Electronic Journal of Differential Equations, Vol. 2023 (2023), No. 35, pp. 1–21. ISSN: 1072-6691. URL: https://ejde.math.txstate.edu or https://ejde.math.unt.edu ASYMPTOTIC BEHAVIOR OF STOCHASTIC FUNCTIONAL DIFFERENTIAL EVOLUTION EQUATION JASON CLARK, OLEKSANDR MISIATS, VIKTORIIA MOGYLOVA, OLEKSANDR STANZHYTSKYI Abstract. In this work we study the long time behavior of nonlinear sto- chastic 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. Introduction In this work we study the asymptotic behavior of the solutions of stochastic functional-differential equations. In a bounded domain, the equation reads as du = [Au+ f(ut)] dt+ σ(ut) dW (t) in D, t > 0; u(t, x) = φ(t, x), t ∈ [−h, 0), u(0, x) = ϕ0(x) in D; u(t, x) = 0, x ∈ ∂D, t ≥ 0. (1.1) The corresponding problem in the entire space has the form du = [Au+ f(ut)] dt+ σ(ut) dW (t) in Rd, t > 0; u(t, x) = φ(t, x), t ∈ [−h, 0), u(0, x) = ϕ0(x) in Rd. (1.2) Here A is the elliptic operator A = A(x) = d∑ i,j=1 aij(x) ∂2 ∂xi∂xj + d∑ i=1 bi(x) ∂ ∂xi + c(x), (1.3) the interval [−h, 0] is the interval of delay, and ut = u(t+ θ) with θ ∈ [−h, 0]. Functional differential equations of types (1.1) and (1.2) are mathematical mod- els 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 ut = ∆u has an essential shortcoming: it predicts infi- nite speed of propagation of thermal fluctuations in Fourier heat conductors. This 2020 Mathematics Subject Classification. 35R60, 60H15, 92C35. Key words and phrases. Stochastic integral; mild solution; semigroup; white noise; delay differential equation; invariant measure. ©2023. This work is licensed under a CC BY 4.0 license. Submitted June 22, 2022. Published April 12, 2023. 1 2 J. CLARK, O. MISIATS, V. MOGYLOVA, O. STANZHYTSKYI EJDE-2023/35 observation suggests that the Fourier’s law of heat conduction may be an approxi- mation 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 as u̇(x, t) + ∫ ∞ 0 β(s)u̇(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] u̇(x, t)− Cuxx(t, x) = u(t, x) [ 1− u(t, x)− ∫ 0 −1 u(t+ r(s), x) ds ] , 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 di- mensional 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 signifi- cantly 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 non- local 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 equa- tion 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. EJDE-2023/35 