Asymptotic behaviour of mild solution of nonlinear stochastic partial functional equations driven by Rosenblatt process

. This paper presents conditions to assure existence, uniqueness and stability for impulsive neutral stochastic integrodifferential equations with delay driven by Rosenblatt process and Poisson jumps. The Banach ﬁxed point theorem and the theory of resolvent operator developed by Grimmer [R.C. Grimmer, Resolvent operators for integral equations in a Banach space, Trans. Am. Math. Soc. , 273(1):333–349, 1982] are used. An example illustrates the potential beneﬁts of these results.


Introduction
Stochastic differential equations (SDEs) arise in many areas of science and engineering, wherein, quite often the future state of such systems depends not only on present state, but also on its history leading to stochastic functional differential equations with delays rather than SDEs. However, many stochastic dynamical systems depend not only on present and past states, but also contain the derivatives with delays. Neutral stochastic differential equations with delays are often used to describe such systems (see, e.g., [19,30]). On the other hand, the stability of impulsive differential equations has been discussed by several authors (see, e.g., [2,10,26,32,33,38]).
Stochastic integrodifferential equations with delay are important for investigating several problems raised from natural phenomena. As far as applications are concerned, stochastic evolution equations have been motivated by such phenomena as wave propagation in random media [16] and turbulence [12]. Important motivations came also from biological sciences, in particular from population biology; see Dawson [27] and Fleming [7]. One has finally to mention early control theoretic application of Wang [8], Kushner [11], Bensoussan and Viot [36]. In addition, the study of neutral stochastic functional differential equations (SFDEs) driven by jumps process also have begin to gain attention and strong growth in recent years (see [15,21,31] and references therein).
In this work, we shall prove the existence, uniqueness and asymptotic behavior of mild solution for a class of impulsive stochastic neutral functional integrodifferential equation with delays driven by Rosenblatt process and Poisson jumps described in the form: ∆u(t k ) = x t + k − x t − k = I k u(t k ) , t = t k , k = 1, 2, . . . , where A, which is the infinitesimal generator of a strongly continuous semigroup (S(t)) t>0 on a Hilbert space H with domain D(A), (Θ(t)) t 0 , is a closed linear operator is a Borel set, and N will be defined later.
Let D F0 = D F0 ([−τ, 0], H) be the space of càdlàg F 0 -measurable functions almost surely bounded from [−τ, 0] × Ω into H, equipped with the supremum norm ζ D = sup −τ θ 0 ζ(θ) H , and ζ has finite second moment. Z H is a Rosenblatt process on a real and separable Hilbert space K 0 . r, ρ, k : [0, ∞) → [0, τ ] (τ > 0) are continuous functions, and G, F Here L 0 2 (K, H) denotes the space of all Q-Hilbert-Schmidt operators from K into H. Moreover, the fixed moments of time t k satisfy 0 < t 1 < t 2 < · · · < t k < · · · , and lim k→∞ t k = ∞, x(t − k ) and x(t + k ) represent the left and right limits of u(s) at time t k with I k (·) : H → H determining the size of the jump.
The analysis of (1) when B ≡ 0 was initiated in Ouahra et al. [28], where the authors proved the existence and stability of solutions by using a strict contraction principle. The main contribution is towards this direction by presenting conditions to assure existence, uniqueness and stability for such a class of system with the integrodifferential term. Our paper expands the usefulness of stochastic integrodifferential equations since the literature shows results for existence and exponential stability for such equations under semigroup theory.
The remaining of the paper is organized as follows. Section 2 presents notation and preliminary results. Section 3 shows the main results for existence, uniqueness of mild solutions for impulsive neutral stochastic integrodifferential equations driven by Rosenblatt process and Poisson processes and conditions for the exponential stability in mean square. Finally, Section 4 presents an example that illustrates our results.

Preliminaries
In this section, we provide some basic results about Poisson process, resolvent operator and Rosenblatt process.

Poisson jumps process
We denote by B σ (H) the Borel σ-algebra of H. Let (p(t)) t 0 be an H-valued, σ-finite stationary F t -adapted Poisson point process on (Ω, F, {F t } t 0 , P). The counting random measure N defined by for any Λ ∈ B σ (H − {0}); where 0 / ∈Λ (the closure of Λ) is called the Poisson random measure associated to the Poisson point process p. The following notation is used: Then it is known that there exists a σ-finite measure ν such that This measure ν is said the Lévy measure. Then the measure N is defined by This measure N (dt, dy) is called the compensated Poisson random measure, and dtν(Λ) is called the compensator (see [18]).

