Exponential stability for neutral stochastic functional partial differential equations driven by Brownian motion and fractional Brownian motion

In this paper, we study the exponential stability in the pth moment of mild solutions to neutral stochastic functional partial differential equations driven by Brownian motion and fractional Brownian motion: \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ d \bigl[x(t)+g(t,x_{t}) \bigr]= \bigl[Ax(t)+f(t,x_{t}) \bigr] \,dt+h(t,x_{t})\,dW(t)+\sigma(t)\,dB^{H}(t), $$\end{document}d[x(t)+g(t,xt)]=[Ax(t)+f(t,xt)]dt+h(t,xt)dW(t)+σ(t)dBH(t), where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$H\in(1/2,1)$\end{document}H∈(1/2,1). Our method for investigating the stability of solutions is based on the Banach fixed point theorem. The obtained results generalize and improve the results due to Boufoussi and Hajji (Stat. Probab. Lett. 82:1549–1558, 2012), Caraballo et al. (Nonlinear Anal. 74:3671–3684, 2011), and Luo (J. Math. Anal. Appl. 355:414–425, 2009).


Introduction
Many dynamical systems not only depend on present and past states but also involve derivatives with delays. Neutral stochastic functional partial differential equations (NSF-PDEs) are often used to describe such kind of systems. In recent years, NSFPDEs have been extensively studied in the literature, we can refer to [6,9,[12][13][14]19] for those only driven by Brownian motion and also refer to [1,2,4,5,11] for those only driven by fractional Brownian motion (fBm). For example, Luo [13] studied the exponential stability in mean square of mild solution for NSFPDE only driven by Brownian motion; Boufoussi and Hajji [2] discussed the exponential stability in mean square of mild solution for NSPDE only driven by fBm with finite delay. Furthermore, the stochastic processes in hydrodynamics, telecommunications, and finance demonstrate the availability of random noise that can be modeled by Brownian motion and also the so-called long memory that can be modeled with the help of fBm with Hurst index 1/2 < H < 1. Since the seminal paper [7], mixed stochastic models containing both standard Brownian motion and fBm have gained a lot of attention. Very recently, there has been considerable interest in studying this class of SDEs (see [3,10,16,17,20,21]).
However, to the best of our knowledge, there is no paper which investigates the exponential stability in the pth moment of mild solutions to neutral stochastic functional partial differential equations driven by Brownian motion and fractional Brownian motion. Motivated by the above, in this work, we consider the following mixed NSFPDE: under suitable conditions on the operator A, the coefficient functions g, f , h, σ , and the initial value ϕ. Here W (t) denotes Brownian motion and B H (t) denotes fBm with H ∈ (1/2, 1).
The purpose of this paper is to investigate the exponential stability in the pth moment of mild solution of mixed NSFPDE (1.1) by means of the Banach fixed point theory.
The rest of this paper is organized as follows. In Sect. 2, we first recall some necessary preliminaries on the stochastic differential equations with respect to Brownian motion and fractional Brownian motion. In Sect. 3, the exponential stability in the pth moment of mild solution of mixed NSFPDE (1.1) is proved, the results in [2,5,13] are generalized and improved.

Preliminaries
Let T > 0 be a fixed time horizon and ( , F, P) be a complete probability space equipped with a normal filtration F = {F t } t≥0 satisfying the usual assumptions. Let W = {W (t), t ∈ [0, T]} be a standard Brownian motion and B = {B H (t), t ∈ [0, T]} be a fractional Brownian motion with Hurst parameter H ∈ (1/2, 1) on the complete probability space ( , F, P). We denote by C([-r, T]; U) the space of all continuous functions from [-r, T] to U. Let (U, · U , (·, ·) U ) and (K i , · K i , (·, ·) K i ) be two separable Hilbert spaces, and let L(K i , U) denote the space of all bounded linear operators from K i to U, i = W , B. We assume that {e (i) n } n∈N + are two complete orthonormal bases in K i and Q (i) ∈ L 0 i (K i , U) are two operators defined by Q (i) e (i) n = λ (i) n e (i) n with finite trace tr Q (i) = ∞ n=1 λ (i) n < ∞, where {λ (i) n } n∈N + are nonnegative real numbers and i = W , B. Then there exists a real-valued sequence {ω n (t)} n∈N + of one-dimensional standard Brownian motions mutually independent over ( , F, P) such that The infinite dimensional cylindrical K B -valued fBm B H (t) is defined by the formal sum where the sequence {w H n (t)} n∈N + are stochastically independent scalar fBms with Hurst parameter H ∈ (1/2, 1). Let L 0 i (K i , U) be the space of all Q (i) -Hilbert-Schmidt operators from K i to U, i = W , B. Now we can show the following two definitions of norms. Definition 2.1 (Chen et al. [6]) Let ξ ∈ L(K W , U) and define n is a separable Hilbert space.