STOCHASTIC FUNCTIONAL DIFFERENTIAL EQUATIONS 3 [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 Rd×L2(−h, 0;Rd). In contrast, for stochastic partial differential equations, the natural phase space for the mild solutions of (1.2) is L2 ρ(Rd)×L2(−h, 0;L2 ρ(Rd)), where L2 ρ(Rd) is a weighted space. The equations of type (1.1) and (1.2) were studied in the space C([−h, 0];L2 ρ(Rd)), 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 L2 ρ(Rd) × L2(−h, 0;L2 ρ(Rd)), 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 integral- differential equations. Section 4 establishes the uniqueness of invariant measure, and the convergence to the stationary solution. 2. Preliminaries and main results Throughout this article, the domain D is either a bounded domain with ∂D satisfying the Lyapunov condition, or D = Rd. Denote ρ(x) := 1 1 + |x|r (2.1) where r > d if D = Rd and r = 0 (i.e. no weight) for bounded D. We introduce the following spaces: Bρ0 := L2 ρ(D), Bρ1 := L2(−h, 0;L2 ρ(D)), Bρ := Bρ0 ×B ρ 1 , H := L2(D), (2.2) with the norms ‖u‖2B0 ρ := ‖u(·)‖2ρ := ∫ D u2(x)ρ(x) dx, ‖u(θ, ·)‖2B1 ρ := ∫ 0 −h ∫ D u2(θ, x)ρ(x) dx dθ, ‖(u(·), u1(θ, ·))‖2Bρ = ‖u(x)‖2ρ + ‖u1(θ, x)‖2B1 ρ , ‖u(·)‖2H = ∫ D u2(x) dx. The coefficients aij of the operator A defined in (1.3) are Holder continuous with the exponent β ∈ (0, 1), symmetric, bounded and satisfying the elipticity condition d∑ i,j=1 ai,jηiηj ≥ C0|η|, ∀η ∈ Rd for some C0 > 0. The coefficients bi and c are also bounded and Holder continuous with some positive Holder exponent. If D is bounded, we impose homogeneous Dirichlet boundary conditions on ∂D. In this case, D(A) = H2(D) ∩H1 0 (D). 4 J. CLARK, O. MISIATS, V. MOGYLOVA, O. STANZHYTSKYI EJDE-2023/35 If D = Rd, then D(A) = H2(Rd). 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 C1(T ), C2(T ) > 0 such that 0 ≤ G(t, x, y) ≤ C1(T )t−d/2e−C2(T ) |x−y|2 t (2.3) for t ∈ [0, T ] and x, y ∈ D. Note that in (2.3), C1 and C2 depend not only on T , but on the constants C0, 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 g1(t, x− y) ≤ G(t, x, y) ≤ g2(t, x− y), (2.4) where gi(t, x) = K(C0, d)t−d/2e−K(C0,d) |x|2 t , i = 1, 2, t ≥ 0, x, y ∈ Rd. In this case, in contrast with (2.3), the constant K(C0, d) is independent of t. Lemma 2.1. For each T > 0 there exists a positive C(r, T ) > 0 such that∫ D G(t, x, y)ρ(y) dy ≤ C(r, T )ρ(x), t ∈ [0, T ]. (2.5) Proof. Note that the weight (2.1) satisfies ρ(x) ρ(y) ≤ C(r)(1 + |x− y|r) (2.6) for some C(r) > 0. Thus∫ D G(t, x, y)ρ(y) dy ≤ C(r) ∫ D G(t, x, y)ρ−1(x− y)ρ(x) dy ≤ C(r)C1(T ) ∫ Rd t−d/2e−C2(T ) |y|2 t (1 + |y|r) dyρ(x) ≤ C(r, T )ρ(x). � We define (S(t)ϕ)(x) := ∫ D G(t, x, y)ϕ(y) dy, t > 0, x ∈ D,ϕ ∈ L2(D), (2.7) and S(0) = I, where I is the identity map. This is a semigroup on L2(D) with generator A. Then for all ϕ ∈ L2(D) and for t ∈ [0, T ] by Lemma 2.1 we have ‖S(t)ϕ‖2B0 ρ = ∫ D (∫ D G(t, x, y)ϕ(y) dy )2 ρ(x) dx ≤ ∫ D ρ(x) (∫ D G(t, x, y) dy )(∫ D G(t, x, y)ϕ2(y) dy ) dx ≤ C ∫ D (∫ D G(t, x, y) ρ(x) ρ(y) dx ) ρ(y)ϕ2(y) dy ≤ Cρ(T )‖ϕ‖2B0 ρ . (2.8) The above estimate allows the semigroup S(t) to be extended to a linear map from Bρ0 to itself. Since L2(D) is dense in Bρ0 , S(t) is strongly continuous in Bρ0 . Let ai ≥ 0, ∑∞ i=1 ai < ∞, and en be orthonormal basis in H, such that en ∈ L∞(D) and supn ‖en‖L∞(D) < ∞. We introduce the operator Q ∈ L(H) such EJDE-2023/35 STOCHASTIC FUNCTIONAL DIFFERENTIAL EQUATIONS 5 that Q is non-negative, Tr(Q) < ∞, Qen = anen. Let (Ω,F , P ) be a complete probability space. We introduce W (t) := ∞∑ i=1 √ aiβi(t)ei(x), t ≥ 0, which is a Q-Wiener process on t ≥ 0 with values in L2(Q). Here βi(t) are standard, one dimensional, mutually independent Wiener processes. Also let {Ft, t ≥ 0} be a normal filtration satisfying • W (t) is Ft-measurable; • W (t+ h)−W (t) is independent of Ft for all h ≥ 0, t ≥ 0. Denote U = Q 1 2 (H). From [19, Lemma 2.2], U ∈ L∞(D). Following [19] introduce the multiplication operator Φ : U → Bρ0 as follows: for a fixed ϕ ∈ Bρ0 , let Φ(ψ) := ϕψ, ψ ∈ U . Since ϕ ∈ Bρ0 and ϕ ∈ L∞(D), the operator is well defined and hence Φ ◦Q1/2 : L2(D)→ Bρ0 defines a Hilbert-Schmidt operator. The operator Φ is also a Hilbert-Schmidt operator satisfying ‖Φ ◦Q1/2‖2L2 := ∞∑ n=1 ‖Φ ◦Q1/2en‖2Bρ0 = ∞∑ n=1 an ∫ D ϕ2(x)e2 n(x)ρ(x) dx ≤ Tr(Q) sup n ‖en‖2∞‖ϕ‖2ρ, (2.9) where Tr(Q) = ∑∞ n=1 an = a. Hence if Φ : Ω × [0, T ] → L(U,Bρ0) is a predictable process satisfying E ∫ T 0 ‖Φ ◦Q1/2‖2L2 ds <∞, following [9] we can define ∫ t 0 Ψ(s) dW (s) ∈ Bρ0 with the expansion∫ t 0 Ψ(s) dW (s) = ∞∑ i=1 √ ai ∫ t 0 Φ(s, ·)ei(·) dβi(s). Furthermore, E ∥∥∫ t 0 Ψ(s) dW (s) ∥∥2 B0 ρ ≤ a sup n ‖en‖2∞ ∫ t 0 E‖Ψ(s, ·)‖2Bρ0 ds. (2.10) We assume f and σ satisfy the following conditions: (i) The functionals f and σ map Bρ1 to Bρ0 , (ii) There exists a constant L > 0 such that ‖f(ϕ1)− f(ϕ2)‖Bρ0 + ‖σ(ϕ1)− σ(ϕ2)‖Bρ0 ≤ L‖ϕ1 − ϕ2‖Bρ1 for any ϕ1, ϕ2 ∈ Bρ1 . 