Deterministic integrodifferential equations
In this subsection, we recall some knowledge on partial integrodifferential equations and the related resolvent operators. Let H and Y be two Banach spaces such that For any y ∈ (Y, · Y ), the map t → Θ(t)y is bounded, differentiable, and the derivative dΘ(t)y/ dt is bounded and uniformly continuous on [0, +∞[.

By
Grimmer [13], under assumptions (H1) and (H2), the following Cauchy problem has an associated resolvent operator of bounded linear operator valued function R(t) ∈ L(H) for t 0.
Definition 2. (See [13].) A resolvent operator for Eq. (2) is a bounded linear operator valued function R(t) ∈ L(H), t 0, satisfying the following properties: (i) R(0) = I and R(t) M e kt for some constants M and k.
Hereafter, the resolvent operator for (2) is assumed to be continuous and exponentially stable.
(H3) The resolvent operator R(·) is both norm continuous and exponentially stable (the exponential stability means that there exist constants M > 0 and k 1 such that R(t) M e −κt for all t 0). http://www.journals.vu.lt/nonlinear-analysis

Rosenblatt process
Self-similar processes are invariant in distribution under suitable scaling. They are of considerable interest in practice since aspects of the self-similarity appear in different phenomena like telecommunications, turbulence, hydrology or economics (see, e.g., [14,17,24,37]). A self-similar processes can be defined as limits that appear in the so-called noncentral limit theorem (see [34]). We briefly recall the Rosenblatt process as well the Wiener integral with respect to it.
Let us recall the notion of Hermite rank. Denote by H j (x) the Hermite polynomial of degree j given by H j (x) = (−1) j e x 2 /2 (d j /dx j )e −x 2 /2 , and let g be a function on R such that E[g(ζ 0 )] = 0 and E[(g(ζ 0 )) 2 ] < ∞. Assume that g has the following expansion in Hermite polynomials Consider (ζ n ) n∈Z a stationary Gaussian sequence with mean zero and variance 1, which exhibits long range dependence in the sense that the correlation function satisfies with H ∈ (1/2, 1) and L is a slowly varying function at infinity. Since E[g(ζ 0 )] = 0, we have k 1. Then the following family of stochastic processes converges as n → ∞, in the sense of finite dimensional distributions, to the self-similar stochastic process with stationary increments where x + = max(x, 0). The above integral is a Wiener-Itô multiple integral of order k with respect to the standard Brownian motion (B(y)) y∈R , and the constant c(H, k) is a normalizing constant that ensures E(Z k H (1)) 2 = 1. The process (Z k H (t)) t 0 is called the Hermite process. When k = 1, the process given by (3) is nothing else that the fractional Brownian motion (fBm) with Hurst parameter H ∈ (1/2, 1). For k = 2, the process is not Gaussian. If k = 2, then process (3) is known as the Rosenblatt process [34]. The Rosenblatt process is of course the most studied process in the class of Hermite processes due to its significant importance in modelling. The Rosenblatt process is, after fBm, the most well know Hermite process.
We also recall the following properties of the Rosenblatt process: • The process Z k H is H-self-similar in the sense that for any c > 0, where d = means equivalence of all finite dimensional distributions. It has stationary increments and all moments are finite.
• From the stationarity of increments and the self-similar, it follows that, for any As a convergence, the Rosenblatt process has Hölder continuous paths of order γ with 0 < γ < H.
Self-similarity and long-range dependence make this process a useful driving noise in models arising in physics, telecommunication networks, finance and other fields.
Consider a time interval [0, T ] with arbitrary fixed horizon T and let {Z H (t), t ∈ [0, T ]} the on-dimensional Rosenblatt process with parameter H ∈ (1/2, 1). By Tudor [35], it is well known that Z H has the following integral representation: ) is a normalizing constant, and K H (t, s) is a Kernel given by , and β(·, ·) denotes the beta function. We put K H (t, s) = 0 if t s. The basic observation is the fact that the covariance structure of the Rosenblatt process is similar to the one of the Rosenblatt process, and this allows the use of the same classes of deterministic integrands as in the Rosenblatt process. By formula (4) we can write Let us denote by ξ the class of elementary functions on R of the form a j 1 (tj ,tj+1] (·), 0 t j < t j+1 T, a j ∈ R, j = 1, . . . , n.
For f ∈ ξ as above, it is natural to define its Wiener integral with respect to the Rosenblatt process Z H by Let K be the set of functions f such that It holds that and the mapping provides an isometry from ξ to L 2 (Ω). On the other hand, it has been proved in [29] that the set of elementary functions ξ is dense in K. As a consequence, mapping (5) can be extended to an isometry from K to L 2 (Ω). We call this extension as the Wiener integral of f ∈ K with respect to Z H . Let us consider the operator K * H from ξ to L 2 ([0, T ]) defined by where K(·, ·, ·) is the kernel of Rosenblatt process in representation (4) We refer to [35] for the proof of the fact that K * H is an isometry between K and Λ 2 ([0, T ]). It follows from [35] that K contains not only functions, but its elements could be also distributions. In order to obtain a space of functions contained in K, we consider the linear space |K| generated by the measurable functions ψ such that where α H = H(2H − 1). The space |K| is a Banach space with the norm ψ |K| , and we have the following inclusions (see [35]). Let H and K be two real, separable Hilbert spaces, and let L(K, H) be the space of bounded linear operator from K to H. For the sake of convenience, we shall use the same notation to denote the norms in H, K and L(K, H). Let Q ∈ L(K, H) be an operator defined by Qe n = λ n e n with finite trace trQ = ∞ n=1 λ n < ∞, where λ n 0 (n = 1, 2 . . . ) are nonnegative real numbers, and {e n } (n = 1, 2, . . . ) is a complete orthonormal basis of K. We define the infinite dimensional Q-Rosenblatt process on K as where (z n ) n 0 is a family of real independent Rosenblatt process. Note that series (6) is Note also that Z Q has covariance function in the sense that E Z Q (t), x Z Q (s), y = R(s, t) Q(x), y , x, y ∈ K, t, s ∈ [0, T ].
http://www.journals.vu.lt/nonlinear-analysis In order to define Wiener integrals with respect to the Q-Rosenblatt process, we introduce the space L 0 2 := L 0 2 (K, H) of all Q-Hilbert-Schmidt operators ψ : K → H. We recall that ψ ∈ L(K, H) is called a Q-Hilbert-Schmidt operator if and that the space L 0 2 equipped with the inner product ϕ, ψ L 0 2 = ∞ n=1 ϕe n , ψe n is a separable Hilbert space. Now, let φ(s), s ∈ [0, T ], be a function with values in L 0 Now, we end this subsection by stating the following result, which is useful to prove the main result.  Proof. Let {e n } (n = 1, 2, . . . ) be the complete orthonormal basis of K introduced above. Applying Hölder inequality, we have

