematical settings, there exists a constant C > 0, which depends on the parameters of the problem but is independent of n and ν, such that $$\mathbb{E}\left[\underset{t\in [0,T]}{\sup }\right|{\mathbf{v}}_n^{\nu }(t){|}_V^2]\le C.$$

Proof.

Following Lemma 3.2 of [35], let us consider the following elliptic problem: (3.19) $$\begin{array}{c}\Delta {\mathbf{v}}_n^{\nu }={\nabla }^{\perp }{\mathbf{w}}_n^{\nu }\hspace{1em}\text{in}\hspace{1em}{\mathbb{R}}^2,\\ {\mathbf{v}}_n^{\nu }(x,·)\to 0\hspace{1em}\text{as}\hspace{1em}\left|x\right|\to \infty \end{array}$$(3.19) where ${\nabla }^{\perp }=(\frac{\partial }{\partial x_2},-\frac{\partial }{\partial x_1}).$

Multiplying (3.19)(3.19) $$\begin{array}{c}\Delta {\mathbf{v}}_n^{\nu }={\nabla }^{\perp }{\mathbf{w}}_n^{\nu }\hspace{1em}\text{in}\hspace{1em}{\mathbb{R}}^2,\\ {\mathbf{v}}_n^{\nu }(x,·)\to 0\hspace{1em}\text{as}\hspace{1em}\left|x\right|\to \infty \end{array}$$(3.19) by ${\mathbf{v}}_n^{\nu }$ and using integration by parts on ${\mathbb{R}}^2,$ we have $${\int }_{{\mathbb{R}}^2}|\nabla {\mathbf{v}}_n^{\nu }(x){|}_{{\mathbb{R}}^2}^{2}dx=-{\int }_{{\mathbb{R}}^2}{〈{\nabla }^{\perp }{\mathbf{w}}_n^{\nu }(x),{\mathbf{v}}_n^{\nu }(x)〉}_{{\mathbb{R}}^2}dx={\int }_{{\mathbb{R}}^2}|{\mathbf{w}}_n^{\nu }(x){|}_{{\mathbb{R}}^2}^{2}dx.$$ This directly implies $$\underset{0\le s\le t}{\sup }|\nabla {\mathbf{v}}_n^{\nu }(s){|}_H^2=\underset{0\le s\le t}{\sup }|{\mathbf{w}}_n^{\nu }(s){|}_H^2.$$ Then taking expectation we have by (3.6)(3.6) $$\mathbb{E}\left[\underset{0\le s\le T}{\sup }\right|{\mathbf{w}}_n^{\nu }(s){|}_H^2]\le C,$$(3.6) $$\mathbb{E}[{\int }_0^t|\nabla {\mathbf{v}}_n^{\nu }(s\left){|}_H^{2}ds\right]\le CT\mathbb{E}\left(\underset{0\le s\le t}{\sup }\right|\nabla {\mathbf{v}}_n^{\nu }\left(s\right){|}_H^2)=CT\mathbb{E}(\underset{0\le s\le t}{\sup }\left|{\mathbf{w}}_n^{\nu }\right(s\left){|}_H^2\right)\le C.$$ This completes the proof. □

Step IV: (Tightness)

Let the spaces ${\mathcal{Z}}_i; i=1,2$ defined by (3.20) $$\{\begin{array}{c}{\mathcal{Z}}_1:=D([0,T];\mathcal{U}\prime )\cap D([0,T];H_w)\cap {\mathrm{L}}_w^2(0,T;V)\cap {\mathrm{L}}^2(0,T;H_{\mathit{loc}}),\hfill \\ {\mathcal{Z}}_2:=D([0,T];\tilde{\mathcal{U}}\prime )\cap D([0,T];{\tilde{H}}_w)\cap {\mathrm{L}}_w^2(0,T;\tilde{V})\cap {\mathrm{L}}^2(0,T;{\tilde{H}}_{\mathit{loc}}).\hfill \end{array}$$(3.20) and let ${\mathcal{T}}_i$ be the supremum of the corresponding topologies in the spaces ${\mathcal{Z}}_i.$ Let $\mathcal{Z}:={\mathcal{Z}}_1\times {\mathcal{Z}}_2$ and $\mathcal{T}$ be the supremum of ${\mathcal{T}}_1$ and ${\mathcal{T}}_2.$ We note that the solutions $({\mathbf{v}}_n^{\nu },{\mathit{\tau }}_n^{\nu })$ obtained from Galerkin approximation induces sequence of measures $\mathcal{L}({\mathbf{v}}_n^{\nu },{\mathit{\tau }}_n^{\nu })$ on the space $(\mathcal{Z},\mathcal{T}).$ Then we have the following result:

Lemma 3.6.

(Tightness of law) The sequence ${\{\mathcal{L}({\mathbf{v}}_n^{\nu },{\mathit{\tau }}_n^{\nu })\}}_{n\in \mathbb{N}}$ of measures is tight on the space $(\mathcal{Z},\mathcal{T}).$

Proof.

We observe from Step II that there exists a constant $C_T=C_T({\nu }_1,{\nu }_2,\kappa ,a)>0$ such that for all $t\in [0,T], \forall n\in \mathbb{N}$ (3.21a) $$\underset{n\in \mathbb{N}}{\sup }\mathbb{E}\left[\underset{t\in [0,T]}{\sup }\right|{\mathbf{v}}_n^{\nu }(t){|}_H^2],\hspace{1em}\underset{n\in \mathbb{N}}{\sup }\mathbb{E}[{\int }_0^T|{\mathbf{v}}_n^{\nu }(t){|}_V^{2}dt]\le C_T,$$(3.21a) (3.21b) $$\underset{n\in \mathbb{N}}{\sup }\mathbb{E}\left[\underset{t\in [0,T]}{\sup }\right|{\mathit{\tau }}_n^{\nu }(t){|}_{\tilde{H}}^2],\hspace{1em}\underset{n\in \mathbb{N}}{\sup }\mathbb{E}[{\int }_0^T|{\mathit{\tau }}_n^{\nu }(t){|}_{\tilde{V}}^{2}dt]\le C_T.$$(3.21b) Therefore from (3.21)(3.21a) $$\underset{n\in \mathbb{N}}{\sup }\mathbb{E}\left[\underset{t\in [0,T]}{\sup }\right|{\mathbf{v}}_n^{\nu }(t){|}_H^2],\hspace{1em}\underset{n\in \mathbb{N}}{\sup }\mathbb{E}[{\int }_0^T|{\mathbf{v}}_n^{\nu }(t){|}_V^{2}dt]\le C_T,$$(3.21a) we conclude that first two conditions of Theorem A.9 for $({\mathbf{v}}_n^{\nu },{\mathit{\tau }}_n^{\nu })$ are satisfied. Now it is sufficient to prove that ${({\mathbf{v}}_n^{\nu })}_{n\in \mathbb{N}}$ and ${({\mathit{\tau }}_n^{\nu })}_{n\in \mathbb{N}}$ satisfy Aldous condition in the space $\mathcal{U}\prime$ and $\tilde{\mathcal{U}}\prime$ respectively. We start with the former sequence. We will use Lemma A.7. Let ${({\tau }_n)}_{n\in \mathbb{N}}$ be a sequence of stopping times such that $0\le {\tau }_n\le T.$

Let ${({\rho }_n)}_{n\in \mathbb{N}}$ be a sequence of stopping times such that $0\le {\rho }_n\le T.$ Then for $t\in (0,{\rho }_n\wedge T)$ we get $$\begin{array}{c}{\mathbf{v}}_n^{\nu }(t)={\mathbf{v}}_{0n}^{\nu }-\nu {\int }_0^t\mathcal{A}{\mathbf{v}}_n^{\nu }(s)ds-{\int }_0^t{B}_n({\mathbf{v}}_n^{\nu }(s))ds+{\int }_0^{t}Div {\mathit{\tau }}_n^{\nu }(s)ds\\ \hspace{1em}+{\int }_0^t{\int }_Z{F}_n({\mathbf{v}}_n^{\nu }(s-),z){\tilde{N}}_1(ds,dz):=\sum _{i=1}^5{J}_i^n(t).\end{array}$$ We will show that each of ${\{J_i^n(t)\}}_{i=1}^5$ satisfies condition (A.27)(A.27) $$\mathbb{E}\left[\right|X_n({\tau }_n+\theta )-X_n({\tau }_n){|}_E^{\alpha }]\le C{\theta }^{\beta }$$(A.27) in Lemma A.7. Let us fix $\theta >0.$ $J_n^1$ being independent of time clearly satisfies (A.27)(A.27) $$\mathbb{E}\left[\right|X_n({\tau }_n+\theta )-X_n({\tau }_n){|}_E^{\alpha }]\le C{\theta }^{\beta }$$(A.27) in Lemma A.7. Now exploiting the fact $\mathcal{A}:V\to V\prime$ given by $|\mathcal{A}\mathbf{v}{|}_{V\prime }\le |\mathbf{v}{|}_V,$ the embedding $V\prime \hookrightarrow \mathcal{U}\prime$ is continuous, then by Hölder’s inequality and (3.21)(3.21a) $$\underset{n\in \mathbb{N}}{\sup }\mathbb{E}\left[\underset{t\in [0,T]}{\sup }\right|{\mathbf{v}}_n^{\nu }(t){|}_H^2],\hspace{1em}\underset{n\in \mathbb{N}}{\sup }\mathbb{E}[{\int }_0^T|{\mathbf{v}}_n^{\nu }(t){|}_V^{2}dt]\le C_T,$$(3.21a) , $J_2^n$ can be estimated as $$\mathbb{E}\left[\right|J_2^n({\rho }_n+\theta )-J_2^n({\rho }_n){|}_{\mathcal{U}\prime }]\le C_1\mathbb{E}[{\theta }^{\frac{1}{2}}{\left({\int }_0^T|{\mathbf{v}}_n^{\nu }(s){|}_V^{2}ds\right)}^{\frac{1}{2}}]\le c_2{\theta }^{\frac{1}{2}}.$$ Thus $J_2^n$ satisfies (A.27)(A.27) $$\mathbb{E}\left[\right|X_n({\tau }_n+\theta )-X_n({\tau }_n){|}_E^{\alpha }]\le C{\theta }^{\beta }$$(A.27) in Lemma A.7 with α = 1 and $\zeta =\frac{1}{2}.$

Using $B:V\times V\to V\prime$ is continuous and the continuous embedding $V\prime \hookrightarrow \mathcal{U}\prime ,$ and (3.21)(3.21a) $$\underset{n\in \mathbb{N}}{\sup }\mathbb{E}\left[\underset{t\in [0,T]}{\sup }\right|{\mathbf{v}}_n^{\nu }(t){|}_H^2],\hspace{1em}\underset{n\in \mathbb{N}}{\sup }\mathbb{E}[{\int }_0^T|{\mathbf{v}}_n^{\nu }(t){|}_V^{2}dt]\le C_T,$$(3.21a) , we estimate $J_3^n$ as $$\mathbb{E}\left[\right|J_3^n({\rho }_n+\theta )-J_3^n({\rho }_n){|}_{\mathcal{U}\prime }]\le C\mathbb{E}[{\int }_0^T|{\mathbf{v}}_n^{\nu }(s){|}_V^{2}ds]·\theta =:c_3\theta .$$ Thus $J_3^n$ satisfies (A.27)(A.27) $$\mathbb{E}\left[\right|X_n({\tau }_n+\theta )-X_n({\tau }_n){|}_E^{\alpha }]\le C{\theta }^{\beta }$$(A.27) in Lemma A.7 with α = 1 and $\zeta =1.$

Using (3.21)(3.21a) $$\underset{n\in \mathbb{N}}{\sup }\mathbb{E}\left[\underset{t\in [0,T]}{\sup }\right|{\mathbf{v}}_n^{\nu }(t){|}_H^2],\hspace{1em}\underset{n\in \mathbb{N}}{\sup }\mathbb{E}[{\int }_0^T|{\mathbf{v}}_n^{\nu }(t){|}_V^{2}dt]\le C_T,$$(3.21a) and the embedding $H\hookrightarrow V\prime \hookrightarrow \mathcal{U}\prime$ we have $$\mathbb{E}\left[\right|J_4^n({\rho }_n+\theta )-J_4^n({\rho }_n){|}_{\mathcal{U}\prime }]\le \mathbb{E}[|{\nu }_1{\int }_{{\rho }_n}^{{\rho }_n+\theta }|Div {\mathit{\tau }}_n^{\nu }(s){|}_{V\prime }ds]\le c_4{\theta }^{\frac{1}{2}}.$$ Therefore, $J_4^n$ satisfies (A.27)(A.27) $$\mathbb{E}\left[\right|X_n({\tau }_n+\theta )-X_n({\tau }_n){|}_E^{\alpha }]\le C{\theta }^{\beta }$$(A.27) in Lemma A.7 with α = 1 and $\zeta =\frac{1}{2}.$

Using (3.21)(3.21a) $$\underset{n\in \mathbb{N}}{\sup }\mathbb{E}\left[\underset{t\in [0,T]}{\sup }\right|{\mathbf{v}}_n^{\nu }(t){|}_H^2],\hspace{1em}\underset{n\in \mathbb{N}}{\sup }\mathbb{E}[{\int }_0^T|{\mathbf{v}}_n^{\nu }(t){|}_V^{2}dt]\le C_T,$$(3.21a) , the continuous embedding $H\hookrightarrow \mathcal{U}\prime ,$ Itô-Lévy isometry and Assumption 2.3, we have $$\mathbb{E}\left[\right|J_5^n({\rho }_n+\theta )-J_5^n({\rho }_n){|}_{\mathcal{U}\prime }^2]\le C\mathbb{E}[{\int }_{{\rho }_n}^{{\rho }_n+\theta }{\int }_Z|F_n({\mathbf{v}}_n^{\nu }(s),z){|}_H^2{\lambda }_1(dz)ds]\le c_5\theta .$$ Thus $J_5^n$ satisfies (A.27)(A.27) $$\mathbb{E}\left[\right|X_n({\tau }_n+\theta )-X_n({\tau }_n){|}_E^{\alpha }]\le C{\theta }^{\beta }$$(A.27) in Lemma A.7 with α = 2 and ζ = 1.

