Retarded Neutral Stochastic Equations Driven by Multiplicative Fractional Brownian Motion

Abstract

In this article, we develop an existence and uniqueness theory of pathwise mild solutions for a class of stochastic neutral partial functional differential equations that are driven by an infinite-dimensional multiplicative fractional noise. Our existence and uniqueness result is divided into two parts: first we establish the existence and uniqueness of a local solution, which proof rests upon the Banach fixed point theorem; then the existence of a unique global solution is based on certain regularity properties of the local solution as well as a suitable way of extending the local solution to the entire interval in finitely many steps.

Keywords:

• Hilbert-valued fractional Brownian motion
• Pathwise solutions
• Neutral equations

Mathematics Subject Classification:

• Primary: 35R60
• Secondary: 60H15
• 60G22

Appendix

In this section, we collect here the proofs of all the Lemmatas of Section 3 as well as the proof of the regularity of the stochastic integral H_u. We begin with the proof of Lemma 3.1.

Proof.

Performing the change of variable (t − s)x = r − s, the integral in the expression of c(ρ) can be expressed in the following way (A.1)

and taking supremum on the interval [0, T] we obtain that the asymptotic behavior of c(ρ) is just lim_{ρ → ∞}c(ρ) = 0, since this is the asymptotic behavior of the supremum of the right hand side of (A.1(A.1) ), see [17] for a detailed proof of this last statement.

We shall use very often estimates similar to the following ones:

for 0 ⩽ s < r < t ⩽ T.

We continue with the proof of Lemma 3.2:

Proof.

First, due to the Hölder regularity of the fBm and the definition of the fractional derivative, for 0 ⩽ s ⩽ r ⩽ t ⩽ T, it is easy to derive that

and, therefore, together with (2.6(2.6) ) and (2.7(2.7) ) we obtain

and in particular the first statement is true. In the above inequality, we have used (Hh) as follows

To prove the second statement, note that

thus, it suffices to estimate the last term on the right hand side. Regarding the integrand of that term, we realize that the only difference with respect to the expression of |H(s, t)| is that we should replace the estimates for |S(t − r)|_L(V) and |S(t − r) − S(t − q)|_L(V) by the corresponding estimates for |S(t − r) − S(s − r)|_L(V) and |S(t − r) − S(t − q) − (S(s − r) − S(s − q))|_L(V) respectively, for which we use (2.7(2.7) ) and (2.8(2.7) ) for appropriate parameters. In particular, taking 0 < α < α′ < 1 such that α′ + β < α + β′ we obtain

and, thus, the second statement is proven.

Regarding the nonlinearity h, we have thus far only used the fact that it is Lipschitz continuous. However, in what follows we elucidate the role of the rest of assumptions in hypothesis (Hh): let us take u, v ∈ C^β([ − μ, T]; V). Thanks to the assumption (Hh) we have the following splitting

see Lemma 7.1 in [18], and, therefore,

Then we arrive at

The last estimate of this result follows from the previous one and the corresponding to the expression

that can be derived by using the same kind of estimates as above, and it is left to the reader.

We finish giving the proof of Lemma 3.3:

Proof.

We begin proving the first estimate. On account of (Hg) and the continuous embedding of V_γ into V we get

where the last inequality follows from the fact that γ > β since (A.2)

Next we show the second estimate of this lemma. Note that

and then we only have to estimate the last term on the right-hand side of the above inequality. Thanks to the properties of the semigroup and the fact that γ > β,

Now let us take u, v ∈ C^β([ − μ, T]; V). Then, applying again [18, Lemma 7.1] and (A.2(A.2) ),

Finally, because of the splitting

the last estimate of this lemma follows from the previous considerations and (Hg).

We finish this section by proving that H_u(0, t) ∈ V_β, for t > 0, being u the local solution to (2.5(2.5) ).

By using (2.6(2.6) ) and (2.7(2.7) ), and taking α′ > α such that β′ + α > β + α′, we get

The three above integrals are finite since if a, b > −1,

and the integral on the right hand side of the previous expression is equal to the Beta function B(1 + b, 1 + a).