6 J. CLARK, O. MISIATS, V. MOGYLOVA, O. STANZHYTSKYI EJDE-2023/35 Definition 2.2. An Ft measurable random process u(t, ·) ∈ Bρ0 is a mild solution of (1.1) or (1.2), if u(t, ·) = S(t)ϕ(0, ·) + ∫ t 0 S(t− s)f(us) ds+ ∫ t 0 S(t− s)σ(us) dW (s) (2.11) where u(0, ·) = ϕ(0, ·) ∈ Bρ0 , u(t, ·) = ϕ(t, ·) ∈ Bρ1 , t ∈ [−h, 0]. Hence the phase space of the problem is the Hilbert space Bρ. In this case y(t) ∈ Bρ if y(t) = (u(t, ·), ut) ∈ Bρ0 ×B ρ 1 , with ut = u(t+ θ, ·) and θ ∈ [−h, 0]. Theorem 2.3 (Existence and uniqueness). Suppose f and σ satisfy the conditions (i) and (ii), and ϕ(t, ·) is an F0 measurable random process for t ∈ [−h, 0], which is independent of W and such that E‖ϕ(0, ·)‖p Bρ0 <∞, E‖ϕ(·, ·)‖p Bρ1 <∞, p ≥ 2. Then there exists a unique mild solution of (1.1) (or 1.2) on [0, T ], and E‖y(t)‖pBρ ≤ K(T )(1 + E‖y(0)‖pBρ), t ∈ [0, T ]. (2.12) Theorem 2.4 (Continuous dependence on the initial data). Let φ ∈ Bρ1 , φ(0, ·) ∈ Bρ0 , φ1 ∈ Bρ1 , φ1(0, ·) ∈ Bρ0 be two initial sets of data of two solutions y(t) = y(t, φ) = ( u(t, φ) ut(φ) ) , y1(t) = y(t, φ1) = ( u(t, φ1) ut(φ1) ) respectively. Then under the conditions of Theorem 2.3 there exists a constant C(T ) such that sup t∈[0,T ] E‖y(t)− y1(t)‖2Bρ ≤ C(T )E‖φ(t)− φ1(t)‖2Bρ . (2.13) The following proposition shows the that the solution u(t, ·) has continuous tra- jectories. Proposition 2.5. Let u(t, ·) be a mild solution of (1.1) or (1.2). Then, under the conditions of Theorem 2.3, ut is continuous at t = 0 in probability with respect to the norm ‖ · ‖Bρ1 , i.e. ‖ut − u0‖2Bρ1 = ∫ 0 −h E‖u(t+ θ)− ϕ(θ)‖2Bρ0 dθ → P 0, t→ 0. Proof. Note that E‖ut − u0‖2Bρ1 ≤ ∫ −t −h E‖ϕ(t+ θ)− ϕ(θ)‖2Bρ0 dθ + ∫ 0 −t E‖u(t+ θ)− ϕ(θ)‖2Bρ0 dθ. The convergence of the first term to 0 follows from the density of C([−h, 0], Bρ0 × L2(Ω)) in L2([−h, 0], Bρ0×B ρ 0×L2(Ω)). The second term converges to zero as t→ 0 since the integrand is bounded. � Let Bb(B ρ) be the Banach space of bounded real Borel functions from Bρ to R, and Cb(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 EJDE-2023/35 STOCHASTIC FUNCTIONAL DIFFERENTIAL EQUATIONS 7 with the initial Fs-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, ut(s, ϕ) = u(t+ θ, s, ϕ), θ ∈ [−h, 0] is a shift of the solution u(t, ϕ), such that us(s, ϕ) = u(s + θ, s, ϕ) = ϕ(θ) and for θ = 0, ϕ(0, ·) ∈ Bρ0 . Following [4], we define the family of shift operators U tsϕ := u(t+ θ, s, ϕ) = ut(s, ϕ). (2.14) Let F ts(dW ) be the minimal σ-algebra containing W (τ)−W (s), τ ∈ [s, t]. Note that ut(s, ϕ) is independent of the σ-algebra Gt, which is the minimal sigma-algebra containing W (τ)−W (t) for τ ≥ t. For any nonrandom ϕ ∈ Bρ with s ≥ 0 and t ≥ s, U tsϕ := ut(s, ϕ) is an F ts(dW ) measurable random function taking values in Bρ1 , with u(t, s, ϕ) ∈ Bρ0 for θ = 0. Defining y(t, s, ϕ) = (u(s, t, ϕ), ut(s, ϕ)), we have that y maps Bρ into itself. The next proposition follows from Theorem 2.3. Proposition 2.6. The family of the operators (2.14) satisfies U tτU τ s ϕ = U tsϕ (2.15) for all t ≥ τ ≥ s ≥ 0 and ϕ ∈ Bρ. Let D be a σ-algebra of Borel subset of Bρ. Then y(t, s, ϕ) naturally denotes the following probability measure µt on D, µt(A) = P{y(t, s, ϕ) ∈ A} = P{U tsϕ ∈ A} = P (s, ϕ, t, A) (2.16) 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) ∈ Bρ is the Markov process on Bρ with the transition function P (s, ϕ, t, A) given by (2.16). Proposition 2.8. For any t ≥ s ≥ 0 we have P (s, ϕ, t, A) = P (0, ϕ, t− s,A) Proof. Let ũ(t) = u(s+t, s, ϕ). Then ũ(0) = ϕ(0, ·) and ũ0 = u(s+θ, s, ϕ) = ϕ(θ, ·). On the other hand, ũ(t) = u(s+ t, s, ϕ) = S(t)ϕ(0, ·) + ∫ s+t s S(s+ t− τ)f(uτ ) dτ + ∫ s+t s S(s+ t− τ)σ(uτ ) dW (τ) = S(t)ϕ(0, ·) + ∫ t 0 S(t− τ)f(uτ+s) dτ + ∫ t 0 S(t− τ)σ(uτ+s)dW̃ (τ) where W̃ (τ) := W (s + τ) −W (s) is once again a Q-Wiener process. This way ũ solves ũ(t) = S(t)ϕ(0, ·) + ∫ t 0 S(t− τ)f(ũ(τ)) dτ + ∫ t 0 S(t− τ)σ(ũ(τ))dW̃ (τ) (2.17) 8 J. CLARK, O. MISIATS, V. MOGYLOVA, O. STANZHYTSKYI EJDE-2023/35 The same equation is satisfied with u(t, 0, ϕ) such that u(0, 0, ϕ) = ϕ(0, ·) and u0 = ϕ(θ, ·). The only difference is that u(t, 0, ϕ) solves (2.17) with a different Wiener process W . However, since the distribution of W is the same as W̃ , the distribution of u(s+ t, s, ϕ) is the same as the distribution of u(t, 0, ϕ), and hence independent of s. Thus the distribution of ut(s, ϕ) = u(t+θ, s, ϕ) = u(t−s+θ+s, sϕ) coincides with the distribution of u(t− s+ θ, 0, ϕ) = ut−s(0, ϕ). Hence P (s, ϕ, t, A) = P{ut(s, ϕ) ∈ A} = P{u(t+ θ, s, ϕ) ∈ A} = P{u(t− s+ θ, 0, ϕ) ∈ A} = P{ut−s(0, ϕ) ∈ A} which yields the desired result. � For g ∈ Bb(Bρ), for ϕ ∈ Bρ and t ≥ s ≥ 0, we define Ps,t(ϕ) := Eg(y(t, s, ϕ)). From