Let us consider the equation for ${\mathit{\tau }}_n^{\nu }$ for $t\in (0,{\rho }_n\wedge T],$ $$\begin{array}{c}{\mathit{\tau }}_n^{\nu }(t)={\mathit{\tau }}_{0n}-{\int }_0^t\tilde{\mathcal{A}}{\mathit{\tau }}_n^{\nu }(s)ds-{\int }_0^t{\tilde{B}}_n({\mathbf{v}}_n^{\nu }(s),{\mathit{\tau }}_n^{\nu }(s))ds-a{\int }_0^t{\mathit{\tau }}_n^{\nu }(s)ds+{\int }_0^t\mathcal{D}({\mathbf{v}}_n^{\nu }(s))ds\\ \hspace{1em}+{\int }_0^t{\int }_B{G}_n(l,{\mathit{\tau }}_n^{\nu }(s-)){\tilde{N}}_2(dl,ds)+{\int }_0^t{b}_n({\mathit{\tau }}_n^{\nu }(s))ds:=\sum _{i=1}^7{K}_i^n(t).\end{array}$$ $K_n^1$ being independent of time, (A.27)(A.27) $$\mathbb{E}\left[\right|X_n({\tau }_n+\theta )-X_n({\tau }_n){|}_E^{\alpha }]\le C{\theta }^{\beta }$$(A.27) in Lemma A.7 is automatically satisfied for any $\alpha ,\beta >0.$ Now exploiting the fact $\tilde{\mathcal{A}}:\tilde{V}\to \tilde{V}\prime$ given by $|\tilde{\mathcal{A}}\mathit{\tau }{|}_{\tilde{V}\prime }\le |\mathit{\tau }{|}_{\tilde{V}},$ the continuous embedding $\tilde{V}\prime \hookrightarrow \tilde{\mathcal{U}}\prime ,$ Hölder’s inequality and (3.21)(3.21a) $$\underset{n\in \mathbb{N}}{\sup }\mathbb{E}\left[\underset{t\in [0,T]}{\sup }\right|{\mathbf{v}}_n^{\nu }(t){|}_H^2],\hspace{1em}\underset{n\in \mathbb{N}}{\sup }\mathbb{E}[{\int }_0^T|{\mathbf{v}}_n^{\nu }(t){|}_V^{2}dt]\le C_T,$$(3.21a) , $K_2^n$ can be estimated as $$\mathbb{E}\left[\right|K_2^n({\rho }_n+\theta )-K_2^n({\rho }_n){|}_{\tilde{\mathcal{U}}\prime }]\le C_1\mathbb{E}[{\theta }^{\frac{1}{2}}{\left({\int }_0^T|{\mathit{\tau }}_n^{\nu }(s){|}_{\tilde{V}}^{2}ds\right)}^{\frac{1}{2}}]\le {\tilde{c}}_2{\theta }^{\frac{1}{2}}.$$ Thus $K_2^n$ satisfies (A.27)(A.27) $$\mathbb{E}\left[\right|X_n({\tau }_n+\theta )-X_n({\tau }_n){|}_E^{\alpha }]\le C{\theta }^{\beta }$$(A.27) in Lemma A.7 with α = 1 and $\zeta =\frac{1}{2}.$

Using $|\tilde{B}(\mathbf{v},\mathit{\tau }){|}_{\widetilde{V_s}\prime }\le |\mathbf{v}{|}_H|\mathit{\tau }{|}_{\tilde{H}}$ and the embedding $\widetilde{V_s}\prime \hookrightarrow \tilde{\mathcal{U}}\prime$ we have $$\mathbb{E}\left[\right|K_3^n({\rho }_n+\theta )-K_3^n\left({\rho }_n\right){|}_{\tilde{\mathcal{U}}\prime }]\le C\mathbb{E}[{\left({\int }_{{\rho }_n}^{{\rho }_n+\theta }|{\mathbf{v}}_n^{\nu }\right(s\left){|}_H^{2}ds\right)}^{\frac{1}{2}}{\left({\int }_{{\rho }_n}^{{\rho }_n+\theta }|{\mathit{\tau }}_n^{\nu }\right(s\left){|}_{\tilde{H}}^{2}ds\right)}^{\frac{1}{2}}]\le {\tilde{c}}_3\theta .$$ Thus $K_3^n$ satisfies (A.27)(A.27) $$\mathbb{E}\left[\right|X_n({\tau }_n+\theta )-X_n({\tau }_n){|}_E^{\alpha }]\le C{\theta }^{\beta }$$(A.27) in Lemma A.7 with α = 1 and $\zeta =1.$

Using the embedding $\tilde{H}\hookrightarrow \tilde{\mathcal{U}}\prime$ and (3.21)(3.21a) $$\underset{n\in \mathbb{N}}{\sup }\mathbb{E}\left[\underset{t\in [0,T]}{\sup }\right|{\mathbf{v}}_n^{\nu }(t){|}_H^2],\hspace{1em}\underset{n\in \mathbb{N}}{\sup }\mathbb{E}[{\int }_0^T|{\mathbf{v}}_n^{\nu }(t){|}_V^{2}dt]\le C_T,$$(3.21a) we have $$\mathbb{E}\left[\right|K_i^n({\rho }_n+\theta )-K_i^n({\rho }_n){|}_{\tilde{\mathcal{U}}\prime }]\le {\tilde{c}}_i{\theta }^{\frac{1}{2}},$$ for $i=4,5.$ Thus $K_i^n:i=4,5$ satisfies (A.27)(A.27) $$\mathbb{E}\left[\right|X_n({\tau }_n+\theta )-X_n({\tau }_n){|}_E^{\alpha }]\le C{\theta }^{\beta }$$(A.27) in Lemma A.7 with α = 1 and $\zeta =\frac{1}{2}.$

Using Itô-Lévy isometry, embedding $\tilde{H}\hookrightarrow \tilde{\mathcal{U}}\prime ,$ Lemma A.3 and (3.21)(3.21a) $$\underset{n\in \mathbb{N}}{\sup }\mathbb{E}\left[\underset{t\in [0,T]}{\sup }\right|{\mathbf{v}}_n^{\nu }(t){|}_H^2],\hspace{1em}\underset{n\in \mathbb{N}}{\sup }\mathbb{E}[{\int }_0^T|{\mathbf{v}}_n^{\nu }(t){|}_V^{2}dt]\le C_T,$$(3.21a) we have $$\mathbb{E}\left[\right|K_6^n({\rho }_n+\theta )-K_6^n({\rho }_n){|}_{\tilde{\mathcal{U}}\prime }^2]\le C\mathbb{E}[{\int }_{{\rho }_n}^{{\rho }_n+\theta }{\int }_B|G_n(l,{\mathit{\tau }}_n^{\nu }(s-)){\tilde{N}}_2(dl,ds){|}_{\tilde{H}}^2]\le {\tilde{c}}_6\theta .$$ Thus $K_6^n$ satisfies (A.27)(A.27) $$\mathbb{E}\left[\right|X_n({\tau }_n+\theta )-X_n({\tau }_n){|}_E^{\alpha }]\le C{\theta }^{\beta }$$(A.27) in Lemma A.7 with α = 2 and ζ = 1.

Since $\tilde{H}\hookrightarrow \tilde{\mathcal{U}}\prime$ is continuous, using Schwarz’s inequality, from Lemma A.3 and (3.21)(3.21a) $$\underset{n\in \mathbb{N}}{\sup }\mathbb{E}\left[\underset{t\in [0,T]}{\sup }\right|{\mathbf{v}}_n^{\nu }(t){|}_H^2],\hspace{1em}\underset{n\in \mathbb{N}}{\sup }\mathbb{E}[{\int }_0^T|{\mathbf{v}}_n^{\nu }(t){|}_V^{2}dt]\le C_T,$$(3.21a) we get, $$\mathbb{E}\left[\right|K_7^n({\rho }_n+\theta )-K_7^n({\rho }_n){|}_{\tilde{\mathcal{U}}\prime }]\le C \mathbb{E}[{\int }_{{\rho }_n}^{{\rho }_n+\theta }|b_n({\mathit{\tau }}_n^{\nu }(s)){|}_{\tilde{H}} ds]\le {\tilde{c}}_7\theta .$$ Thus $K_7^n(t)$ satisfies the condition (A.27)(A.27) $$\mathbb{E}\left[\right|X_n({\tau }_n+\theta )-X_n({\tau }_n){|}_E^{\alpha }]\le C{\theta }^{\beta }$$(A.27) in Lemma A.7 with α = 1 and $\zeta =1.$ This proves that the sequences ${({\mathbf{v}}_n^{\nu })}_{n\in \mathbb{N}}$ and ${({\mathit{\tau }}_n^{\nu })}_{n\in \mathbb{N}}$ satisfy the Aldous condition in the space $\mathcal{U}\prime$ and $\tilde{\mathcal{U}}\prime$ respectively. Thus the proof is complete. □

Step V: (Use of Skorokhod Representation Theorem)

In separable metric spaces, one can apply Prokhorov Theorem (see [36], Theorem II.6.7) and classical Skorokhod Representation Theorem (see [37], Theorem 6.7) to obtain a.s. convergence from tightness. However, for our purpose, we need a modified version of Skorokhod Representation Theorem due to Jakubowski [29, 30], which can be used for non-metrizable topological space. The version used in this paper is stated below (see also [31], Proposition 9).

Proposition 3.7.

Let ${\mathcal{X}}_1$ be a complete separable metric space and ${\mathcal{X}}_2$ a topological space such that there is a sequence of continuous functions $f_m:{\mathcal{X}}_2\to \mathbb{R}$ that separates points of ${\mathcal{X}}_2$. Denote $\mathcal{X}:={\mathcal{X}}_1\times {\mathcal{X}}_2$ and equip $\mathcal{X}$ with the topology induced by the canonical projections ${\pi }_j:{\mathcal{X}}_1\times {\mathcal{X}}_2\to {\mathcal{X}}_j$. Let $(\Omega ,\mathcal{F},\mathbb{P})$ be a probability space and ${({\chi }_n)}_{n\in \mathbb{N}}$ be a tight sequence of random variables in $(\mathcal{X},\mathcal{B}({\mathcal{X}}_1)\otimes \mathcal{A})$, where $\mathcal{A}$ is the $\sigma -$algebra generated by $f_m, m\in \mathbb{N}$. Assume that there is a random variable η in ${\mathcal{X}}_1$ such that ${\mathbb{P}}^{{\pi }_1°{\chi }_{\eta }}={\mathbb{P}}^{\eta }$. Then there are a subsequence ${\chi }_{n_k} k\in \mathbb{N}$ and random variables ${\widehat{\chi }}_k, \widehat{\chi }$ in $\mathcal{X}$ for $k\in \mathbb{N}$ on a common probability space $(\widehat{\Omega },\widehat{\mathcal{F}},\widehat{\mathbb{P}})$ with

• ${\widehat{\mathbb{P}}}^{{\widehat{\chi }}_k}={\mathbb{P}}^{{\chi }_{n_k}}$ for $k\in \mathbb{N},$

• ${\widehat{\chi }}_k\to \widehat{\chi }$ in $\mathcal{X}$ almost surely for $k\to \infty ,$

• ${\pi }_2°{\widehat{\chi }}_k={\pi }_1°\widehat{\chi }$ almost surely.

Note that, by $M_{\widehat{\mathbb{N}}}([0,T]\times Z),$ we denote the set of all $\widehat{\mathbb{N}}$-valued Borel measures ξ on $[0,T]\times Z$ with $\xi ({\mathcal{S}}_n)<\infty$ for all $n\in \mathbb{N},$ for some sequence ${\mathcal{S}}_n\subset [0,T]\times Z$ of Borel sets with ${\mathcal{S}}_n\uparrow [0,T]\times Z$ and $\mathit{Leb}\otimes {\lambda }_1({\mathcal{S}}_n)<\infty ,$ for all $n\in \mathbb{N}.$ It is well known that $M_{\widehat{\mathbb{N}}}([0,T]\times Z)$ is a complete separable metric space (see [31]). Similarly, $M_{\widehat{\mathbb{N}}}([0,T]\times B)$ is a complete separable metric spaces.

