ASYMPTOTIC BEHAVIOR OF STOCHASTIC FUNCTIONAL DIFFERENTIAL EVOLUTION EQUATION

. In this work we study the long time behavior of nonlinear stochastic functional-diﬀerential equations in Hilbert spaces. In particular, we start with establishing the existence and uniqueness of mild solutions. We proceed with deriving a priory uniform in time bounds for the solutions in the appropriate Hilbert spaces. These bounds enable us to establish the existence of invariant measure based on Krylov-Bogoliubov theorem on the tightness of the family of measures. Finally, under certain assumptions on nonlinearities, we establish the uniqueness of invariant measures.


1.2)
Here A is the elliptic operator the interval [−h, 0] is the interval of delay, and u t = u(t + θ) with θ ∈ [−h, 0]. Functional differential equations of types (1.1) and (1.2) are mathematical models of processes, the evolution of which depends on the previous states. One of the natural examples of such behavior is heat conduction. In particular, the classic model of heat conduction u t = ∆u has an essential shortcoming: it predicts infinite speed of propagation of thermal fluctuations in Fourier heat conductors. This observation suggests that the Fourier's law of heat conduction may be an approximation to a more general constitutive assumption relating the heat conduction to the material's thermal history. Gurtin and Pipkin [14] have proposed a memory theory of heat conduction, which has finite heat propagation speeds, and, in its linearized version reads aṡ u(x, t) + ∞ 0 β(s)u(x, t − s) ds = C∆u(x, t), which is a particular example of a functional-differential equation. Furthermore, [25] provides an example of temperature regularization through heat injection or extraction, controlled by a thermostat, which creates additional memory and delay effects. A closely related problem arises emerges in modeling partially diffused population dynamics with delay in the birth process [25] u(x, t) − Cu xx (t, x) = u(t, x) 1 − u(t, x) − where r(s) is a continuous delay function. In [27] we used a functional-differential equation to take into account the delay effects in modeling Performance-on-Demand Micro-electromechanical systems (POD MEMS). Similar memory effects emerge in Hodgkin-Huxley model, Dawson-Fleming model of population genetics [11], among others.
The classic results for deterministic functional-differential equations in finite dimensional spaces can be found in [13] and references therein. Stochastic functional differential equation in finite dimensions have be studies extensively as well. In particular, the existence of invariant measures for stochastic ordinary differential equations was established in [3,12]. The work [15] addressed the stochastic stability, as well as various applications of stochastic delay equations in finite dimensions.
The results on functional differential equations in infinite dimensions are significantly more sparse. One example of analysis and applications of functional partial differential equations may be found in [1]. In this work, the authors study the nonlocal reaction-diffusion model of population dynamics. They establish the existence of time stationary solution and show that all other solutions converge to it.
The results on stochastic functional differential equations include [29,8], which establish the existence of solutions and their stability. Stochastic differential equation of neutral type were studied in [26,16,31]. The work [28] established the comparison principle for such equations.
The main goal of the present work is to establish the existence and uniqueness of invariant measures for the equations (1.1) and (1.2) based on Krylov-Bogoliubov theorem on the tightness of the family of measures [17]. More precisely, we will use the compactness approach of Da Parto and Zabczyk [9], which involves the following key steps: (i) Establishing the existence of a Markovian solution of (1.1) or (1.2) in a certain functional space, in which the corresponding transition semigroup is Feller; (ii) Showing that the semigroup S(t) generated by A is compact; (iii) Showing that the corresponding equation with a suitable initial condition has a solution, which is bounded in probability. This approach was used in establishing the existence of invariant measure for a large class of stochastic nonlinear partial differential equations without delay, e.g. [2,5,7,10,21,22] and references therein. For functional differential equations in finite dimensions, the approach above was used in [6]. In this work, the author established the existence of an invariant measure in R d ×L 2 (−h, 0; R d ). In contrast, for stochastic partial differential equations, the natural phase space for the mild solutions of (1.