proposition 2.8 we have P0,t−s(ϕ) and denote Ptϕ = P0,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 Pt, t ≥ 0 is stochastically continuous an satisfies the Feller property Pt : Cb(B ρ)→ Cb(B ρ), lim t→0 Ptϕ(θ) = ϕ(θ). We define ρ̄(x) = (1 + |x|r̄)−1. The main result of the paper is the following theorem. Theorem 2.10. Let the assumptions of Theorem 2.3 hold. Assume the equation (2.11) has a solution in Bρ̄ which is bounded in probability for t ≥ 0 with r > d+ r̄. (2.18) Then there exists an invariant measure µ on Bρ, i.e.∫ Bρ Ptϕ(x)dµ(x) = ∫ Bρ ϕ(x)dµ, for all t ≥ 0 and ϕ ∈ Cb(Bρ). Remark 2.11. Condition (2.18) is equivalent to∫ Rd ρ(x) ρ̄(x) dx <∞. The key condition in Theorem 2.10 is the existence of a globally bounded solu- tion. 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. Theorem 2.12. Assume • D = Rd, d ≥ 3; • the conditions of Theorem 2.3 hold; • for some σ0 > 0, we have |σ(u)| ≤ σ0, for all u ∈ Bρ1 ; • there exists Ψ ∈ L1(Rd)∩L∞(Rd) such that |f(u(·))| ≤ Ψ(·) for all u ∈ Bρ1 ; • u(t, ·) = ϕ(t, ·), t ∈ [−h, 0], u(0, ·) = ϕ(0, x) satisfy E ∫ Rd |ϕ(0, x)|2 dx <∞ and E ∫ Rd ∫ 0 −h |ϕ(θ, x)|2 dx dθ <∞. EJDE-2023/35 STOCHASTIC FUNCTIONAL DIFFERENTIAL EQUATIONS 9 Then sup t≥0 E‖y(t)‖2Bρ <∞, 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 B0 := L2(D), B1 := L2(−h, 0;B0), B := B0 ×B1. The semigroup (2.7) now satisfies the exponential estimate ‖S(t)u0‖2B0 ≤ e−2λ1t‖u0‖2B0 , where λ1 > 0 is the principle eigenvalue of −A. In a standard way, we can extend the Q-Weiner process W (t) to t ∈ R as W (t) = { W (t), t ≥ 0; V (−t), t ≤ 0. Here V is another Q-Weiner process, independent of W . Definition 2.13. A B0-valued process u(t) is a mild solution of (1.1) for t ∈ R if (1) for all t ∈ R, u(t) is Ft measurable; (2) for all t ∈ R E‖u(t)‖2B0 <∞; (3) for all −∞ < t0 < t <∞ with probability 1 we have u(t) = S(t− t0)u(t0) + ∫ t t0 S(t− s)f(us) ds+ ∫ t t0 S(t− s)σ(us) dW (s) 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)‖2B <∞. Furthermore, this solution is exponentially attractive, that is exist K, γ > 0 such that for all t0 ∈ R and t > t0 + h, and for any other solution η(t) with η(t0) ∈ B0 and ηt0 ∈ B1 we have E‖u(·, t)− η(·, t)‖2B ≤ Ke−γ(t−t0)E‖u(·, t0)− η(·, t0)‖2B . 