In the line of notations used in Proposition 3.7, we denote $${\mathcal{X}}_1=M_{\widehat{\mathbb{N}}}([0,T]\times Z)\times M_{\widehat{\mathbb{N}}}([0,T]\times B),\hspace{1em}{\mathcal{X}}_2={\mathcal{Z}}_1\times {\mathcal{Z}}_2=\mathcal{Z}.$$ Clearly, ${\mathcal{X}}_1$ is a complete separable metric space. Since any infinite dimensional normed space with respect to weak topology in that space is not metrizable, both ${\mathcal{Z}}_1$ and ${\mathcal{Z}}_2$ are not metrizable, and so is ${\mathcal{X}}_2$ or $\mathcal{Z}.$

Lemma 3.6 ensures that the sequence of measures ${\{\mathcal{L}({\mathbf{v}}_n^{\nu },{\mathit{\tau }}_n^{\nu })\}}_{n\in \mathbb{N}}$ is tight on the space $(\mathcal{Z},\mathcal{T}).$ Let $N_i^n:=N_i$ for $i=1,2.$ Then the set of measures $\{\mathcal{L}({\mathbf{v}}_n^{\nu },{\mathit{\tau }}_n^{\nu },N_1^n,N_2^n),$ $n\in \mathbb{N}\}$ is tight on $\mathcal{Z}\times M_{\widehat{\mathbb{N}}}([0,T]\times Z)\times M_{\widehat{\mathbb{N}}}([0,T]\times B).$

We choose $\nu =\frac{1}{n}.$ By the Skorokhod-Jakubowski theorem (see Proposition 3.7 and also Lemma A.10), there exists a subsequence ${(n_k)}_{k\in \mathbb{N}},$ a filtered probability space $\widehat{\mathfrak{L}}=(\widehat{\Omega },\widehat{\mathcal{F}},\widehat{\mathbb{F}},\widehat{\mathbb{P}})$ with a filtration $\widehat{\mathbb{F}}={({\widehat{\mathcal{F}}}_t)}_{t\ge 0},$ and on this space, $\mathcal{Z}\times M_{\widehat{\mathbb{N}}}([0,T]\times Z)\times M_{\widehat{\mathbb{N}}}([0,T]\times B)$-valued random variables $(\widehat{\mathbf{v}},\widehat{\mathit{\tau }},{\widehat{N}}_1,{\widehat{N}}_2),({\widehat{\mathbf{v}}}_n^{\nu },{\widehat{\mathit{\tau }}}_n^{\nu },{\widehat{N}}_1^n,{\widehat{N}}_2^n),n\in \mathbb{N}$ such that

• $\mathcal{L}(({\widehat{\mathbf{v}}}_n^{\nu },{\widehat{\mathit{\tau }}}_n^{\nu },{\widehat{N}}_1^n,{\widehat{N}}_2^n))=\mathcal{L}(({\mathbf{v}}_{m_n}^{\nu },{\mathit{\tau }}_{m_n}^{\nu },N_1^{m_n},N_2^{m_n}))$ for all $n\in \mathbb{N};$

• $({\widehat{\mathbf{v}}}_n^{\nu },{\widehat{\mathit{\tau }}}_n^{\nu },{\widehat{N}}_1^n,{\widehat{N}}_2^n)\to (\widehat{\mathbf{v}},\widehat{\mathit{\tau }},{\widehat{N}}_1,{\widehat{N}}_2)$ in $\mathcal{Z}\times M_{\widehat{\mathbb{N}}}([0,T]\times Z)\times M_{\widehat{\mathbb{N}}}([0,T]\times B)$ with probability 1 on $\widehat{\mathfrak{L}}$ as $n\to \infty ;$

• $({\widehat{N}}_1^n(\widehat{\omega }),{\widehat{N}}_2^n(\widehat{\omega }))=(N_1(\widehat{\omega }),N_2(\widehat{\omega }))$ for all $\widehat{\omega }\in \widehat{\Omega }.$

We will denote these sequences again by ${(({\mathbf{v}}_n^{\nu },{\mathit{\tau }}_n^{\nu },N_1^n,N_2^n))}_{n\in \mathbb{N}}$ and ${(({\widehat{\mathbf{v}}}_n^{\nu },{\widehat{\mathit{\tau }}}_n^{\nu },{\widehat{N}}_1^n,{\widehat{N}}_2^n))}_{n\in \mathbb{N}}.$ Using the definiton of $\mathcal{Z},$ we have $\widehat{\mathbb{P}}$-a.s. (3.22a) $${\widehat{\mathbf{v}}}_n^{\nu }\to \widehat{\mathbf{v}}\mathrm{ }in\mathrm{ }{\mathrm{L}}_w^2(0,T;V)\cap {\mathrm{L}}^2(0,T;H_{\mathit{loc}})\cap D([0,T];\mathcal{U}\prime )\cap D([0,T];H_w),$$(3.22a) (3.22b) $${\widehat{\mathit{\tau }}}_n^{\nu }\to \widehat{\mathit{\tau }}\mathrm{ }\text{in}\mathrm{ }{\mathrm{L}}_w^2(0,T;\tilde{V})\cap {\mathrm{L}}^2(0,T;{\tilde{H}}_{\mathit{loc}})\cap D([0,T];\tilde{\mathcal{U}}\prime )\cap D([0,T];{\tilde{H}}_w).$$(3.22b) It is easy to verify that the spaces ${\mathcal{Z}}_i,i=1,2$ are not a Polish spaces. The space $D([0,T];\mathcal{U}\prime )\cap L^2(0,T;H)$ is a Polish space. Then by Kuratowski theorem, $D([0,T];H_n)$ is a Borel subset of $D([0,T];\mathcal{U}\prime )\cap L^2(0,T;H).$ Hence $D([0,T];H_n)\cap {\mathcal{Z}}_1$ is a Borel subset of $D([0,T];\mathcal{U}\prime )\cap L^2(0,T;H)\cap {\mathcal{Z}}_1,$ which is equal to ${\mathcal{Z}}_1$ (see Lemma 4.2 in [38]). Thus we have the following proposition.

Proposition 3.8.

The set $D([0,T];H_n)\cap {\mathcal{Z}}_1$ is a Borel subset of ${\mathcal{Z}}_1$ and the corresponding embedding transforms Borel sets into Borel subsets. A similar result is true for ${\mathcal{Z}}_2.$

The above result leads us to the following conclusion:

Corollary 3.9.

${\widehat{\mathbf{v}}}_n^{\nu }$ and ${\widehat{\mathit{\tau }}}_n^{\nu }$ take values in H_n and ${\tilde{H}}_n$ respectively. The laws of ${\mathbf{v}}_n^{\nu }$ and ${\widehat{\mathbf{v}}}_n^{\nu }$ are equal on $D([0,T];H_n)$ and the laws of ${\mathit{\tau }}_n^{\nu }$ and ${\widehat{\mathit{\tau }}}_n^{\nu }$ are equal on $D([0,T];{\tilde{H}}_n).$

Thus one may conclude that the limiting processes $\widehat{\mathbf{v}}$ and $\widehat{\mathit{\tau }}$ satisfy the following estimates for $r\ge 2,$ $$\widehat{\mathbb{E}}\left[\underset{s\in [0,T]}{\sup }\right|\widehat{\mathbf{v}}(s){|}_H^r]\le C_r,\hspace{1em}\widehat{\mathbb{E}}[\underset{s\in [0,T]}{\sup }|\widehat{\mathit{\tau }}(s){|}_{\tilde{H}}^r]\le C_r,$$ for some constant C_r (depending on r).

Step VI: (Convergence)

Let us define the following functionals for $t\in [0,T],$ and for all $\varphi \in \mathcal{U}, \psi \in \tilde{\mathcal{U}},$ (3.23a) $$\begin{array}{c}{\mathfrak{S}}_n({\widehat{\mathbf{v}}}_n^{\nu },{\widehat{\mathit{\tau }}}_n^{\nu },{\widehat{N}}_1^n,\varphi )(t):={({\widehat{\mathbf{v}}}_n^{\nu }(0),\varphi )}_H-\nu {\int }_0^t〈\mathcal{A}{\widehat{\mathbf{v}}}_n^{\nu }(s),\varphi 〉 ds-{\int }_0^t〈B_n({\widehat{\mathbf{v}}}_n^{\nu }(s)),\varphi 〉 ds\\ \hspace{1em}+{\int }_0^t〈Div {\widehat{\mathit{\tau }}}_n^{\nu }(s),\varphi 〉 ds+{\int }_0^t{\int }_Z(F_n({\widehat{\mathbf{v}}}_n^{\nu }(s),z),\varphi {)}_H {\tilde{\widehat{N}}}_1^n(ds,dz),\end{array}$$(3.23a) (3.23b) $$\begin{array}{c}{\Lambda }_n({\widehat{\mathbf{v}}}_n^{\nu },{\widehat{\mathit{\tau }}}_n^{\nu },{\widehat{N}}_2^n,\psi )(t):={({\widehat{\mathit{\tau }}}_n^{\nu }(0),\psi )}_{\tilde{H}}-{\int }_0^t〈\tilde{\mathcal{A}}{\widehat{\mathit{\tau }}}_n^{\nu }(s),\psi 〉 ds-{\int }_0^t〈{\tilde{B}}_n({\widehat{\mathbf{v}}}_n^{\nu }(s),{\widehat{\mathit{\tau }}}_n^{\nu }(s)),\psi 〉 ds\\ \hspace{1em}-a{\int }_0^t〈{\widehat{\mathit{\tau }}}_n^{\nu }(s)),\psi 〉 ds+{\int }_0^t〈\mathcal{D}({\widehat{\mathbf{v}}}_n^{\nu }(s)),\psi 〉 ds\\ \hspace{1em}+{\int }_0^t{\int }_B(G_n(l,{\widehat{\mathit{\tau }}}_n^{\nu }(s)),\psi {)}_{\tilde{H}} {\tilde{\widehat{N}}}_2^n(ds,dl)+{\int }_0^t〈b_n({\widehat{\mathit{\tau }}}_n^{\nu }(s)),\psi 〉 ds,\end{array}$$(3.23b) (3.23c) $$\begin{array}{c}\mathfrak{S}(\widehat{\mathbf{v}},\widehat{\mathit{\tau }},{\widehat{N}}_1,\varphi )(t):={(\widehat{\mathbf{v}}(0),\varphi )}_H-{\int }_0^t〈B(\widehat{\mathbf{v}}(s)),\varphi 〉 ds+{\int }_0^t〈Div \widehat{\mathit{\tau }}(s),\varphi 〉 ds\\ \hspace{1em}+{\int }_0^t{\int }_Z(F(\widehat{\mathbf{v}}(s),z),\varphi {)}_H {\tilde{\widehat{N}}}_1(ds,dz),\end{array}$$(3.23c) (3.23d) $$\begin{array}{c}\Lambda (\widehat{\mathbf{v}},\widehat{\mathit{\tau }},{\widehat{N}}_2,\psi )(t):={(\widehat{\mathit{\tau }}(0),\psi )}_{\tilde{H}}-{\int }_0^t〈\tilde{\mathcal{A}}\widehat{\mathit{\tau }}(s),\psi 〉 ds-{\int }_0^t〈\tilde{B}(\widehat{\mathbf{v}}(s),\widehat{\mathit{\tau }}(s)),\psi 〉 ds\\ \hspace{1em}-a{\int }_0^t〈\widehat{\mathit{\tau }}(s)),\psi 〉 ds+{\int }_0^t〈\mathcal{D}(\widehat{\mathbf{v}}(s)),\psi 〉 ds\\ \hspace{1em}+{\int }_0^t{\int }_B(G_n(l,\widehat{\mathit{\tau }}(s)),\psi {)}_{\tilde{H}} {\tilde{\widehat{N}}}_2(ds,dl)+{\int }_0^t〈b(\widehat{\mathit{\tau }}(s)),\psi 〉 ds.\end{array}$$(3.23d) Next, we claim that (3.24a) $$1. \underset{n\to \infty }{\lim }\Vert {({\widehat{\mathbf{v}}}_n^{\nu }(·),\varphi )}_H-{(\widehat{\mathbf{v}}(·),\varphi )}_H{\Vert }_{{\mathrm{L}}^2([0,T]\times \widehat{\Omega })}=0,$$(3.24a) (3.24b) $$2. \underset{n\to \infty }{\lim }\Vert {\mathfrak{S}}_n({\widehat{\mathbf{v}}}_n^{\nu },{\widehat{\mathit{\tau }}}_n^{\nu },{\widehat{N}}_1^n,\varphi )-\mathfrak{S}(\widehat{\mathbf{v}},\widehat{\mathit{\tau }},{\widehat{N}}_1,\varphi ){\Vert }_{{\mathrm{L}}^2([0,T]\times \widehat{\Omega })}=0,$$(3.24b) (3.24c) $$3. \underset{n\to \infty }{\lim }\Vert {({\widehat{\mathit{\tau }}}_n^{\nu }(·),\psi )}_{\tilde{H}}-{(\widehat{\mathit{\tau }}(·),\psi )}_{\tilde{H}}{\Vert }_{{\mathrm{L}}^2([0,T]\times \widehat{\Omega })}=0,$$(3.24c) (3.24d) $$4. \underset{n\to \infty }{\lim }\Vert {\Lambda }_n({\widehat{\mathbf{v}}}_n^{\nu },{\widehat{\mathit{\tau }}}_n^{\nu },{\widehat{N}}_2^n,\psi )-\Lambda (\widehat{\mathbf{v}},\widehat{\mathit{\tau }},{\widehat{N}}_2,\psi ){\Vert }_{{\mathrm{L}}^2([0,T]\times \widehat{\Omega })}=0,$$(3.24d) which is a direct consequence of the following Proposition.

Proposition 3.10.