Definition 2.2 (Boufoussi and Hajji [2]) In order to define Wiener integrals with respect to the
and that the space n is a separable Hilbert space.

Lemma 2.1 (Prato and Zabczyk [8]) For any p ≥ 0 and for arbitrary L
Let {w H (t)} t∈[0,T] be the one-dimensional fBm with Hurst parameter H ∈ (1/2, 1). This means by definition that w H is a centered Gaussian process with covariance function: Moreover, w H has the following Wiener integral representation: is a Wiener process and K H (t, s) is the kernel given by (2.3)

Lemma 2.4 (Pazy [18]) Suppose that A is the infinitesimal generator of an analytic semigroup of uniformly bounded linear operators {S(t)} t≥0 on the separable Hilbert space U.
It is well known that there exist some constants We consider the exponential stability of mild solution to the following mixed NSFPDE: t ∈ [0, T], (iii) x(t) ∈ U has càdlàg paths on t ∈ [0, T] almost surely, and for arbitrary t ∈ [0, T],

Definition 2.4
Let p be an integer p ≥ 2. Equation (2.5) is said to be exponentially stable in the pth moment if, for any initial value ϕ, there exists a pair of constants γ > 0 and C > 0 such that In order to set the stability problem, we suppose that the following assumptions hold: where β ∈ (0, 1] and satisfies pβ > 1, p is an integer p ≥ 2. We further assume g(t, 0) ≡ 0 for t ≥ 0.

Main results
In this section, we consider the exponential stability in the pth moment of mild solution of mixed NSFPDE (2.4) by means of the Banach fixed point theory. Define an operator π : S − → S by (πx)(t) = ψ(t) for t ∈ [-r, 0] and for t ≥ 0,

Theorem 1 Suppose that conditions (H1)-(H4) hold. Then Eq. (2.4) is exponentially stable in the pth moment if
Firstly, we verify the continuity in the pth moment of π on [0, ∞). Let x ∈ S, t 1 ≥ 0, and r be positive and small enough, then Obviously, Since the operator (-A) -β is bounded and by (H3) we know the mapping (-A) β g is continuous in the pth moment, so E I 2 (t 1 + r) -I 2 (t 1 ) p U − → 0, as r − → 0.
As for the third term on the right-hand side of (3.2), we get By using Lemma 2.4 and the fact that 0 < β ≤ 1, we have since β ∈ (0, 1] and by the Lebesgue dominated theorem, we obtain lim r→0 I 31 (r) = 0.
Secondly, we show that π(S) ⊂ S. It follows from (3.2) that Now we estimate the terms on the right-hand side of (3.3). First, by condition (2.7), we can obtain For any x(t) ∈ S and any ε 1 > 0, there exists t 1 > 0 such that e αt E x(t) p U < ε 1 for tr > t 1 . Thus we can get -r≤s≤0 x(t + s) p U ≤ 6 p-1 (-A) -β p U C p g e αt e -α(t+s) ε 1 ≤ 6 p-1 (-A) -β p U C p g e αs ε 1 .
Remark 3.3 When σ ≡ 0, p = 2, then Eq. (2.4) reduces to a NSFPDE only driven by Brownian motion in which the exponential stability in mean square of mild solution has been studied by Luo [13]. Obviously, the given result in [13] can be seen as a special case of our result. In this sense, we generalized the result given in [13].

Funding
This research is supported by the National Natural Science Project of China (Grant No. 17BQNS01004).