Almost sure asymptotic stability for some stochastic partial functional integrodifferential equations on Hilbert spaces

Abstract: In this work, we study the asymptotic behavior of the mild solutions of a class of stochastic partial functional integrodifferential equation on Hilbert spaces. Using the stochastic convolution developed, we establish the exponential stability in p mean square with p ≥ 2. Also, pathwise exponential stability is proved for p> 2. We extend the result of an example is provided for illustration.

ABOUT THE AUTHORS Moustapha Dieye received her PHD degree in Mathematics from Gaston Berger University (Saint-Louis, Sénégal) in 2018. Currently, he is a post-doc student in AIMS-GHANA . His research area includes controllability of deterministic and stochastic systems.
Mamadou Abdoul Diop received his BSc (Maths) degree from University of Gaston Berger, Sénégal in 1995 and MSc (Maths) from the same university in 1997 and PhD(Maths) from University of Provence, Marseille India in 2002. At present, he is working as a professor in the Department of Mathematics, Gaston Berger University. His research interests include nonlinear analysis, control theory and stochastic differential equations.
Khalil Ezzinbi received his PhD (Maths) from Cadi Ayyad University in Morroco. He is presently working as a research senior in the Department of Mathematics, University of Caddi Ayyad in Marrakech in Morroco. His research area includes functional and ordinary differential equations, partial functional differential equations,linear operators theory and evolution equation, infinite dynamical systems, applied mathematics controllability of deterministic and stochastic systems.

PUBLIC INTEREST STATEMENT
One of the important problems in many branches of science and industry, e.g. engineering, management, finance, social science, is the specification of the stochastic process governing the behavior of an underlying quantity. We here use the term underlying quantity to describe any interested object whose value is known at present but is liable to change in the future. Typical examples are the number of cancer cells, number of HIV infected individuals, share price in a company, price of gold, oil or electricity. All these stochastic processes are studied by stochastic differential equations. Therefore, in this paper, we discuss the stability of stochastic partial integrodifferential equations using stochastic convolution.

Introduction
Stochastic delay differential equations (SDDEs) play an important role in many branches of science and industry. Such models have been used with great success in a variety of application areas, including biology, epidemiology, mechanics, economics, and finance. In the past few decades, qualitative theory of SDDEs has been studied intensively by many scholars. Here, we refer to Da Prato and Zabczyk (1992a) and references therein. In recent years, existence, uniqueness, stability, and other quantitative and qualitative properties of solutions to stochastic partial differential equations have been extensively investigated by several authors.
The existence, uniqueness and asymptotic behavior of mild solutions of some stochastic partial differential equations on Hilbert spaces were considered by applying the comparison theorem in Govindan (2002Govindan ( , 2003. Taniguchi (1995) and Taniguchi, Liu, and Truman (2002) have studied the existence, uniqueness and asymptotic behavior of mild solutions of stochastic partial functional differential equation on Hilbert space by using the semigroup approach. Liu and Truman (2000) and Taniguchi (1998) have proved the almost sure exponential stability of mild solution for stochastic partial functional differential equation by using the analytic technique. Liu and Shi (2006), and Liu (2006) have considered the exponential stability for stochastic partial functional differential equations by means of the Razuminkhin-type theorem.
The stochastic integrodifferential equations are more general and still in a state of flux, with new basic results continuously emerging. Integrodifferential equations are important for investigating some problems raised from natural phenomena. They have applications in many areas such as physics, chemistry, economics, social sciences, finance, population dynamics, electrical engineering, medicine biology, ecology and other areas of science and engineering. Qualitative properties such as existence, uniqueness, optimality conditions, controllability and stability for various linear and nonlinear stochastic partial integrodifferential equations have been extensively studied by many researchers, see for instance (Balachandran & Sakthivel, 2001;Diagana, Hernàndez, & Dos Santos, 2009;Dieye, Diop, & Ezzinbi, 2016a, 2016b, 2017Diop, Ezzinbi, & Lo, 2012;Dos Santos, Guzzo, & Rabelo, 2010;Ezzinbi & Ghnimi, 2010;Sathya & Balachandran, 2012) and the references therein.
As the motivation of above-discussed works, we consider the following stochastic partial functional integrodifferential equation: In this work, our main aim is to study the exponential stability in p-th mean and also almost sure stability property of mild solutions for the system (1) by using the theory of resolvent operator as developed by Grimmer (1982) and the properties of stochastic convolution developed in Dieye, Diop, and Ezzinbi (2016c). The analysis of (1) when B;0 was initiated in Taniguchi (1995), where the authors proved the existence and stability of solutions by using a strict contraction principle. The main contribution of this paper is on finding conditions to assure the existence, uniqueness, and stability of impulsive neutral stochastic integrodifferential equations. Our paper expands the usefulness of stochastic integrodifferential equations since the literature shows results for existence and stability for such equations under semigroup theory.
The remaining of the paper is organized as follows. Section 2, presents notations and preliminary results. We study also the existence of the mild solutions of Equation (1). Section 3, shows the stability of the mild solutions. Finally, Section 4, presents an example that illustrates our results.