If ${\mathfrak{S}}_n, {\Lambda }_n, \mathfrak{S}, \Lambda$ are defined by (3.23)(3.23a) $$\begin{array}{c}{\mathfrak{S}}_n({\widehat{\mathbf{v}}}_n^{\nu },{\widehat{\mathit{\tau }}}_n^{\nu },{\widehat{N}}_1^n,\varphi )(t):={({\widehat{\mathbf{v}}}_n^{\nu }(0),\varphi )}_H-\nu {\int }_0^t〈\mathcal{A}{\widehat{\mathbf{v}}}_n^{\nu }(s),\varphi 〉 ds-{\int }_0^t〈B_n({\widehat{\mathbf{v}}}_n^{\nu }(s)),\varphi 〉 ds\\ \hspace{1em}+{\int }_0^t〈Div {\widehat{\mathit{\tau }}}_n^{\nu }(s),\varphi 〉 ds+{\int }_0^t{\int }_Z(F_n({\widehat{\mathbf{v}}}_n^{\nu }(s),z),\varphi {)}_H {\tilde{\widehat{N}}}_1^n(ds,dz),\end{array}$$(3.23a) , then we have the following convergences: (3.25) $$1. \underset{n\to \infty }{\lim }\Vert {\mathfrak{S}}_n({\widehat{\mathbf{v}}}_n^{\nu },{\widehat{\mathit{\tau }}}_n^{\nu },{\widehat{N}}_1^n,\varphi )-\mathfrak{S}({\mathbf{v}}_{\ast },{\mathit{\tau }}_{\ast },N_1^{\ast },\varphi ){\Vert }_{{\mathrm{L}}^2([0,T]\times \widehat{\Omega })}=0,$$(3.25) (3.26) $$2. \underset{n\to \infty }{\lim }\Vert {\Lambda }_n({\widehat{\mathbf{v}}}_n^{\nu },{\widehat{\mathit{\tau }}}_n^{\nu },{\widehat{N}}_2^n,\varphi )-\Lambda ({\mathbf{v}}_{\ast },{\mathit{\tau }}_{\ast },N_2^{\ast },\varphi ){\Vert }_{{\mathrm{L}}^2([0,T]\times \widehat{\Omega })}=0.$$(3.26)

We postpone the proof of the Proposition in Appendix A.1. Hence from (3.24)(3.24a) $$1. \underset{n\to \infty }{\lim }\Vert {({\widehat{\mathbf{v}}}_n^{\nu }(·),\varphi )}_H-{(\widehat{\mathbf{v}}(·),\varphi )}_H{\Vert }_{{\mathrm{L}}^2([0,T]\times \widehat{\Omega })}=0,$$(3.24a) , we infer that for Leb-almost all $t\in [0,T],$ all $\varphi \in \mathcal{U}$ and $\psi \in \tilde{\mathcal{U}},$ and $\widehat{\mathbb{P}}$-almost all $\omega \in \widehat{\Omega },$ we infer that $(\widehat{\mathbf{v}},\widehat{\mathit{\tau }})$ satisfies $$\begin{array}{c}{(\widehat{\mathbf{v}}(t),\varphi )}_H+{\int }_0^t〈B(\widehat{\mathbf{v}}(s)),\varphi 〉 ds+{\int }_0^t〈Div \widehat{\mathit{\tau }}(s),\varphi 〉 ds={({\widehat{\mathbf{v}}}_0,\varphi )}_H+{\int }_0^t{\int }_Z(F(\widehat{\mathbf{v}}(s-),z),\varphi {)}_H{\tilde{\widehat{N}}}_1(ds,dz),\\ {(\widehat{\mathit{\tau }}(t),\psi )}_{\tilde{H}}+{\int }_0^t〈\tilde{\mathcal{A}}\widehat{\mathit{\tau }}(s),\psi 〉 ds+{\int }_0^t〈\tilde{B}(\widehat{\mathbf{v}}(s),\widehat{\mathit{\tau }}(s)),\psi 〉 ds+a{\int }_0^t〈\widehat{\mathit{\tau }}(s),\psi 〉 ds\\ ={(\widehat{\mathit{\tau }}(0),\psi )}_{\tilde{H}}+{\int }_0^t〈\mathcal{D}(\widehat{\mathbf{v}}(s)),\psi 〉 ds+{\int }_0^t〈b(\widehat{\mathit{\tau }}(s)),\psi 〉 ds+{\int }_0^t{\int }_B(G(l,\widehat{\mathit{\tau }}(s)),\psi {)}_{\tilde{H}} {\tilde{\widehat{N}}}_2(ds,dl).\end{array}$$ Finally, since $\mathcal{U}$ is dense in V and $\tilde{\mathcal{U}}$ is dense in $\tilde{V},$ we infer that the above two equalities hold for all $t\in [0,T]$ and all $\varphi \in V$ and $\psi \in \tilde{V}$ respectively. This concludes that the system $(\widehat{\mathfrak{L}},\widehat{\mathbf{v}},\widehat{\mathit{\tau }},{\widehat{N}}_1,{\widehat{N}}_2)$ is a weak martingale solution of (3.1)(3.1a) $$d\mathbf{v}(t)+[B(\mathbf{v}(t))-{\nu }_{1}Div \mathit{\tau }(t)]dt={\int }_{Z}F(\mathbf{v}(t-),z){\tilde{N}}_1(dt,dz),\hspace{1em}t\ge 0$$(3.1a) . Thus the proof of Theorem 3.1 is complete. □

Acknowledgements

The first author would like to thank Professor Zdzisław Brzeźniak for many useful discussions on Marcus canonical SDEs. The second author would like to thank Professor Erika Hausenblas for useful discussions and partial support from the FWF-Project P32295.

Disclosure statement

Both the authors declare that they have no conflict of interest.

Additional information

Funding

The research of the first author is partially supported by his Royal Society International Exchange Grant entitled “Stochastic Landau-Lifshitz-Gilbert equation with Lévy noise and ferromagnetism” (ref: IE140328).

Notes

1 Supremum topology is the topology generated by the collection $\mathcal{S}={\cup }_{i=1}^4{\mathcal{T}}_i,$ i.e., the topology generated from the collection $\mathcal{S}$ as a subbasis. Let ${\mathcal{K}}_i$ be the supremum of the corresponding topologies in the spaces ${\mathcal{Z}}_i,i=1,2.$ This supremum topology is different from the subspace topology.

Appendix A.

Technical lemmas

We now state some useful results obtained by the first author and his collaborator [32].

Lemma A.1.

• There exists $M_1>0$ such that for any $l\in B$ and $\mathit{\tau }\in {\tilde{H}}_n,$ $$|G_n(l,\mathit{\tau }){|}_{{\tilde{H}}_n}\le M_1|l{|}_{{\mathbb{R}}^k}(1+|\mathit{\tau }{|}_{{\tilde{H}}_n}).$$

• There exists $M_2>0$ such that for any $l\in B$ and ${\mathit{\tau }}_1,{\mathit{\tau }}_2\in {\tilde{H}}_n,$ $$|G_n(l,{\mathit{\tau }}_2)-G_n(l,{\mathit{\tau }}_1){|}_{{\tilde{H}}_n}\le M_2|l{|}_{{\mathbb{R}}^k}|{\mathit{\tau }}_2-{\mathit{\tau }}_1{|}_{{\tilde{H}}_n}.$$

• There exists $M_3>0$ such that for any $l\in B$ and $\mathit{\tau }\in {\tilde{H}}_n,$ $$|K_n(l,\mathit{\tau }){|}_{{\tilde{H}}_n}\le M_3|l{|}_{{\mathbb{R}}^k}^2(1+|\mathit{\tau }{|}_{{\tilde{H}}_n}).$$

• There exists $M_4>0$ such that for any $l\in B$ and ${\mathit{\tau }}_1,{\mathit{\tau }}_2\in {\tilde{H}}_n,$ $$|K_n(l,{\mathit{\tau }}_2)-K_n(l,{\mathit{\tau }}_1){|}_{{\tilde{H}}_n}\le M_4|l{|}_{{\mathbb{R}}^k}^2|{\mathit{\tau }}_2-{\mathit{\tau }}_1{|}_{{\tilde{H}}_n}.$$

Lemma A.2.

There exists a constant $C_1>0$ such that for any ${\mathit{\tau }}_1,{\mathit{\tau }}_2\in {\tilde{H}}_n,$ $$|b_n({\mathit{\tau }}_2)-b_n({\mathit{\tau }}_1){|}_{{\tilde{H}}_n}^2+{\int }_B|G_n(l,{\mathit{\tau }}_2)-G_n(l,{\mathit{\tau }}_2){|}_{{\tilde{H}}_n}^2 {\lambda }_2(dl)\le C_1|{\mathit{\tau }}_2-{\mathit{\tau }}_1{|}_{{\tilde{H}}_n}^2.$$

Lemma A.3.

There exists a constant $C_2>0$ such that for any $\mathit{\tau }\in {\tilde{H}}_n,$ (A.1) $$|b_n(\mathit{\tau }){|}_{{\tilde{H}}_n}^2+{\int }_B|G_n(l,\mathit{\tau }){|}_{{\tilde{H}}_n}^2 {\lambda }_2(dl)\le C_2|\mathit{\tau }{|}_{{\tilde{H}}_n}^2.$$(A.1)

Below we provide Proof of Lemma 2.4.

Proof of Lemma 2.4.

Proof. Since $\psi (z)=|z{|}_{\tilde{H}}^p,$ for all $g,k\in \tilde{H},$ $${\psi }^{\prime }(z)g=p|z{|}_{\tilde{H}}^{p-2} {〈z,g〉}_{\tilde{H}},\mathrm{ }\text{and}{\psi }^{″}(z)(g,k)=[p(p-2)|z{|}_{\tilde{H}}^{p-4} (z\otimes z)+p|z{|}_{\tilde{H}}^{p-2}]{〈g,k〉}_{\tilde{H}}.$$

1. Note that $$\psi (\Phi (l,\mathit{\tau }))-\psi (\mathit{\tau })=\psi (y(1))-\psi (y(0))={\int }_0^1\frac{d}{ds}[\psi °y](s)ds={\int }_0^1{\psi }^{\prime }(y(s))(\mathcal{R}y(s))ds.$$ Hence $$\begin{array}{c}|\psi (\Phi (l,\mathit{\tau }))-\psi (\mathit{\tau })|\le p{\int }_0^1|y(s){|}_{\tilde{H}}^{p-1}|\mathcal{R}y(s){|}_{\tilde{H}}ds\le p\Vert \mathcal{R}{\Vert }_{\mathcal{L}(\tilde{H})}{\int }_0^1|y(s){|}_{\tilde{H}}^{p}ds\\ \le \mathfrak{N}p\left|l{|}_{{\mathbb{R}}^k}\Vert g{\Vert }_{\mathcal{L}(\tilde{H})}\right|\mathit{\tau }{|}_{\tilde{H}}^p.\end{array}$$ This proves first part of the Lemma.

2. We begin with the observation that (A.2) $$\begin{array}{c}\psi (\Phi (l,\mathit{\tau }))-\psi (\mathit{\tau })-{\psi }^{\prime }(\mathit{\tau })\mathcal{R}\mathit{\tau }=\psi (y(1))-\psi (y(0))-{\psi }^{\prime }(y(0))\mathcal{R}(y(0))\\ :={\int }_0^1{(\psi °y)}^{\prime }(s)ds-{(\psi °y)}^{\prime }(0)\\ ={\int }_0^1{\int }_0^s{\psi }^{″}(y(r))(\mathcal{R}y(r),\mathcal{R}y(r))\mathit{drds}+{\int }_0^1{\int }_0^s{\psi }^{\prime }(y(r))({\mathcal{R}}^{2}y(r))\mathit{drds}\\ :=I_1+I_2.\end{array}$$(A.2) Hence as $0\le s\le 1,$ (A.3) $$\begin{array}{c}\left|I_1\right|\le p(p-1){\int }_0^1|y(s){|}_{\tilde{H}}^{p-2} |\mathcal{R}y(s){|}_{\tilde{H}}^{2}ds\le p(p-1)\Vert \mathcal{R}{\Vert }_{\mathcal{L}(\tilde{H})}^2{\int }_0^1|y(s){|}_{\tilde{H}}^{p}ds\\ \le p(p-1)\Vert \mathcal{R}{\Vert }_{\mathcal{L}(\tilde{H})}^2|\mathit{\tau }{|}_{\tilde{H}}^p{\int }_0^1\Vert e^{s\mathcal{R}}{\Vert }^{p}ds\le \mathfrak{N}p(p-1)\Vert \mathcal{R}{\Vert }_{\mathcal{L}(\tilde{H})}^2|\mathit{\tau }{|}_{\tilde{H}}^p.\end{array}$$(A.3) Similarly, one obtains (A.4) $$\left|I_2\right|\le p\Vert \mathcal{R}{\Vert }_{\mathcal{L}(\tilde{H})}^2{\int }_0^1|y(s){|}_{\tilde{H}}^{p}ds\le \mathfrak{N}p\Vert \mathcal{R}{\Vert }_{\mathcal{L}(\tilde{H})}^2|\mathit{\tau }{|}_{\tilde{H}}^p.$$(A.4) Therefore using the estimates (A.3)(A.3) $$\begin{array}{c}\left|I_1\right|\le p(p-1){\int }_0^1|y(s){|}_{\tilde{H}}^{p-2} |\mathcal{R}y(s){|}_{\tilde{H}}^{2}ds\le p(p-1)\Vert \mathcal{R}{\Vert }_{\mathcal{L}(\tilde{H})}^2{\int }_0^1|y(s){|}_{\tilde{H}}^{p}ds\\ \le p(p-1)\Vert \mathcal{R}{\Vert }_{\mathcal{L}(\tilde{H})}^2|\mathit{\tau }{|}_{\tilde{H}}^p{\int }_0^1\Vert e^{s\mathcal{R}}{\Vert }^{p}ds\le \mathfrak{N}p(p-1)\Vert \mathcal{R}{\Vert }_{\mathcal{L}(\tilde{H})}^2|\mathit{\tau }{|}_{\tilde{H}}^p.\end{array}$$(A.3) and (A.4)(A.4) $$\left|I_2\right|\le p\Vert \mathcal{R}{\Vert }_{\mathcal{L}(\tilde{H})}^2{\int }_0^1|y(s){|}_{\tilde{H}}^{p}ds\le \mathfrak{N}p\Vert \mathcal{R}{\Vert }_{\mathcal{L}(\tilde{H})}^2|\mathit{\tau }{|}_{\tilde{H}}^p.$$(A.4) in (A.2)(A.2) $$\begin{array}{c}\psi (\Phi (l,\mathit{\tau }))-\psi (\mathit{\tau })-{\psi }^{\prime }(\mathit{\tau })\mathcal{R}\mathit{\tau }=\psi (y(1))-\psi (y(0))-{\psi }^{\prime }(y(0))\mathcal{R}(y(0))\\ :={\int }_0^1{(\psi °y)}^{\prime }(s)ds-{(\psi °y)}^{\prime }(0)\\ ={\int }_0^1{\int }_0^s{\psi }^{″}(y(r))(\mathcal{R}y(r),\mathcal{R}y(r))\mathit{drds}+{\int }_0^1{\int }_0^s{\psi }^{\prime }(y(r))({\mathcal{R}}^{2}y(r))\mathit{drds}\\ :=I_1+I_2.\end{array}$$(A.2) we have $$|\psi (\Phi (l,\mathit{\tau }))-\psi (\mathit{\tau })-{\psi }^{\prime }(\mathit{\tau })\mathcal{R}\mathit{\tau }|\le \mathfrak{N}{p}^2\Vert \mathcal{R}{\Vert }_{\mathcal{L}(\tilde{H})}^2|\mathit{\tau }{|}_{\tilde{H}}^p\le \mathfrak{N}{p}^2|l{|}_{{\mathbb{R}}^k}^2\Vert g{\Vert }_{\mathcal{L}(\tilde{H})}^2|\mathit{\tau }{|}_{\tilde{H}}^p.$$