, which is a significantly easier problem [26,28,29]. In these spaces the authors studied the conditions for the existence and uniqueness of the solution, as well as their Markov's and Feller properties. However, in order to apply the compactness approach one needs to work in L 2 , which is done in this work. We also establish the existence and uniqueness of the stationary solution, and the convergence of other solutions to it in square mean, which is the stochastic analog of the main result of [1].
This article is structured as follows. In Section 2 we introduce the notation and formulate the main results. Section 3 is devoted to the proof of the existence of invariant measure, as well as an example of application of this result to integraldifferential equations. Section 4 establishes the uniqueness of invariant measure, and the convergence to the stationary solution.

Preliminaries and main results
Throughout this article, the domain D is either a bounded domain with ∂D satisfying the Lyapunov condition, or D = R d . Denote where r > d if D = R d and r = 0 (i.e. no weight) for bounded D. We introduce the following spaces: The coefficients a ij of the operator A defined in (1.3) are Holder continuous with the exponent β ∈ (0, 1), symmetric, bounded and satisfying the elipticity condition If D = R d , then D(A) = H 2 (R d ). Denote G(t, x, y) to be the fundamental solution (or the Green's function in the case of bounded D) for ∂ ∂t − A. It follows from, e.g., [18, p. 468], that there are positive constants C 1 (T ), C 2 (T ) > 0 such that for t ∈ [0, T ] and x, y ∈ D. Note that in (2.3), C 1 and C 2 depend not only on T , but on the constants C 0 , d, T , maximum values of the coefficients of A, and the Holder constants. If the operator is in the divergence form Au = div(a∇u), the estimates are of a different type, see e.g. [17], namely In this case, in contrast with (2.3), the constant K(C 0 , d) is independent of t. Proof. Note that the weight (2.1) satisfies We define and S(0) = I, where I is the identity map. This is a semigroup on L 2 (D) with generator A. Then for all ϕ ∈ L 2 (D) and for t ∈ [0, T ] by Lemma 2.1 we have (2.8) The above estimate allows the semigroup S(t) to be extended to a linear map from B ρ 0 to itself. Since L 2 (D) is dense in B ρ 0 , S(t) is strongly continuous in B ρ 0 . Let a i ≥ 0, ∞ i=1 a i < ∞, and e n be orthonormal basis in H, such that e n ∈ L ∞ (D) and sup n e n L ∞ (D) < ∞. We introduce the operator Q ∈ L(H) such that Q is non-negative, T r(Q) < ∞, Qe n = a n e n . Let (Ω, F, P ) be a complete probability space. We introduce which is a Q-Wiener process on t ≥ 0 with values in L 2 (Q). Here β i (t) are standard, one dimensional, mutually independent Wiener processes. Also let {F t , t ≥ 0} be a normal filtration satisfying . Following [19] introduce the multiplication operator Φ : U → B ρ 0 as follows: for a fixed ϕ ∈ B ρ 0 , let Φ(ψ) := ϕψ, ψ ∈ U . Since ϕ ∈ B ρ 0 and ϕ ∈ L ∞ (D), the operator is well defined and hence Φ • Q 1/2 : L 2 (D) → B ρ 0 defines a Hilbert-Schmidt operator. The operator Φ is also a Hilbert-Schmidt operator satisfying following [9] we can define Furthermore, We assume f and σ satisfy the following conditions: (i) The functionals f and σ map B ρ 1 to B ρ 0 , (ii) There exists a constant L > 0 such that . Hence the phase space of the problem is the Hilbert space B ρ . In this case 3 (Existence and uniqueness). Suppose f and σ satisfy the conditions (i) and (ii), and ϕ(t, ·) is an F 0 measurable random process for t ∈ [−h, 0], which is independent of W and such that Then there exists a unique mild solution of (1.1) (or 1.2) on [0, T ], and respectively. Then under the conditions of Theorem 2.3 there exists a constant C(T ) such that sup 13) The following proposition shows the that the solution u(t, ·) has continuous trajectories.