Stochastic processes and integrodifferential equations
Let X and Y be Banach spaces. LðX; YÞ denotes the space of bounded linear operator from X to Y, simply LðXÞ when X ¼ Y. We will assume that ðΩ; F ; ðF t Þ t!0 ; PÞ is a complete filtered probability space.
We are given a Q-Wiener process the probability space and having value in U a separable Hilbert space, one can construct wðtÞ as follows, wðtÞ :¼ ∑ þ1 n¼1 ffiffiffiffi ffi λ n p B n ðtÞe n t ! 0; where B n ðtÞðn ¼ 1; 2; 3; Á Á ÁÞ is a sequence of real-valued standard Brownian motions mutually independent of ðΩ; F ; ðF t Þ t!0 ; PÞ, λ n ! 0ðn ¼ 1; 2; 3; Á Á ÁÞ are positive real numbers such that ∑ þ1 n¼1 λ n < þ 1; ðe n Þ n!1 is a complete orthonormal basis in U, and Q 2 LðUÞ is the incremental covariance operator of the w which is a symmetric nonnegative trace class operator defined by Qe n ¼ λ n e n n ¼ 1; 2; 3; Á Á Á For this analysis, we recall the definition of H-valued stochastic integral with respect to the Uvalued Q-Wiener process w. Let L 0 2 ¼ L 2 ðU 0 ; HÞ denote the space of all Hilbert-Schmidt operator from U 0 ¼ Q 1=2 ðUÞ to H which is a separable Hilbert space, equipped with the following norm: Clearly, for any bounded operators φ 2 LðU; HÞ, this norm is given by Then, define the H-valued stochastic integral which is a continuous square integrable martingale. For more details on stochastic integrals, we refer to Da Prato and Zabczyk (1992a).
Next, we recall conditions that guarantee the existence of solution for the deterministic, integrodifferential equation (2) Definition 2.1. (Grimmer, 1982) A resolvent operator for Equation (2) is a bounded linear operatorvalued function RðtÞ 2 LðHÞ for t ! 0, having the following properties: (i) Rð0Þ ¼ I(the identity map of H) and k RðtÞk LðHÞ Ne ηt for some constants N > 0 and η 2 R: (ii) For each x 2 H, RðtÞx is strongly continuous for t ! 0.
(A2) For all t ! 0, ΥðtÞ is closed linear operator from DðAÞ to H and ΥðtÞ 2 LðY; HÞ. For any y 2 Y, the map t7 !ΥðtÞy is bounded, differentiable and the derivative t7 !Υ 0 ðtÞy is bounded uniformly The following theorem gives a satisfactory answer to the problem of existence of solutions.
In the following, we give some results for the existence of solutions for the following integrodifferential equation: ( 3) where q : R þ ! H is a continuous function.
Theorem 2.2. (Grimmer, 1982) Assume that ðA1Þ À ðA2Þ hold. If v is a strict solution of Equation (3), then Rðt À sÞqðsÞds for t ! 0: In order to set our problem, we make the following assumptions (A3) There exist γ > 0 and M ! 1 such that the resolvent operator ðRðtÞÞ t!0 of Equation (2) satisfies The exponential stability of the resolvent operator will play a crucial role to prove the main results of this work. It has been studied in Grimmer (1982).
(A4) The functions f and g are Lipschitz continuous. Let L 1 ; L 2 ; K > 0 be such that for every x; y 2 H and t ! 0 the following conditions are satisfied: k gðt; xÞ À gðt; yÞk 2 L 2 k x À yk H ; k f ðt; xÞ k 2 H þ k gðt; xÞ k 2 2 Kð1þ k x k 2 H Þ: 2.1. Existence of the mild solution of Equation (1) The next definition introduces the concept of solution for the stochastic system (1). (1) on ½0; T is a stochastic process x : ½0; T ! H defined on the probability space ðΩ; F ; PÞ such that x is F t -adapted predictable on ½0; T satisfies with probability one,