A.1. Proof of Proposition 3.10

We now see using Fubini’s Theorem that (A.5) $$\begin{array}{c}\Vert {\mathfrak{S}}_n({\widehat{\mathbf{v}}}_n^{\nu },{\widehat{\mathit{\tau }}}_n^{\nu },{\widehat{N}}_1^n,\varphi )-\mathfrak{S}({\mathbf{v}}_{\ast },{\mathit{\tau }}_{\ast },N_1^{\ast },\varphi ){\Vert }_{{\mathrm{L}}^2([0,T]\times \widehat{\Omega })}^2\\ ={\int }_0^T\widehat{\mathbb{E}}\left[\right|{\mathfrak{S}}_n({\widehat{\mathbf{v}}}_n^{\nu },{\widehat{\mathit{\tau }}}_n^{\nu },{\widehat{N}}_1^n,\varphi )(t)-\mathfrak{S}({\mathbf{v}}_{\ast },{\mathit{\tau }}_{\ast },N_1^{\ast },\varphi )(t){|}^2] dt.\end{array}$$(A.5) Lemma A.4 (the following Lemma) ensures that each term on the right hand side of (3.23a)(3.23a) $$\begin{array}{c}{\mathfrak{S}}_n({\widehat{\mathbf{v}}}_n^{\nu },{\widehat{\mathit{\tau }}}_n^{\nu },{\widehat{N}}_1^n,\varphi )(t):={({\widehat{\mathbf{v}}}_n^{\nu }(0),\varphi )}_H-\nu {\int }_0^t〈\mathcal{A}{\widehat{\mathbf{v}}}_n^{\nu }(s),\varphi 〉 ds-{\int }_0^t〈B_n({\widehat{\mathbf{v}}}_n^{\nu }(s)),\varphi 〉 ds\\ \hspace{1em}+{\int }_0^t〈Div {\widehat{\mathit{\tau }}}_n^{\nu }(s),\varphi 〉 ds+{\int }_0^t{\int }_Z(F_n({\widehat{\mathbf{v}}}_n^{\nu }(s),z),\varphi {)}_H {\tilde{\widehat{N}}}_1^n(ds,dz),\end{array}$$(3.23a) converges to the right hand side of corresponding term in (3.23c)(3.23c) $$\begin{array}{c}\mathfrak{S}(\widehat{\mathbf{v}},\widehat{\mathit{\tau }},{\widehat{N}}_1,\varphi )(t):={(\widehat{\mathbf{v}}(0),\varphi )}_H-{\int }_0^t〈B(\widehat{\mathbf{v}}(s)),\varphi 〉 ds+{\int }_0^t〈Div \widehat{\mathit{\tau }}(s),\varphi 〉 ds\\ \hspace{1em}+{\int }_0^t{\int }_Z(F(\widehat{\mathbf{v}}(s),z),\varphi {)}_H {\tilde{\widehat{N}}}_1(ds,dz),\end{array}$$(3.23c) in ${\mathrm{L}}^2([0,T]\times \widehat{\Omega })$ which further assures that right hand side of (A.5)(A.5) $$\begin{array}{c}\Vert {\mathfrak{S}}_n({\widehat{\mathbf{v}}}_n^{\nu },{\widehat{\mathit{\tau }}}_n^{\nu },{\widehat{N}}_1^n,\varphi )-\mathfrak{S}({\mathbf{v}}_{\ast },{\mathit{\tau }}_{\ast },N_1^{\ast },\varphi ){\Vert }_{{\mathrm{L}}^2([0,T]\times \widehat{\Omega })}^2\\ ={\int }_0^T\widehat{\mathbb{E}}\left[\right|{\mathfrak{S}}_n({\widehat{\mathbf{v}}}_n^{\nu },{\widehat{\mathit{\tau }}}_n^{\nu },{\widehat{N}}_1^n,\varphi )(t)-\mathfrak{S}({\mathbf{v}}_{\ast },{\mathit{\tau }}_{\ast },N_1^{\ast },\varphi )(t){|}^2] dt.\end{array}$$(A.5) goes to zero as $n\to \infty .$ This verifies (3.25)(3.25) $$1. \underset{n\to \infty }{\lim }\Vert {\mathfrak{S}}_n({\widehat{\mathbf{v}}}_n^{\nu },{\widehat{\mathit{\tau }}}_n^{\nu },{\widehat{N}}_1^n,\varphi )-\mathfrak{S}({\mathbf{v}}_{\ast },{\mathit{\tau }}_{\ast },N_1^{\ast },\varphi ){\Vert }_{{\mathrm{L}}^2([0,T]\times \widehat{\Omega })}=0,$$(3.25) . Lemma A.5 (which is proven below) ensures that each term on the right hand side of (3.23b)(3.23b) $$\begin{array}{c}{\Lambda }_n({\widehat{\mathbf{v}}}_n^{\nu },{\widehat{\mathit{\tau }}}_n^{\nu },{\widehat{N}}_2^n,\psi )(t):={({\widehat{\mathit{\tau }}}_n^{\nu }(0),\psi )}_{\tilde{H}}-{\int }_0^t〈\tilde{\mathcal{A}}{\widehat{\mathit{\tau }}}_n^{\nu }(s),\psi 〉 ds-{\int }_0^t〈{\tilde{B}}_n({\widehat{\mathbf{v}}}_n^{\nu }(s),{\widehat{\mathit{\tau }}}_n^{\nu }(s)),\psi 〉 ds\\ \hspace{1em}-a{\int }_0^t〈{\widehat{\mathit{\tau }}}_n^{\nu }(s)),\psi 〉 ds+{\int }_0^t〈\mathcal{D}({\widehat{\mathbf{v}}}_n^{\nu }(s)),\psi 〉 ds\\ \hspace{1em}+{\int }_0^t{\int }_B(G_n(l,{\widehat{\mathit{\tau }}}_n^{\nu }(s)),\psi {)}_{\tilde{H}} {\tilde{\widehat{N}}}_2^n(ds,dl)+{\int }_0^t〈b_n({\widehat{\mathit{\tau }}}_n^{\nu }(s)),\psi 〉 ds,\end{array}$$(3.23b) converges to the right hand side of corresponding term in (3.23d)(3.23d) $$\begin{array}{c}\Lambda (\widehat{\mathbf{v}},\widehat{\mathit{\tau }},{\widehat{N}}_2,\psi )(t):={(\widehat{\mathit{\tau }}(0),\psi )}_{\tilde{H}}-{\int }_0^t〈\tilde{\mathcal{A}}\widehat{\mathit{\tau }}(s),\psi 〉 ds-{\int }_0^t〈\tilde{B}(\widehat{\mathbf{v}}(s),\widehat{\mathit{\tau }}(s)),\psi 〉 ds\\ \hspace{1em}-a{\int }_0^t〈\widehat{\mathit{\tau }}(s)),\psi 〉 ds+{\int }_0^t〈\mathcal{D}(\widehat{\mathbf{v}}(s)),\psi 〉 ds\\ \hspace{1em}+{\int }_0^t{\int }_B(G_n(l,\widehat{\mathit{\tau }}(s)),\psi {)}_{\tilde{H}} {\tilde{\widehat{N}}}_2(ds,dl)+{\int }_0^t〈b(\widehat{\mathit{\tau }}(s)),\psi 〉 ds.\end{array}$$(3.23d) in ${\mathrm{L}}^2([0,T]\times \widehat{\Omega })$ which further verifies (3.26)(3.26) $$2. \underset{n\to \infty }{\lim }\Vert {\Lambda }_n({\widehat{\mathbf{v}}}_n^{\nu },{\widehat{\mathit{\tau }}}_n^{\nu },{\widehat{N}}_2^n,\varphi )-\Lambda ({\mathbf{v}}_{\ast },{\mathit{\tau }}_{\ast },N_2^{\ast },\varphi ){\Vert }_{{\mathrm{L}}^2([0,T]\times \widehat{\Omega })}=0.$$(3.26) . This completes the proof.

Lemma A.4.

For all $\varphi \in \mathcal{U},$

1. $\underset{n\to \infty }{\lim }\widehat{\mathbb{E}}[{\int }_0^T|{({\widehat{\mathbf{v}}}_n^{\nu }(t)-{\mathbf{v}}_{\ast }(t),\varphi )}_H{|}^2 dt]=0,$

2. $\underset{n\to \infty }{\lim }{\int }_0^T\widehat{\mathbb{E}}[|{({\widehat{\mathbf{v}}}_n^{\nu }(0)-{\mathbf{v}}_{\ast }(0),\varphi )}_H{|}^2]dt=0,$

3. $\underset{n\to \infty }{\lim }{\int }_0^T\widehat{\mathbb{E}}[|{\int }_0^t〈\mathcal{A}{\widehat{\mathbf{v}}}_n^{\nu }(s)-\mathcal{A}{\mathbf{v}}_{\ast }(s),\varphi 〉 ds{|}^2] dt=0,$

4. $\underset{n\to \infty }{\lim }{\int }_0^T\widehat{\mathbb{E}}[|{\int }_0^t〈B_n({\widehat{\mathbf{v}}}_n^{\nu }(s))-B({\mathbf{v}}_{\ast }(s)),\varphi 〉 ds{|}^2] dt=0,$

5. $\underset{n\to \infty }{\lim }{\int }_0^T\widehat{\mathbb{E}}[|{\int }_0^t〈Div {\widehat{\mathit{\tau }}}_n^{\nu }(s)-\mathit{Div} {\mathit{\tau }}_{\ast }(s),\varphi 〉 ds{|}^2] dt=0,$

6. $\underset{n\to \infty }{\lim }{\int }_0^T\widehat{\mathbb{E}}[|{\int }_0^t{\int }_Z〈F_n({\widehat{\mathbf{v}}}_n^{\nu }(s),z)-F({\mathbf{v}}_{\ast }(s),z),\varphi 〉 {\lambda }_1(dz)ds{|}^2]dt=0.$

7. $\underset{n\to \infty }{\lim }{\int }_0^T\widehat{\mathbb{E}}[|{\int }_0^t{\int }_Z〈F_n({\widehat{\mathbf{v}}}_n^{\nu }(s),z)-F({\mathbf{v}}_{\ast }(s),z),\varphi 〉 {\tilde{N}}_1^{\ast }(ds,dz){|}^2]dt=0.$

Proof.

Let $\varphi \in \mathcal{U}$ be fixed.

1. Owing to (3.22)(3.22a) $${\widehat{\mathbf{v}}}_n^{\nu }\to \widehat{\mathbf{v}}\mathrm{ }in\mathrm{ }{\mathrm{L}}_w^2(0,T;V)\cap {\mathrm{L}}^2(0,T;H_{\mathit{loc}})\cap D([0,T];\mathcal{U}\prime )\cap D([0,T];H_w),$$(3.22a) , in particular for almost all $t\in [0,T],$ $\underset{n\to \infty }{\lim }{({\widehat{\mathbf{v}}}_n^{\nu }(t),\varphi )}_H={({\mathbf{v}}_{\ast }(t),\varphi )}_H \widehat{\mathbb{P}}-\mathrm{a}.\mathrm{s}.$ Also using $\widehat{\mathbb{E}}[{\int }_0^T|{({\widehat{\mathbf{v}}}_n^{\nu }(t)-{\mathbf{v}}_{\ast }(t),\varphi )}_H{|}^{2}dt]\le C,$ and by employing Vitali Theorem we have $$\underset{n\to \infty }{\lim }\Vert {({\widehat{\mathbf{v}}}_n^{\nu },\varphi )}_H-{({\mathbf{v}}_{\ast },\varphi )}_H{\Vert }_{{\mathrm{L}}^2([0,T]\times \widehat{\Omega })}^2=\underset{n\to \infty }{\lim }\widehat{\mathbb{E}}\left[{\int }_0^T|{({\widehat{\mathbf{v}}}_n^{\nu }(t)-{\mathbf{v}}_{\ast }(t),\varphi )}_H{|}^{2}dt\right]=0.$$

2. Equation (3.22)(3.22a) $${\widehat{\mathbf{v}}}_n^{\nu }\to \widehat{\mathbf{v}}\mathrm{ }in\mathrm{ }{\mathrm{L}}_w^2(0,T;V)\cap {\mathrm{L}}^2(0,T;H_{\mathit{loc}})\cap D([0,T];\mathcal{U}\prime )\cap D([0,T];H_w),$$(3.22a) ensures that ${({\widehat{\mathbf{v}}}_n^{\nu }(0),\varphi )}_H\to {({\mathbf{v}}_{\ast }(0),\varphi )}_H \widehat{\mathbb{P}}--\mathrm{a}.\mathrm{s}.,$ which further with Vitali Theorem infer $\underset{n\to \infty }{\lim }\Vert {({\widehat{\mathbf{v}}}_n^{\nu }(0)-{\mathbf{v}}_{\ast }(0),\varphi )}_H{\Vert }_{{\mathrm{L}}^2([0,T]\times \widehat{\Omega })}^2=0.$

