Ensemble Averaging for Dynamical Systems Under Fast Oscillating Random Boundary Conditions

Abstract

This article is devoted to providing a theoretical underpinning for ensemble forecasting with rapid fluctuations in body forcing and in boundary conditions. Ensemble averaging principles are proved under suitable “mixing” conditions on random boundary conditions and on random body forcing. The ensemble averaged model is a nonlinear stochastic partial differential equation, with the deviation process (i.e., the approximation error process) quantified as the solution of a linear stochastic partial differential equation.

Keywords:

• Multiscale modeling
• Ensemble averaging
• Random partial differential equations
• Stochastic partial differential equations
• Random boundary conditions
• Martingale.

Mathematics Subject Classifications (2010):

• 60H15
• 60F10
• 60G17.

Appendix

Proof of Lemma 4

A similar result has been given in [11, Proposition 7]. Here we present another proof which gives a stronger convergence, together with the convergence rate in probability.

First, under the assumption (H_g), we show that for almost all ω ∈ Ω, (A.1) for every q ∈ H.

Noticing we get Consider By a mixing property [4, Proposition 7.2.2 ], we have (A.2) Noticing that by estimate (43(A.2) ) we have (42(A.1) ).

By the estimate in Section 2.1, for every κ > 0, there is C^κ_T > 0, which is independent of ε, such that (A.3) for every t ⩾ s ⩾ 0. Furthermore, by the tightness of the distributions of {u^ε} in space C(0, T; H), for every κ > 0, there is a compact set K_κ⊂C(0, T; H) such that (A.4) So we define Due to the compactness of K_κ, for every ϵ > 0, we only need to consider a finite ϵ-net {q₁ , q₂ , … , q_N} in C(0, T; H), which covers {u^ε}. Without loss of generality, we assume that q_j, j = 1, 2, …, N, are simple functions [11].

Now we consider all ω ∈ Ω_κ. By the construction of and boundedness of f, we have for ω ∈ Ω_κ for some constant C > 0.

For every δ > 0, we partition the interval [0, T] into subintervals of length of δ. Then for t ∈ [kδ, (k + 1)δ), , for some q_j. Notice that, by the assumption (H_g) and the definition of , is also Lipschitz continuous in u with the same Lipschitz constant L_g. Then by the assumption (H_g) and the definition of Ω_κ, Due to the arbitrary choice of δ, ϵ, and κ, and notice a similar discussion as that in [11], we thus complete the proof.