Main results
In this part, conditions guaranteeing existence, uniqueness and exponential stability of mild solution to Eq. (1) are presented. The next definition introduces the concept of solution for the stochastic system (1).
that evolves on H is referred to as mild solution of (1) if it satisfies the next four conditions: In order to prove the main result in this section, we require the following assumptions: (H4) The functions F (t, ·), G(t, ·) and σ satisfy global Lipschitz conditions, that is, there exists K,K > 0 such that for any η, ζ ∈ H and t 0, The main result of this paper is given in the next theorem.
Theorem 1. Suppose that (H1)-(H7) hold and that Then, for all T > 0, Eq. (1) has a unique mild solution on [−τ, T ] and is exponential decay to zero in mean square, i.e., there exists a pair of positive constants a > 0 and Proof. In this proof, we let M ζ be the set all càdlàg processes u(t, ω) : [−τ, T ]×Ω → H satisfying the next two conditions: (i) u(t) = ζ(t), t ∈ [−τ, 0], and (ii) there exist some constants µ = µ(ζ, a) > 0 and a > 0 such that E u(t) 2 µe −at for all t 0. We point out that M ζ equipped with the norm · M ζ is a Banach space.
To clarify the next reasoning, take the operator Φ on M ζ as and for t ∈ [0, T ], Now, we prove that Φ has a fixed point in M ζ .
The remaining arguments for proving the above theorem are divided into the following three main steps.
Moreover, ifb is bounded and a C 1 function such that its derivativeb is bounded and uniformly continuous, then (H1) and (H2) are satisfied, and hence, by Theorem 2.1 in [9], Eq. (2) has a resolvent operator (r(t)) t 0 on H.