3. Owing to (3.22)(3.22a) $${\widehat{\mathbf{v}}}_n^{\nu }\to \widehat{\mathbf{v}}\mathrm{ }in\mathrm{ }{\mathrm{L}}_w^2(0,T;V)\cap {\mathrm{L}}^2(0,T;H_{\mathit{loc}})\cap D([0,T];\mathcal{U}\prime )\cap D([0,T];H_w),$$(3.22a) , we have for any $\tilde{\varphi }\in {\mathrm{L}}^2(0,T;V)$ (A.6) $$\underset{n\to \infty }{\lim }{\int }_0^T〈\mathcal{A}{\widehat{\mathbf{v}}}_n^{\nu }(s)-\mathcal{A}{\mathbf{v}}_{\ast }(s),\tilde{\varphi }(s)〉ds=0.$$(A.6) Let $\varphi \in \mathcal{U}.$ Let $t\in [0,T]$ be fixed. We choose $\tilde{\varphi }(s)={\chi }_{(0,t)}(s)\varphi$ and note that $\tilde{\varphi }\in {\mathrm{L}}^2(0,T;V).$ Hence for $r>2,$ using (A.6)(A.6) $$\underset{n\to \infty }{\lim }{\int }_0^T〈\mathcal{A}{\widehat{\mathbf{v}}}_n^{\nu }(s)-\mathcal{A}{\mathbf{v}}_{\ast }(s),\tilde{\varphi }(s)〉ds=0.$$(A.6) and Hölder’s inequality, we achieve for all $t\in [0,T],$ and $n\in \mathbb{N},$ (A.7) $$\widehat{\mathbb{E}}[|{\int }_0^t〈\mathcal{A}{\widehat{\mathbf{v}}}_n^{\nu }(s),\varphi 〉\mathrm{ }ds{|}^r]\le \widehat{\mathbb{E}}[{\int }_0^t|{({\widehat{\mathbf{v}}}_n^{\nu }(s),\varphi )}_V{|}^r\mathrm{ }ds]\le 2^r{T}^r\Vert \varphi {\Vert }_{\mathcal{U}}^r \widehat{\mathbb{E}}[\underset{0\le s\le t}{\sup }|{\widehat{\mathbf{v}}}_n^{\nu }(s){|}_V^r]\le c_r$$(A.7) for some positive constant c_r (depending on r). Therefore using (A.7)(A.7) $$\widehat{\mathbb{E}}[|{\int }_0^t〈\mathcal{A}{\widehat{\mathbf{v}}}_n^{\nu }(s),\varphi 〉\mathrm{ }ds{|}^r]\le \widehat{\mathbb{E}}[{\int }_0^t|{({\widehat{\mathbf{v}}}_n^{\nu }(s),\varphi )}_V{|}^r\mathrm{ }ds]\le 2^r{T}^r\Vert \varphi {\Vert }_{\mathcal{U}}^r \widehat{\mathbb{E}}[\underset{0\le s\le t}{\sup }|{\widehat{\mathbf{v}}}_n^{\nu }(s){|}_V^r]\le c_r$$(A.7) and by Vitali Theorem, we conclude for all $t\in [0,T]$ (A.8) $$\underset{n\to \infty }{\lim }\widehat{\mathbb{E}}[|{\int }_0^t〈\mathcal{A}{\widehat{\mathbf{v}}}_n^{\nu }(s)-\mathcal{A}{\mathbf{v}}_{\ast }(s),\varphi 〉ds{|}^2]=0.$$(A.8) Hence (A.7)(A.7) $$\widehat{\mathbb{E}}[|{\int }_0^t〈\mathcal{A}{\widehat{\mathbf{v}}}_n^{\nu }(s),\varphi 〉\mathrm{ }ds{|}^r]\le \widehat{\mathbb{E}}[{\int }_0^t|{({\widehat{\mathbf{v}}}_n^{\nu }(s),\varphi )}_V{|}^r\mathrm{ }ds]\le 2^r{T}^r\Vert \varphi {\Vert }_{\mathcal{U}}^r \widehat{\mathbb{E}}[\underset{0\le s\le t}{\sup }|{\widehat{\mathbf{v}}}_n^{\nu }(s){|}_V^r]\le c_r$$(A.7) , (A.8)(A.8) $$\underset{n\to \infty }{\lim }\widehat{\mathbb{E}}[|{\int }_0^t〈\mathcal{A}{\widehat{\mathbf{v}}}_n^{\nu }(s)-\mathcal{A}{\mathbf{v}}_{\ast }(s),\varphi 〉ds{|}^2]=0.$$(A.8) and the Dominated Convergence Theorem proves (iii).

4. Let $\varphi \in \mathcal{U}.$ We have, $B({\widehat{\mathbf{v}}}_n^{\nu })-B({\mathbf{v}}_{\ast }):=B({\widehat{\mathbf{v}}}_n^{\nu },{\widehat{\mathbf{v}}}_n^{\nu })-B({\mathbf{v}}_{\ast },{\mathbf{v}}_{\ast })=B({\widehat{\mathbf{v}}}_n^{\nu }-{\mathbf{v}}_{\ast },{\widehat{\mathbf{v}}}_n^{\nu })+B({\mathbf{v}}_{\ast },{\widehat{\mathbf{v}}}_n^{\nu }-{\mathbf{v}}_{\ast }).$ Now, using Hölder’s inequality and ${\widehat{\mathbf{v}}}_n^{\nu }\to {\mathbf{v}}_{\ast }$ in $L^2(0,T;H),$ we have $$\underset{n\to \infty }{\lim }{\int }_0^t〈B({\widehat{\mathbf{v}}}_n^{\nu }(s))-B({\mathbf{v}}_{\ast }(s)),\varphi 〉ds=0\hspace{1em}\widehat{\mathbb{P}}-\mathrm{a}.\mathrm{s}.$$ For every $\varphi \in \mathcal{U},$ since $P_n\varphi \to \varphi$ in V_s and $\mathcal{U}\subset V_1,$ we conclude that for all $\varphi \in \mathcal{U}$ and all $t\in [0,T]$, (A.9) $$\underset{n\to \infty }{\lim }{\int }_0^t〈B_n({\widehat{\mathbf{v}}}_n^{\nu }(s))-B({\mathbf{v}}_{\ast }(s)),\varphi 〉ds=0\hspace{1em}\widehat{\mathbb{P}}-\mathrm{a}.\mathrm{s}.$$(A.9) Using Hölder’s inequality we obtain for all $t\in [0,T], r>2$ and $n\in \mathbb{N},$ (A.10) $$\widehat{\mathbb{E}}[|{\int }_0^t〈B_n({\widehat{\mathbf{v}}}_n^{\nu }(s)),\varphi 〉ds{|}^r]\le \widehat{\mathbb{E}}\left[{\left({\int }_0^t|B_n\right({\widehat{\mathbf{v}}}_n^{\nu }\left(s\right)\left){|}_{V_s^{\prime }}\right|\varphi {|}_{V_s}ds)}^r\right]\le C{C}_1(r).$$(A.10) Considering (A.9)(A.9) $$\underset{n\to \infty }{\lim }{\int }_0^t〈B_n({\widehat{\mathbf{v}}}_n^{\nu }(s))-B({\mathbf{v}}_{\ast }(s)),\varphi 〉ds=0\hspace{1em}\widehat{\mathbb{P}}-\mathrm{a}.\mathrm{s}.$$(A.9) and (A.10) (A.10) $$\widehat{\mathbb{E}}[|{\int }_0^t〈B_n({\widehat{\mathbf{v}}}_n^{\nu }(s)),\varphi 〉ds{|}^r]\le \widehat{\mathbb{E}}\left[{\left({\int }_0^t|B_n\right({\widehat{\mathbf{v}}}_n^{\nu }\left(s\right)\left){|}_{V_s^{\prime }}\right|\varphi {|}_{V_s}ds)}^r\right]\le C{C}_1(r).$$(A.10) and by the Vitali Theorem we achieve for all $t\in [0,T]$ (A.11) $$\underset{n\to \infty }{\lim }\widehat{\mathbb{E}}[|{\int }_0^t〈B_n({\widehat{\mathbf{v}}}_n^{\nu }(s))-B({\mathbf{v}}_{\ast }(s)),\varphi 〉ds{|}^2]=0.$$(A.11) Hence, in view of (A.10)(A.10) $$\widehat{\mathbb{E}}[|{\int }_0^t〈B_n({\widehat{\mathbf{v}}}_n^{\nu }(s)),\varphi 〉ds{|}^r]\le \widehat{\mathbb{E}}\left[{\left({\int }_0^t|B_n\right({\widehat{\mathbf{v}}}_n^{\nu }\left(s\right)\left){|}_{V_s^{\prime }}\right|\varphi {|}_{V_s}ds)}^r\right]\le C{C}_1(r).$$(A.10) , (A.11)(A.11) $$\underset{n\to \infty }{\lim }\widehat{\mathbb{E}}[|{\int }_0^t〈B_n({\widehat{\mathbf{v}}}_n^{\nu }(s))-B({\mathbf{v}}_{\ast }(s)),\varphi 〉ds{|}^2]=0.$$(A.11) and the Dominated Convergence Theorem, we infer that (A.12) $$\underset{n\to \infty }{\lim }{\int }_0^T\widehat{\mathbb{E}}[|{\int }_0^t〈B_n({\widehat{\mathbf{v}}}_n^{\nu }(s))-B({\mathbf{v}}_{\ast }(s)),\varphi 〉ds{|}^2]dt=0.$$(A.12) Hence we have (iv).

5. As $\varphi \in \tilde{\mathcal{U}},$ so $\nabla \varphi \in \tilde{H}.$ Now using (3.22)(3.22a) $${\widehat{\mathbf{v}}}_n^{\nu }\to \widehat{\mathbf{v}}\mathrm{ }in\mathrm{ }{\mathrm{L}}_w^2(0,T;V)\cap {\mathrm{L}}^2(0,T;H_{\mathit{loc}})\cap D([0,T];\mathcal{U}\prime )\cap D([0,T];H_w),$$(3.22a) we have (A.13) $$|{\int }_0^t〈Div {\widehat{\mathit{\tau }}}_n^{\nu }(s)-\mathit{Div} {\mathit{\tau }}_{\ast }(s),\varphi 〉 ds{|}^2=|-{\int }_0^t〈{\widehat{\mathit{\tau }}}_n^{\nu }(s)-{\mathit{\tau }}_{\ast }(s),\nabla \varphi 〉ds{|}^2\to 0\hspace{1em}\text{as}\hspace{1em}n\to \infty .$$(A.13) Let $r\ge 2.$ Now using (3.22)(3.22a) $${\widehat{\mathbf{v}}}_n^{\nu }\to \widehat{\mathbf{v}}\mathrm{ }in\mathrm{ }{\mathrm{L}}_w^2(0,T;V)\cap {\mathrm{L}}^2(0,T;H_{\mathit{loc}})\cap D([0,T];\mathcal{U}\prime )\cap D([0,T];H_w),$$(3.22a) we have (A.14) $${\int }_0^T\widehat{\mathbb{E}}[|{\int }_0^t〈\mathit{Div} {\widehat{\mathit{\tau }}}_n^{\nu }(s)-\mathit{Div} {\mathit{\tau }}_{\ast }(s),\varphi 〉ds{|}^r]={\int }_0^T\widehat{\mathbb{E}}[|-{\int }_0^t〈{\widehat{\mathit{\tau }}}_n^{\nu }(s)-{\mathit{\tau }}_{\ast }(s),\nabla \varphi 〉ds{|}^r]\le C_r,$$(A.14) for some positive constant $C_r.$ Hence by Vitali Theorem, using (A.13)(A.13) $$|{\int }_0^t〈Div {\widehat{\mathit{\tau }}}_n^{\nu }(s)-\mathit{Div} {\mathit{\tau }}_{\ast }(s),\varphi 〉 ds{|}^2=|-{\int }_0^t〈{\widehat{\mathit{\tau }}}_n^{\nu }(s)-{\mathit{\tau }}_{\ast }(s),\nabla \varphi 〉ds{|}^2\to 0\hspace{1em}\text{as}\hspace{1em}n\to \infty .$$(A.13) and (A.14)(A.14) $${\int }_0^T\widehat{\mathbb{E}}[|{\int }_0^t〈\mathit{Div} {\widehat{\mathit{\tau }}}_n^{\nu }(s)-\mathit{Div} {\mathit{\tau }}_{\ast }(s),\varphi 〉ds{|}^r]={\int }_0^T\widehat{\mathbb{E}}[|-{\int }_0^t〈{\widehat{\mathit{\tau }}}_n^{\nu }(s)-{\mathit{\tau }}_{\ast }(s),\nabla \varphi 〉ds{|}^r]\le C_r,$$(A.14) we have (v).