Proposition 2.5. Let u(t, ·) be a mild solution of (1.1) or (1.2). Then, under the conditions of Theorem 2.3, u t is continuous at t = 0 in probability with respect to the norm · B ρ 1 , i.e.
Proof. Note that The convergence of the first term to 0 follows from the density of ). The second term converges to zero as t → 0 since the integrand is bounded.
Let B b (B ρ ) be the Banach space of bounded real Borel functions from B ρ to R, and C b (B ρ ) be the space of bounded continuous functions. Since the choice of T > 0 in Theorem 2.3 is arbitrary, the solution exists for all t ≥ 0, thus y(t) also exists for all t ≥ 0. Replacing the initial interval [−h, 0] with [−h + s, s] for all s ≥ 0, we can guarantee the existence and uniqueness of the solutions for t ≥ s ≥ 0 with the initial F s -measurable functions ϕ(θ, ·), ϕ(0, ·), which satisfy the conditions of Theorem 2.3 on [s − h, s]. This solution will be denoted with u(t, s, ϕ). Similarly, is a shift of the solution u(t, ϕ), such that u s (s, ϕ) = u(s + θ, s, ϕ) = ϕ(θ) and for θ = 0, ϕ(0, ·) ∈ B ρ 0 . Following [4], we define the family of shift operators For any nonrandom ϕ ∈ B ρ with s ≥ 0 and t ≥ s, U t s ϕ : Defining y(t, s, ϕ) = (u(s, t, ϕ), u t (s, ϕ)), we have that y maps B ρ into itself. The next proposition follows from Theorem 2.3.
Proposition 2.6. The family of the operators (2.14) satisfies Let D be a σ-algebra of Borel subset of B ρ . Then y(t, s, ϕ) naturally denotes the following probability measure µ t on D, The measure µ is the transition function corresponding to the random process y(t, s, ϕ). In a similar way as in the finite dimensional case [4], one can show that this function satisfies the properties of the transition probability. This way we have Theorem 2.7 (Markov property). Under the assumptions of Theorem 2.3, the process y(t, s, ϕ) ∈ B ρ is the Markov process on B ρ with the transition function P (s, ϕ, t, A) given by (2.16).
From proposition 2.8 we have P 0,t−s (ϕ) and denote P t ϕ = P 0,t (ϕ). From Theorem 2.4 and Proposition 2.5 we have the following result.
Proposition 2.9. Under the assumptions of Theorem 2.3 the transition semigroup P t , t ≥ 0 is stochastically continuous an satisfies the Feller property We defineρ(x) = (1 + |x|r) −1 . The main result of the paper is the following theorem. Remark 2.11. Condition (2.18) is equivalent to The key condition in Theorem 2.10 is the existence of a globally bounded solution. The next theorem provides the sufficient conditions for the existence of such solution in terms of the coefficients, in the case when A is in the divergence form.
which is a sufficient condition for the boundedness in probability.
Finally, for a bounded domain D we establish the uniqueness of the stationary solution as well as its stability. In this section, the weight ρ ≡ 1, thus The semigroup (2.7) now satisfies the exponential estimate In a standard way, we can extend the Q-Weiner process W (t) to t ∈ R as Here V is another Q-Weiner process, independent of W .
Theorem 2.14. Assume the Lipschitz constant L is sufficiently small (see (4.2) for the exact condition), then equation (1.1) has a unique solution u * (t, x), defined for t ∈ R, and sup t∈R E u * (t) 2 B < ∞.
Furthermore, this solution is exponentially attractive, that is exist K, γ > 0 such that for all t 0 ∈ R and t > t 0 + h, and for any other solution η(t) with η(t 0 ) ∈ B 0 and η t0 ∈ B 1 we have

Proofs of main results
Proof of Theorem 2.3. Let B p,T , p ≥ 2 be the space of F t -measurable for t ∈ [0, T ] processes, equipped with the norm Φ p B p,T := E  This way, It follows from (2.8) that Next, using the conditions (i) and (ii) for f , we have To estimate I 3 , we use [9, Lemma 7.2] and (2.10). Using the definition of Hilbert-Schmidt norm given in (2.9), we have the same way as in (3.2). Combining these estimates, we have Ψ : B p,T → B p,T . We next show that Ψ is contractive. For any Φ,Φ ∈ B p,t we have Now from estimate (3.3), we have (3.6) Consequently, fort small enough, (3.5) and (3.6) imply that the map Ψ has a unique fixed point in B p,t , which is the solution of (2.11 (3.7) We consider two separate cases t ∈ [0, h] and t ∈ [h, T ]: If t ∈ [0, h], then Estimating the last term separately, we have ].