Definition 2.3. A mild solution of the integrodifferential Equation
Rðt À sÞg s; xðsÞ ð Þ dwðsÞ for t 2 ½0; T; Let I ¼ ½0; T and PðI; HÞ ; P denote the space of F t -adapted predictable random process with values in the Hilbert space H satisfying sup t2I E k xðtÞ k 2 H < 1: We define the following norm on P by Theorem 2.4. If hypotheses ðA1Þ; ðA2Þ and ðA4Þ hold, then for each initial datum x 0 F 0 -measurable X-valued square-integrable random variable, the integrodifferential Equation (1) has a unique mild solution on ½0; T.
Proof. Let T > 0. We define for x; y 2 P the following applications on P Â P by δ t ðx; yÞ :¼ sup Since sup t2I ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi E k xðtÞ k 2 H q ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sup t2I E k xðtÞ k 2 H q , then dðx; yÞ ¼ x À y j j, therefore ðP; dÞ becomes a complete metric space. Now, we define the following map: Rðt À sÞg s; xðsÞ ð Þ dwðsÞ for t 2 ½0; T: Note that a fixed point of G is a mild solution of Equation (1). Using the same arguments developed in Dieye et al. (2016c), we obtain that G applied P to itself. Moreover, we have the following estimation: E k ðGxÞðtÞ À ðGyÞðtÞk 2 CðtÞ ð t 0 δ s ðx; yÞds; where CðtÞ ¼ 2ðM T L 1 Þ 2 t þ 2ðML 2 Þ 2 with M T ¼ sup t2½0;T k RðtÞk LðHÞ . Taking the supremum over ½0; t, we get that δ t ðGx; GyÞ CðTÞ ð t 0 δ s ðx; yÞds: By iterative process involving repeated substitution of the expression (13) into itself, we obtain after n iterations the following inequality: δ t ðG n x; G n yÞ ðC n t n =n!Þδ t ðx; yÞ where C ¼ CðTÞ and G n denotes the n-fold composition of the operator G. Hence, by taking t ¼ T the above inequality gives that dðG n x; G n yÞ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðCTÞ n =n! q dðx; yÞ: The constants C and T being finite, then for n large enough: 0 < ðCTÞ n =n! < 1 and hence the n-th iterate G n of the operator G is a contraction on the metric space ðPðI; HÞ; dÞ. Since this is a complete metric space it follows from Banach fixed point Theorem that for n sufficiently large, G n has a unique fixed point, that is a unique fixed point also of G. We deduce that the mild solution exists on ½0; T, this is true for any T > 0 which means, we have a global existence.
3. Exponential stability of the mild solutions Equation (1) We are mainly interested in the stability properties of the mild solutions, we consider the following equation: instead of Equation (1).