6. Let us consider $\varphi \in \mathcal{U}.$ Using Assumption 2.3 we have (A.15) $${\int }_0^t{\int }_Z|{(F({\widehat{\mathbf{v}}}_n^{\nu }(s),z)-F({\mathbf{v}}_{\ast }(s),z),\varphi )}_H{|}^2{\lambda }_1(dz)ds\to 0\hspace{1em}\text{as}\hspace{1em}n\to \infty .$$(A.15) Furthermore, using Assumption 2.3, for every $t\in [0,T], r>2$ and $n\in \mathbb{N},$ we have the following inequality (A.16) $$\widehat{\mathbb{E}}[{\int }_0^t{\int }_Z|{(F({\widehat{\mathbf{v}}}_n^{\nu }(s),z)-F({\mathbf{v}}_{\ast }(s),z),\varphi )}_H{|}^2{\lambda }_1(dz)ds{|}^r]\le {\tilde{C}}_r,$$(A.16) for some constant ${\tilde{C}}_r>0.$ Since the restriction of P_n to the space H is the ${(·,·)}_H$-projection onto $H_n,$ hence by (A.15)(A.15) $${\int }_0^t{\int }_Z|{(F({\widehat{\mathbf{v}}}_n^{\nu }(s),z)-F({\mathbf{v}}_{\ast }(s),z),\varphi )}_H{|}^2{\lambda }_1(dz)ds\to 0\hspace{1em}\text{as}\hspace{1em}n\to \infty .$$(A.15) , (A.16)(A.16) $$\widehat{\mathbb{E}}[{\int }_0^t{\int }_Z|{(F({\widehat{\mathbf{v}}}_n^{\nu }(s),z)-F({\mathbf{v}}_{\ast }(s),z),\varphi )}_H{|}^2{\lambda }_1(dz)ds{|}^r]\le {\tilde{C}}_r,$$(A.16) and by the Vitali Theorem we infer that for all $t\in [0,T], \forall \varphi \in \mathcal{U}$ (A.17) $$\underset{n\to \infty }{\lim }\widehat{\mathbb{E}}[{\int }_0^t{\int }_Z|{(F_n({\widehat{\mathbf{v}}}_n^{\nu }(s),z)-F({\mathbf{v}}_{\ast }(s),z),\varphi )}_H{|}^2{\lambda }_1(dz)ds]=0,\hspace{1em}\forall \varphi \in H.$$(A.17) Since $\mathcal{U}\subset H,$ (A.17)(A.17) $$\underset{n\to \infty }{\lim }\widehat{\mathbb{E}}[{\int }_0^t{\int }_Z|{(F_n({\widehat{\mathbf{v}}}_n^{\nu }(s),z)-F({\mathbf{v}}_{\ast }(s),z),\varphi )}_H{|}^2{\lambda }_1(dz)ds]=0,\hspace{1em}\forall \varphi \in H.$$(A.17) holds for all $\varphi \in \mathcal{U}.$ Moreover, $\widehat{\mathbb{E}}[{\int }_0^t{\int }_Z|{(F_n({\widehat{\mathbf{v}}}_n^{\nu }(s),z)-F({\mathbf{v}}_{\ast }(s),z),\varphi )}_H{|}^2{\lambda }_1(dz)ds]\le {\tilde{C}}_2$ and (A.17)(A.17) $$\underset{n\to \infty }{\lim }\widehat{\mathbb{E}}[{\int }_0^t{\int }_Z|{(F_n({\widehat{\mathbf{v}}}_n^{\nu }(s),z)-F({\mathbf{v}}_{\ast }(s),z),\varphi )}_H{|}^2{\lambda }_1(dz)ds]=0,\hspace{1em}\forall \varphi \in H.$$(A.17) and the Dominated Convergence Theorem assures assertion (vi).

7. Using Itô-Lévy isometry, and the fact that ${\widehat{N}}_i^n=N_i^{\ast }$ for $i=1,2,$ and (A.17)(A.17) $$\underset{n\to \infty }{\lim }\widehat{\mathbb{E}}[{\int }_0^t{\int }_Z|{(F_n({\widehat{\mathbf{v}}}_n^{\nu }(s),z)-F({\mathbf{v}}_{\ast }(s),z),\varphi )}_H{|}^2{\lambda }_1(dz)ds]=0,\hspace{1em}\forall \varphi \in H.$$(A.17) and then exploiting Dominated Convergence Theorem we ensure assertion (vii).

Lemma A.5.

For all $\psi \in \tilde{\mathcal{U}},$

1. $\underset{n\to \infty }{\lim }\widehat{\mathbb{E}}[{\int }_0^T|{({\widehat{\mathit{\tau }}}_n^{\nu }(t)-{\mathit{\tau }}_{\ast }(t),\psi )}_{\tilde{H}}{|}^2 dt]=0,$

2. $\underset{n\to \infty }{\lim }{\int }_0^T\widehat{\mathbb{E}}[|{({\widehat{\mathit{\tau }}}_n^{\nu }(0)-{\mathit{\tau }}_{\ast }(0),\psi )}_{\tilde{H}}{|}^2] dt=0,$

3. $\underset{n\to \infty }{\lim }{\int }_0^T\widehat{\mathbb{E}}[|{\int }_0^t〈\tilde{\mathcal{A}}{\widehat{\mathit{\tau }}}_n^{\nu }(s)-\tilde{\mathcal{A}}{\mathit{\tau }}_{\ast }(s),\psi 〉 ds{|}^2] dt=0,$

4. $\underset{n\to \infty }{\lim }{\int }_0^T\widehat{\mathbb{E}}[|{\int }_0^t〈{\tilde{B}}_n({\widehat{\mathbf{v}}}_n^{\nu }(s),{\widehat{\mathit{\tau }}}_n^{\nu }(s))-\tilde{B}({\mathbf{v}}_{\ast }(s),{\mathit{\tau }}_{\ast }(s)),\psi 〉 ds{|}^2] dt=0,$

5. $\underset{n\to \infty }{\lim }{\int }_0^T\widehat{\mathbb{E}}[|{\int }_0^t〈\mathcal{D}({\widehat{\mathbf{v}}}_n^{\nu }(s))-\mathcal{D}({\mathbf{v}}_{\ast }(s)),\psi 〉 ds{|}^2] dt=0,$

6. $\underset{n\to \infty }{\lim }{\int }_0^T\widehat{\mathbb{E}}[|{\int }_0^t{\int }_B〈G_n(l,{\widehat{\mathit{\tau }}}_n^{\nu }(s))-G(l,{\mathit{\tau }}_{\ast }(s)),\psi 〉 {\tilde{N}}_2^{\ast }(dl,ds){|}^2 dt]=0,$

7. $\underset{n\to \infty }{\lim }{\int }_0^T\widehat{\mathbb{E}}[|{\int }_0^t〈b_n({\widehat{\mathit{\tau }}}_n^{\nu }(s))-b({\mathit{\tau }}_{\ast }(s)),\psi 〉 ds{|}^2] dt=0.$

Proof.

Proofs of (i)–(iv) are similar to the proofs of Lemma A.4 (i)–(iv), and hence omitted.

(v) Let $\psi \in \tilde{\mathcal{U}}.$ Now using integration by parts and (3.22) we note that (A.18) $$|{\int }_0^t〈\mathcal{D}({\widehat{\mathbf{v}}}_n^{\nu }(s))-\mathcal{D}({\mathbf{v}}_{\ast }(s)),\psi 〉ds{|}^2\to 0\hspace{1em}\text{as}\hspace{1em}n\to \infty .$$(A.18) Let $r\ge 2.$ Now using (3.22)(3.22a) $${\widehat{\mathbf{v}}}_n^{\nu }\to \widehat{\mathbf{v}}\mathrm{ }in\mathrm{ }{\mathrm{L}}_w^2(0,T;V)\cap {\mathrm{L}}^2(0,T;H_{\mathit{loc}})\cap D([0,T];\mathcal{U}\prime )\cap D([0,T];H_w),$$(3.22a) we have ${\int }_0^T\widehat{\mathbb{E}}[|{\int }_0^t〈\mathcal{D}({\widehat{\mathbf{v}}}_n^{\nu }(s))-\mathcal{D}({\mathbf{v}}_{\ast }(s)),\psi 〉\mathrm{ }ds{|}^r]\mathrm{ }dt\le C_r,$ for some positive constant C_r depending on r. Hence by Vitali Theorem, using (A.18)(A.18) $$|{\int }_0^t〈\mathcal{D}({\widehat{\mathbf{v}}}_n^{\nu }(s))-\mathcal{D}({\mathbf{v}}_{\ast }(s)),\psi 〉ds{|}^2\to 0\hspace{1em}\text{as}\hspace{1em}n\to \infty .$$(A.18) we ensure assertion (v).

(vi) Using Lipschitz property of G and (3.22)(3.22a) $${\widehat{\mathbf{v}}}_n^{\nu }\to \widehat{\mathbf{v}}\mathrm{ }in\mathrm{ }{\mathrm{L}}_w^2(0,T;V)\cap {\mathrm{L}}^2(0,T;H_{\mathit{loc}})\cap D([0,T];\mathcal{U}\prime )\cap D([0,T];H_w),$$(3.22a) , we obtain for all $\psi \in \tilde{H},$ for all $t\in [0,T],$ $\widehat{\mathbb{P}}$-a.s. (A.19) $${\int }_0^t{\int }_B|{(G(l,{\widehat{\mathit{\tau }}}_n^{\nu }(s))-G(l,{\mathit{\tau }}_{\ast }(s)),\psi )}_{\tilde{H}}{|}^2 {\lambda }_2(dl)ds\le C{\int }_0^T|{\widehat{\mathit{\tau }}}_n^{\nu }(s)-{\mathit{\tau }}_{\ast }(s){|}_{\tilde{H}}^2 ds\to 0\hspace{1em}\text{as}\hspace{1em}n\to \infty .$$(A.19) Moreover, for every $t\in [0,T],$ every $r\ge 1$ and every $n\in \mathbb{N},$ (A.20) $$\widehat{\mathbb{E}}[|{\int }_0^t{\int }_B|{(G(l,{\widehat{\mathit{\tau }}}_n^{\nu }(s))-G(l,{\mathit{\tau }}_{\ast }(s)),\psi )}_{\tilde{H}}{|}^2 {\lambda }_2(dl)ds{|}^r]\le C.$$(A.20) Since the restriction of ${\tilde{P}}_n$ to the space $\tilde{H}$ is the ${(·,·)}_{\tilde{H}}$-projection onto ${\tilde{H}}_n,$ then by (A.19)(A.19) $${\int }_0^t{\int }_B|{(G(l,{\widehat{\mathit{\tau }}}_n^{\nu }(s))-G(l,{\mathit{\tau }}_{\ast }(s)),\psi )}_{\tilde{H}}{|}^2 {\lambda }_2(dl)ds\le C{\int }_0^T|{\widehat{\mathit{\tau }}}_n^{\nu }(s)-{\mathit{\tau }}_{\ast }(s){|}_{\tilde{H}}^2 ds\to 0\hspace{1em}\text{as}\hspace{1em}n\to \infty .$$(A.19) , (A.20)(A.20) $$\widehat{\mathbb{E}}[|{\int }_0^t{\int }_B|{(G(l,{\widehat{\mathit{\tau }}}_n^{\nu }(s))-G(l,{\mathit{\tau }}_{\ast }(s)),\psi )}_{\tilde{H}}{|}^2 {\lambda }_2(dl)ds{|}^r]\le C.$$(A.20) and by Vitali’s Theorem, we obtain (A.21) $$\underset{n\to \infty }{\lim }\widehat{\mathbb{E}}[{\int }_0^t{\int }_B|{(G_n(l,{\widehat{\mathit{\tau }}}_n^{\nu }(s))-G(l,{\mathit{\tau }}_{\ast }(s)),\psi )}_{\tilde{H}}{|}^2 {\lambda }_2(dl)ds]=0,\hspace{1em}\psi \in \tilde{H}.$$(A.21) Since $\tilde{\mathcal{U}}\hookrightarrow \tilde{H},$ (A.21)(A.21) $$\underset{n\to \infty }{\lim }\widehat{\mathbb{E}}[{\int }_0^t{\int }_B|{(G_n(l,{\widehat{\mathit{\tau }}}_n^{\nu }(s))-G(l,{\mathit{\tau }}_{\ast }(s)),\psi )}_{\tilde{H}}{|}^2 {\lambda }_2(dl)ds]=0,\hspace{1em}\psi \in \tilde{H}.$$(A.21) holds for all $\psi \in \tilde{\mathcal{U}}.$

As ${\widehat{N}}_2^n=N_2^{\ast },$ for all $n\in \mathbb{N}.$ From (A.21)(A.21) $$\underset{n\to \infty }{\lim }\widehat{\mathbb{E}}[{\int }_0^t{\int }_B|{(G_n(l,{\widehat{\mathit{\tau }}}_n^{\nu }(s))-G(l,{\mathit{\tau }}_{\ast }(s)),\psi )}_{\tilde{H}}{|}^2 {\lambda }_2(dl)ds]=0,\hspace{1em}\psi \in \tilde{H}.$$(A.21) and the Itô isometry we have, (A.22) $$\underset{n\to \infty }{\lim }\widehat{\mathbb{E}}[|{\int }_0^t{\int }_B〈G_n(l,{\widehat{\mathit{\tau }}}_n^{\nu }(s))-G(l,{\mathit{\tau }}_{\ast }(s)),\psi 〉 {\tilde{N}}_2^{\ast }(ds,dl){|}^2]=0.$$(A.22) Moreover, from (A.20)(A.20) $$\widehat{\mathbb{E}}[|{\int }_0^t{\int }_B|{(G(l,{\widehat{\mathit{\tau }}}_n^{\nu }(s))-G(l,{\mathit{\tau }}_{\ast }(s)),\psi )}_{\tilde{H}}{|}^2 {\lambda }_2(dl)ds{|}^r]\le C.$$(A.20) and the Itô isometry, with $r=1,$ we obtain, (A.23) $$\widehat{\mathbb{E}}[|{\int }_0^t{\int }_B〈G_n(l,{\widehat{\mathit{\tau }}}_n^{\nu }(s))-G(l,{\mathit{\tau }}_{\ast }(s)),\psi 〉 {\tilde{N}}_2^{\ast }(ds,dl){|}^2]\le C.$$(A.23) Finally, from (A.22)(A.22) $$\underset{n\to \infty }{\lim }\widehat{\mathbb{E}}[|{\int }_0^t{\int }_B〈G_n(l,{\widehat{\mathit{\tau }}}_n^{\nu }(s))-G(l,{\mathit{\tau }}_{\ast }(s)),\psi 〉 {\tilde{N}}_2^{\ast }(ds,dl){|}^2]=0.$$(A.22) , (A.23)(A.23) $$\widehat{\mathbb{E}}[|{\int }_0^t{\int }_B〈G_n(l,{\widehat{\mathit{\tau }}}_n^{\nu }(s))-G(l,{\mathit{\tau }}_{\ast }(s)),\psi 〉 {\tilde{N}}_2^{\ast }(ds,dl){|}^2]\le C.$$(A.23) and using Dominated Convergence Theorem we obtain (vi).