Combining the estimates above, we obtain Proof of Theorem 2.4. By definition of y 1 and y 2 , (3.10) The first term in (3.10) can be estimated as follows As for the second term in (3.10), once again we consider separately the cases t ∈ which completes the proof.
For the proof of Theorem 2.10, we need the following auxiliary lemmas. Proof. By [23], there exists an orthonormal basis {h n , n ≥ 1} in B ρ 0 such that sup n h n L ∞ (D) < ∞. It is straightforward to verify that if {e n , n ≥ 1} is an orthonormal basis in H = L 2 (D), then { en ρ 1/2 , n ≥ 1} is an orthonormal basis in Bρ 0 . Therefore which completes the proof. (3.13) The arguments in [9,Theorem 11.29] can be applied to (3.12) directly.
Lemma 3.3. For p > 2 and α ≥ 1 p , the operator We will use the infinite dimensional version of Arzela-Ascoli Theorem. To this, we need to show Using Corollary 3.2, S(ε) is a compact operator from Bρ 0 to B ρ 0 . Then, following [9, p.227], G ε α converges to G α strongly as ε → 0, hence G α is compact and (i) follows.
The rest of the proof of Theorem 2.10 follows the lines of the proof of [9,Theorem 11.29].
Proof of Theorem 2.12. This proof a lot in common with the proof of [20,Theorem 1]. Let us point out the differences caused by the presence of the delay. We have By definition of a mild solution (2.11), we have where It follows from (2.4) that for all t ≥ 0, which completes the proof. Example 3.6. Letf : R → R andσ : R → R be Lipschitz functions with Lipschitz constants L. We define Then for all ϕ 1 , ϕ 2 ∈ B ρ 1 we have Thus f and σ are examples of Lipschits maps from B ρ 1 to B ρ 0 , for which the theorems above apply.

Uniqueness of the invariant measure
Proof of Theorem 2.14. Let B be the class of F t measurable B 0 -valued processes ξ(t), such that sup we follow the procedure in [20] and define the successive approximations u (0) ≡ 0 and Then sup Thus by Theorem 5 [20], equation (4.1) has the unique solution u (n+1) (t) such that and therefore, sup Hence for in a similar way to [20] we can argue that the sequence is in fact Cauchy, and there exists a unique u * (t) such that sup t∈R E u * (t) B < ∞ and sup t∈R E u n (t) − u * (t) 2 B → 0, n → ∞.
Furthermore, we can argue that u * satisfies Consider any other solution (4.3) such that η(t 0 ) is F t0 -measurable, and E η(t 0 ) 2 B < ∞. Here η t0 = ϕ(θ, x) is defined on [−h, 0]. Let us show that the solution η converges to u * exponentially. Since we are interested in the behavior of the solutions for large t, suppose t > t 0 + h. Then t + θ > t 0 and η(t) is defined via the formula (4.3). Hence ds.

Conclusions
In summary, we completed the analysis of the long time behavior of nonlinear stochastic functional-differential equations in Hilbert spaces is several steps. In Theorem 2.3 and Theorem 2.4 we establish the existence and uniqueness of mild solutions, as well as their continuous dependence on the initial data. Next, in Theorem 2.12 we obtain a priory, uniform in time bounds for these solutions in the appropriate Hilbert spaces, which were further used to deduce the main result, namely, the existence of invariant measure, in Theorem 2.10. Furthermore, in Theorem 2.14 we exploit the further properties of the problem, which enable us to deduce the exponential stability and thus the uniqueness of invariant measures.