Exponential stability in the -th mean
In the next, we discuss the exponential asymptotic stability in the p-th mean of mild solutions of Equation (1). From now on, let xðtÞ ¼ xðt; x 0 Þ denote the solution of Equation (16) with an initial value x 0 random variable and we always assume that x 0 is F 0 measurable with E k x 0 k p H < 1ðp ! 2Þ and x 0 is independent of wðtÞ.
Definition 3.1. Let p ! 2 be an integer. The mild solution xðt; x 0 Þ of Equation (1) is said to be globally exponentially asymptotically stable in the p-th mean if there exist ρ > 0 and L ! 1 such that, for any mild solution of Equation (1), yðt; y 0 Þ corresponding to an initial value y 0 with E k y 0 k p H < 1, the following inequality holds: Theorem 3.1. Let p ! 2 be an integer and let xðt; x 0 Þ and yðt; y 0 Þ be solutions of Equation (16) with initial values x 0 and y 0 respectively. Supposse that ðA1Þ À ðA4Þ hold true. Then, the following inequality holds: Proof. Let x and y be solutions of Equation (16) with initial values x 0 and y 0 respectively. Then, we have Now, we compute the terms on the right-hand side of the above inequality. From assumption (A3), we have From the assumptions (A3) (20), (21) and (23), one can see that the inequality (19) becomes Hence Gronwall's inequality yields that is, which completes the proof.
Consequently, we have the following result as corollary.
Corollary 3.3. Suppose that all hypotheses of Theorem 3.1 hold and let γ > α=p. Then mild solutions of Equation (1) are globally exponentially asymptotically stable in the p-th mean.

Almost sure asymptotic stability
In this subsection, we state the pathwise asymptotic stability for the mild solutions of equation (1). Due to the properties of the stochastic convolution, we study the case p > 2 . At first, we need the following lemma Theorem 3.6. Supposse that ðA1Þ À ðA4Þ hold. Let p > 2 be an integer, xðt; x 0 Þ and yðt; y 0 Þ be solutions of Equation (16) with initial values x 0 and y 0 respectively. If γ > α=p then there exists TðωÞ > 0 such that for t ! TðωÞ, we have k xðt; x 0 Þ À yðt; y 0 Þ k p H E k x 0 À y 0 k p H e ÀðpγÀαÞt=2 P a:s: Proof. Let n be a sufficiently large integer and I n denote the interval ½n; n þ 1. Then for t 2 I n , we have It follows that where m ¼ ð3MÞ p β þ ð3ML 1 Þ p ðβ=ðγ À αÞÞ þ ð3ML 2 Þ p ðβκ p =ðγ À αÞÞ: Now, for each integer n, we set n ¼ E k x 0 À y 0 k p H À Á 1=p e ÀðpγÀαÞðnþ1Þ=ð2pÞ . Then, it follows where δ ¼ e ðpγÀαÞ=2 .
By using inequality (31) and applying Borel-Cantelli's Lemma it follows that Pðlim supE n Þ ¼ 0: Then, the set of all ω such that there exists an infinite number index n with ω 2 E n is a negligible set. Hence, for almost sure, there exists infinite number index nðωÞ such that ω‚E nðωÞ i.e sup t2I nðωÞ k xðtÞ À yðtÞ k ðE k x 0 À y 0 k p Þ 1=p e ÀðγpÀαÞðnðωÞþ1Þ=ð2pÞ P a:s:: Let TðωÞ be the lower bound of index nðωÞ almost surely.

Application
Consider the following stochastic partial functional integrodifferential equation: Clearly f and g satisfy the assumption ðA4Þ with L 1 ¼ L k 1 and L 2 ¼ L k 2 where L k 1 and L k 2 are the Lipschitz constants of the functions k 1 and k 2 , respectively.
Moreover, if b is bounded and C 1 function such that b 0 is bounded and uniformly continuous, then ðA1Þ and ðA2Þ are satisfied, and hence, by Theorem 2.1, Equation (2) has a resolvent operator ðRðtÞÞ t!0 on H.
Proposition 4.1. (Dieye et al., 2016c) Suppose that b is bounded and C 1 function such that b 0 is bounded and uniformly continuous and bðtÞ 1 a e Àβt for all t ! 0 where β > a > 1. Then the resolvent operator of the abstract form of Equation (34) decays exponentially to zero. Specifically k RðtÞ k e Àγt where γ ¼ 1 À 1=a.
In the next, we assume that b is bounded and C 1 function such that b 0 is bounded and uniformly continuous and bðtÞ 1 a e Àβt for all t ! 0 where β > a>1.
Therefore, by Theorem 2.4 the existence and uniqueness of the mild solution of the stochastic partial functional integrodifferential Equation (33) is true.