(vii) Since, ${\widehat{\mathit{\tau }}}_n^{\nu }\to {\mathit{\tau }}_{\ast }$ in ${\mathrm{L}}^2(0,T;\tilde{V}),$ exploiting the Lipschitz property of b, we obtain, (A.24) $$\underset{n\to \infty }{\lim }{\int }_0^t〈b_n({\widehat{\mathit{\tau }}}_n^{\nu }(s))-b({\mathit{\tau }}_{\ast }(s)),\psi 〉 ds=\underset{n\to \infty }{\lim }{\int }_0^t〈b({\widehat{\mathit{\tau }}}_n^{\nu }(s))-b({\mathit{\tau }}_{\ast }(s)),{\tilde{P}}_n\psi 〉 ds=0$$(A.24) Now using Lemma A.1, for $r\ge 1$ we get, $\widehat{\mathbb{E}}[|{\int }_0^t〈b_n({\widehat{\mathit{\tau }}}_n^{\nu }(s)),\psi 〉 ds{|}^r]\le C_r,$ which further with (A.24)(A.24) $$\underset{n\to \infty }{\lim }{\int }_0^t〈b_n({\widehat{\mathit{\tau }}}_n^{\nu }(s))-b({\mathit{\tau }}_{\ast }(s)),\psi 〉 ds=\underset{n\to \infty }{\lim }{\int }_0^t〈b({\widehat{\mathit{\tau }}}_n^{\nu }(s))-b({\mathit{\tau }}_{\ast }(s)),{\tilde{P}}_n\psi 〉 ds=0$$(A.24) , and Vitali’s theorem we get, (A.25) $$\underset{n\to \infty }{\lim }\widehat{\mathbb{E}}[|{\int }_0^t〈b_n({\widehat{\mathit{\tau }}}_n^{\nu }(s))-b({\mathit{\tau }}_{\ast }(s)),\psi 〉 ds{|}^2]=0.$$(A.25) Finally from (A.25)(A.25) $$\underset{n\to \infty }{\lim }\widehat{\mathbb{E}}[|{\int }_0^t〈b_n({\widehat{\mathit{\tau }}}_n^{\nu }(s))-b({\mathit{\tau }}_{\ast }(s)),\psi 〉 ds{|}^2]=0.$$(A.25) and Dominated convergence theorem we get $$\underset{n\to \infty }{\lim }{\int }_0^T\widehat{\mathbb{E}}[|{\int }_0^t〈b_n({\widehat{\mathit{\tau }}}_n^{\nu }(s))-b({\mathit{\tau }}_{\ast }(s)),\psi 〉 ds{|}^2] dt=0.$$

A.2. Compactness and tightness criteria

Let $\mathbb{S}$ be a complete separable metric space with a metric ρ. Let us denote by $D([0,T];\mathbb{S}),$ the set of all $\mathbb{S}$-valued functions defined on $[0,T],$ which are right continuous and have left limits (càdlàg functions) for every $t\in [0,T].$ The space $D([0,T];\mathbb{S})$ is endowed with the Skorokhod J-topology. For more details see Métivier’s [39, Chapter II] and Billingsley’s [37, Chapter 3].

A.2.1. The Aldous condition

Let $(\mathbb{S},\rho )$ be a complete, separable metric space. Let $(\Omega ,\mathcal{F},\mathbb{F},\mathbb{P})$ be a probability space with usual hypotheses.

Definition A.1.

Let $\mathbf{u}\in \mathbb{D}([0,T];\mathbb{S})$ and let $\delta >0$ be given. A modulus of continuity is defined by (A.26) $${\mathcal{W}}_{[0,T],\mathbb{S}}(\mathbf{u};\delta ):=\underset{{\Pi }_{\delta }}{\inf }{\max }_{t_i\in \tilde{\omega }}\underset{t_i\le s<t<t_{i+1}\le T}{\sup }\rho (\mathbf{u}(t),\mathbf{u}(s)),$$(A.26) where ${\Pi }_{\delta }$ is the set of all increasing sequences $\tilde{\omega }=\{0=t_0<t_1<\cdots <t_n=T\}$ with the following property $$t_{i+1}-t_i\ge \delta ,\hspace{1em}i=0,1,\dots ,n-1.$$

Definition A.2.

Let $(X_n)$ be a sequence of $\mathbb{S}$-valued random variables. The sequence of laws of these processes is Tight if and only if

[T] $\forall \epsilon >0\forall \eta >0\exists \delta >0:$ $$\underset{n\in \mathbb{N}}{\sup }\mathbb{P}\{{\mathcal{W}}_{[0,T],\mathbb{S}}(X_n,\delta )>\eta \}\le \epsilon .$$

Definition A.3.

A sequence ${(X_n)}_{n\in \mathbb{N}}$ of $\mathbb{S}$-valued random variables satisfies the Aldous condtion if and only if

[A] $\forall ϵ>0\forall \eta >0\exists \delta >0$ such that for every sequence ${({\tau }_n)}_{n\in \mathbb{N}}$ of $\mathbb{F}$-stopping times with ${\tau }_n\le T$ one has $$\underset{n\in \mathbb{N}}{\sup }\mathrm{ }\underset{0\le \theta \le \delta }{\sup }\mathbb{P}\{\rho (X_n({\tau }_n+\theta ),X_n({\tau }_n))\ge \eta \}\le ϵ.$$

Lemma A.6.

Condition [A] implies condition [T].

Proof.

See Theorem 2.2.2 of [40]. □

Lemma A.7.

Let $(E,|·{|}_E)$ be a separable Banach space and let ${(X_n)}_{n\in \mathbb{N}}$ be a sequence of E-valued random variables. Assume that for every sequence ${({\tau }_n)}_{n\in \mathbb{N}}$ of $\mathbb{F}$-stopping times with ${\tau }_n\le T$ and for every $n\in \mathbb{N}$ and $\theta \ge 0$ the following condition holds (A.27) $$\mathbb{E}\left[\right|X_n({\tau }_n+\theta )-X_n({\tau }_n){|}_E^{\alpha }]\le C{\theta }^{\beta }$$(A.27) for some $\alpha ,\beta >0$ and some constant $C>0.$ Then the sequence ${(X_n)}_{n\in \mathbb{N}}$ satisfies the Aldous condtion in the space E.

Proof.

See [33] for the proof. □

Theorem A.8.

(Compactness criteria for $(\mathbf{v},\mathit{\tau })$) Let ${\mathcal{Z}}_1:=D([0,T];\mathcal{U}\prime )\cap D([0,T];H_w)\cap {\mathrm{L}}_w^2(0,T;V)\cap {\mathrm{L}}^2(0,T;H_{\mathit{loc}})$ and ${\mathcal{T}}_1$ be the supremum of the corresponding topologies. Let ${\mathcal{Z}}_2:=D([0,T];\tilde{\mathcal{U}}\prime )\cap D([0,T];{\tilde{H}}_w)\cap {\mathrm{L}}_w^2(0,T;\tilde{V})\cap {\mathrm{L}}^2(0,T;{\tilde{H}}_{\mathit{loc}})$ and ${\mathcal{T}}_2$ be the supremum of the corresponding topologies. Define $\mathcal{Z}:={\mathcal{Z}}_1\times {\mathcal{Z}}_2$ and $\mathcal{T}$ as the supremum of ${\mathcal{T}}_1$ and ${\mathcal{T}}_2.$ Then $({\mathcal{K}}_1,{\mathcal{K}}_2)\subset \mathcal{Z}$ is $\mathcal{T}-$relatively compact if the following three conditions are satisfied:

1. $\forall (\mathbf{v},\mathit{\tau })\in {\mathcal{K}}_1\times {\mathcal{K}}_2$ and all $t\in [0,T],(\mathbf{v}(t),\mathit{\tau }(t))\in H\times \tilde{H},$ and $$\underset{(\mathbf{v},\mathit{\tau })\in {\mathcal{K}}_1\times {\mathcal{K}}_2}{\sup }\left(\underset{s\in [0,T]}{\sup }\right|\mathbf{v}\left(s\right){|}_H+\underset{s\in [0,T]}{\sup }\left|\mathit{\tau }\right(s\left){|}_{\tilde{H}}\right)<\infty ,$$

2. $\underset{(\mathbf{v},\mathit{\tau })\in {\mathcal{K}}_1\times {\mathcal{K}}_2}{\sup }{\int }_0^T(|\mathbf{v}(s){|}_V^2+|\mathit{\tau }(s){|}_{\tilde{V}}^2)ds<\infty ,$ i.e. $({\mathcal{K}}_1,{\mathcal{K}}_2)$ is bounded in ${\mathrm{L}}^2(0,T;V)\times {\mathrm{L}}^2(0,T;\tilde{V}),$

3. $\underset{\delta \to 0}{\lim }\underset{(\mathbf{v},\mathit{\tau })\in {\mathcal{K}}_1\times {\mathcal{K}}_2}{\sup }(w_{[0,T],V\prime }(\mathbf{v},\delta )+w_{[0,T],\tilde{V}\prime }(\mathit{\tau },\delta ))=0.$

Proof of Theorem A.8 can be found in Lemma 3.3 in Brzeźniak and Motyl [23], Lemma 4.1 in Motyl [41], Theorem 2 of Motyl [33], Lemma 2.7 in Mikulevicius and Rozovskii [42].

Theorem A.9.

Let ${({\mathbf{v}}_n,{\mathit{\tau }}_n)}_{n\in \mathbb{N}}$ be a sequence of càdlàg $\mathbb{F}$-adapted $V\prime \times {\tilde{V}}^{\prime }$-valued processes such that

1. there exists a positive constant C₁ such that $$\underset{n\in \mathbb{N}}{\sup }\mathbb{E}\left[\underset{s\in [0,T]}{\sup }\right(\left|{\mathbf{v}}_n\right(s){|}_H+|{\mathit{\tau }}_n\left(s\right){|}_{\tilde{H}}\left)\right]\le C_1,$$

2. there exists a positive constant C₂ such that $$\underset{n\in \mathbb{N}}{\sup }\mathbb{E}\left[{\int }_0^T(\right|{\mathbf{v}}_n\left(s\right){|}_V^2+\left|{\mathit{\tau }}_n\right(s\left){|}_{\tilde{V}}^2\right)ds]\le C_2,$$

3. ${({\mathbf{v}}_n)}_{n\in \mathbb{N}}$ (respectively ${({\mathit{\tau }}_n)}_{n\in \mathbb{N}}$) satisfies the Aldous condition in $V\prime$ (respectively ${\tilde{V}}^{\prime }$).

Let $({\mathbb{P}}_n^1,{\mathbb{P}}_n^2)$ be the law of ${({\mathbf{v}}_n,{\mathit{\tau }}_n)}_{n\in \mathbb{N}}$ on $\mathcal{Z}:={\mathcal{Z}}_1\times {\mathcal{Z}}_2$. Then, for every $ϵ>0$, there exist compact subset $(K_{ϵ}^1,K_{ϵ}^2)$ of $\mathcal{Z}$ such that ${\mathbb{P}}_n^i(K_{ϵ}^1)\ge 1-ϵ,$ for i = 1, 2 and the sequence of measures $\{({\mathbb{P}}_n^1,{\mathbb{P}}_n^2),n\in \mathbb{N}\}$ is said to be tight on $(\mathcal{Z},\mathcal{T}).$

For a relevant proof see Corollary 1, Motyl [33].

Lemma A.10.

Let ${({\mathbf{v}}_n,{\mathit{\tau }}_n)}_{n\in \mathbb{N}}$ be a sequence of adapted ${\mathcal{U}}^{\prime }\times \widetilde{{\mathcal{U}}^{\prime }}$-valued processes satisfying the Aldous condition in ${\mathcal{U}}^{\prime }\times \widetilde{{\mathcal{U}}^{\prime }}$ and $$\underset{n\in \mathbb{N}}{\sup }\mathbb{E}\left[\left|\right|{\mathbf{v}}_n|{|}_{{\mathrm{L}}^{\infty }(0,T;H)}^2+|\left|{\mathit{\tau }}_n\right|{|}_{{\mathrm{L}}^{\infty }(0,T;\tilde{H})}^2\right]<\infty ,\hspace{1em}\underset{n\in \mathbb{N}}{\sup }\mathbb{E}\left[\left|\right|{\mathbf{v}}_n|{|}_{{\mathrm{L}}^2(0,T;V)}^2+|\left|{\mathit{\tau }}_n\right|{|}_{{\mathrm{L}}^2(0,T;\tilde{V})}^2\right]<\infty .$$ Then, there are a subsequence ${({\mathbf{v}}_{n_k},{\mathit{\tau }}_{n_k})}_{k\in \mathbb{N}}$ and random variables $({\widehat{\mathbf{v}}}_k,{\widehat{\mathit{\tau }}}_k)$ and $(\widehat{\mathbf{v}},\widehat{\mathit{\tau }})$ for $k\in \mathbb{N}$ on a common probability space $\widehat{\mathfrak{L}}$ with ${\widehat{\mathbb{P}}}^{{\widehat{\mathbf{v}}}_k}={\mathbb{P}}^{{\mathbf{v}}_{n_k}}$ and ${\widehat{\mathbb{P}}}^{{\widehat{\mathit{\tau }}}_k}={\mathbb{P}}^{{\mathit{\tau }}_{n_k}}$ for $k\in \mathbb{N},$ and $({\widehat{\mathbf{v}}}_k,{\widehat{\mathit{\tau }}}_k)\to (\widehat{\mathbf{v}},\widehat{\mathit{\tau }})$ $\widehat{\mathbb{P}}$-almost surely in $\mathcal{Z}$ for $k\to \infty .$