3. Proofs of main results Proof of Theorem 2.3. Let Bp,T , p ≥ 2 be the space of Ft-measurable for t ∈ [0, T ] processes, equipped with the norm ‖Φ‖pBp,T := E ∫ T −h ‖Φ(t, ·)‖p Bρ0 dt. We define ΨΦ(t, ·) := S(t)Φ(0, ·) + ∫ t 0 S(t− s)f(Φ(s+ θ, ·)) ds + ∫ t 0 S(t− s)σ(Φ(s+ θ, ·)) dW (s) (3.1) for t ∈ [0, T ], and ΨΦ(t, ·) = ϕ(t, ·), t ∈ [−h, 0], with ΨΦ(0, ·) = ϕ(0, ·). 10 J. CLARK, O. MISIATS, V. MOGYLOVA, O. STANZHYTSKYI EJDE-2023/35 This way, ΨΦ(t, ·)‖pBp,T ≤ E ∫ 0 −h ‖ϕ(t, ·)‖p Bρ0 dt+ 3p−1E ∫ T 0 ‖S(t)ϕ(0, ·)‖p Bρ0 dt + 3p−1E ∫ T 0 ∥∥∫ t 0 S(t− s)f(Φ(s+ θ, ·)) ds ∥∥p Bρ0 dt + 3p−1E ∫ T 0 ∥∥∫ t 0 S(t− s)σ(Φ(s+ θ, ·)) dW (s) ∥∥p Bρ0 dt ≤ C1(T ) + 3p−1(I1 + I2 + I3). It follows from (2.8) that I1 ≤ Cpρ (T ) ∫ T 0 E‖ϕ(0, ·)‖p Bρ0 dt <∞. Next, using the conditions (i) and (ii) for f , we have I2 ≤ Cpρ (T ) ∫ T 0 T p−1 ( E ∫ t 0 ‖f(Φs, ·)‖pBρ0 ds ) dt ≤ C2 ∫ T 0 dt ∫ t 0 ( 1 + E‖Φs‖pBρ1 ) ds ≤ C3 + C2 ∫ T 0 ∫ t 0 E (∫ 0 −h ‖Φ(s+ θ, ·)‖2Bρ0 dθ )p/2 ds dt ≤ C3 + C4E ∫ T −h ‖Φ(t, ·)‖p Bρ0 dt <∞. (3.2) To estimate I3, we use [9, Lemma 7.2] and (2.10). Using the definition of Hilbert- Schmidt norm given in (2.9), we have I3 ≤ C(p) ∫ T 0 E (∫ t 0 ‖S(t− s)σ(Φs(·))‖2L2 ds )p/2 dt ≤ C(p)ap sup n ‖en‖p∞ ∫ T 0 E (∫ t 0 ‖S(t− s)σ(Φs(·))‖2Bρ0 ds )p/2 dt ≤ C4 + C5 ∫ T 0 ∫ t 0 E‖Φs(·)‖pBρ0 ds dt <∞ (3.3) the same way as in (3.2). Combining these estimates, we have Ψ : Bp,T → Bp,T . We next show that Ψ is contractive. For any Φ, Φ̃ ∈ Bp,t we have ‖ΨΦ(s, ·)−ΨΦ̃(s, ·)‖pBp,T ≤ 2p−1 ∫ t 0 E ∥∥∫ s 0 S(s− τ)(f(Φτ (·)− f(Φ̃τ (·)) dτ ∥∥p Bρ0 ds + 2p−1 ∫ t 0 E ∥∥∫ s 0 S(s− τ)(σ(Φτ (·))− σ(Φ̃τ (·)) dτ ∥∥p Bρ0 ds := 2p−1(I4 + I5). (3.4) EJDE-2023/35 STOCHASTIC FUNCTIONAL DIFFERENTIAL EQUATIONS 11 I4 ≤ Cpρ (T )Lp ∫ t 0 E (∫ s 0 ‖Φτ (·)− Φ̃τ (·)‖Bρ1 dτ )p ds ≤ C(ρ, T, p) ∫ t 0 ∫ s 0 E (∫ 0 −h ‖Φ(τ + θ, ·)− Φ̃(τ + θ, ·)‖2Bρ0 dθ )p/2 dτ ds ≤ C5(ρ, T, p)t2‖Φ− Φ̃‖pBp,t (3.5) Now from estimate (3.3), we have I5 ≤ C(p)ap sup n ‖en‖p∞ ∫ t 0 E (∫ s 0 ‖S(s− τ)[σ(Φτ )− σ(Φ̃τ )]‖2Bρ0 )p/2 dt ≤ C6 ∫ t 0 ∫ s 0 E (∫ 0 −h ‖Φ(τ + θ, ·)− Φ̃(τ + θ, ·)‖2Bρ0 dθ )p/2 dτ ds ≤ C6(ρ, T, p, h)t2‖Φ− Φ̃‖pBp,t . (3.6) Consequently, for t̃ small enough, (3.5) and (3.6) imply that the map Ψ has a unique fixed point in Bp,t̃, which is the solution of (2.11). If we consider the problem on [0, t̃], [t̃, 2t̃], ... with C6t̃ 2 < 1. Since the solution is continuous with probability 1 in Bρ0 norm, we obtain the existence and uniqueness of the solution on [0, T ]. It remains to prove estimate (2.12). It follows from (2.11) that for any t ∈ [−h, T ] we have E‖u(t, ·)‖p Bρ0 ≤ 3p−1E‖S(t)ϕ(0, ·)‖p Bρ0 + 3p−1E (∫ t 0 ‖S(t− s)f(us)‖Bρ0 ds )p + 3p−1E ∥∥∫ t 0 S(t− s)σ(us) dW (s) ∥∥p Bρ0 ≤ 3p−1Cρ(T )E‖ϕ(0, ·)‖p Bρ0 + 3p−1C7 ∫ t 0 (1 + E‖us‖pBρ1 ) ds + 3p−1C8E (∫ t 0 ‖S(t− s)σ(us)‖2L2 ds )p/2 ≤ C9 ( E‖ϕ(0, ·)‖p Bρ0 + ∫ t 0 (1 + E‖us‖pBρ1 ) ds ) . (3.7) We consider two separate cases t ∈ [0, h] and t ∈ [h, T ]: If t ∈ [0, h], then E‖ut‖pBρ1 = E (∫ 0 −h ‖u(t+ θ, ·)‖2Bρ0 dθ )p/2 ≤ 2 p 2−1 ( E (∫ −t −h ‖u(s, ·)‖2Bρ0 ds )p/2 + E (∫ 0 −t ‖u(s, ·)‖2Bρ0 ds )p/2) ≤ 2 p 2−1 ( E‖ϕ(t, ·)‖p Bρ1 + h p−2 p ∫ t 0 E‖u(s, ·)‖p Bρ0 ds ) ≤ 2 p 2−1 ( E‖ϕ(t, ·)‖p Bρ1 + C10 sup s∈[0,t] E‖u(s, ·)‖p Bρ0 ) . (3.8) If t ∈ [h, T ], then E‖ut‖pBρ1 = E (∫ 0 −h ‖u(t+ θ, ·)‖2Bρ0 dθ )p/2 ≤ C11(T ) sup s∈[0,t] E‖u(s)‖p Bρ0 . (3.9) 12 J. CLARK, O. MISIATS, V. MOGYLOVA, O. STANZHYTSKYI EJDE-2023/35 From (3.7)–(3.9) we have sup s∈[0,t] E‖u(s, ·)‖p Bρ0 ≤ C12(T ) ( E‖ϕ(0, ·)‖p Bρ0 + E‖ϕ(t, ·)‖p Bρ1 + ∫ t 0 sup τ∈[0,s] E‖u(τ, ·)‖p Bρ0 ds ) . Estimating the last term separately, we have sup s∈[0,t] E‖u(s, ·)‖p Bρ0 ≤ C13(T )[1 + E‖ϕ(0, ·)‖p Bρ0 + E‖ϕ(t, ·)‖p Bρ1 ]. Combining the estimates above, we obtain E‖ut‖pBρ1 ≤ C14(T )(1 + E‖y(0)‖pBρ , which completes the proof. � Proof of Theorem 2.4. By definition of y1 and y2, sup t∈[0,T ] E‖y(t)− y1(t)‖2Bρ ≤ sup t∈[0,T ] E‖u(t, φ)− u(t, φ1)‖2Bρ0 + sup t∈[0,T ] E‖ut(φ)− ut(φ1)‖2Bρ1 . (3.10) The first term in (3.10) can be estimated as follows sup t∈[0,T ] E‖u(t, φ)− u(t, φ1)‖2Bρ0 ≤ C15 sup t∈[0,T ] E‖φ(t)− φ1(t)‖2Bρ0 . (3.11) As for the second term in (3.10), once again we consider separately the cases t ∈ [0, h] and t ∈ [h, T ]. Taking into account the estimate (3.11), we obtain sup t∈[0,T ] E ∫ 0 −h ‖u(t+ θ, φ)− u(t+ θ, φ1)‖2Bρ0 dθ ≤ C16 sup t∈[0,T ] E‖φ(t)− φ1(t)‖2Bρ , which completes the proof. � For the proof of Theorem 2.10, we need the following auxiliary lemmas. Lemma 3.1. For any fixed T0 > 2h, the operator Aϕ0 := S(T0 + θ) : Bρ̄0 → Bρ1 is a Hilbert-Schmidt operator. Proof. By [23], there exists an orthonormal basis {hn, n ≥ 1} in Bρ0 such that supn ‖hn‖L∞(D) < ∞. It is straightforward to verify that if {en, n ≥ 1} is an orthonormal basis in H = L2(D), then { en ρ̄1/2 , n ≥ 1} is an orthonormal basis in Bρ̄0 . Therefore ‖A‖2L2 = ∞∑ i=1 ‖A ei√ ρ̄ ‖2Bρ1 = ∞∑ i=1 ‖S(T0 + θ) ei√ ρ̄ ‖2Bρ1 = ∞∑ i=1 ∫ 0 −h dθ ∫ D ∣∣∣S(T0 + θ) ei√ ρ̄ ∣∣∣2ρ(x) dx EJDE-2023/35 STOCHASTIC FUNCTIONAL DIFFERENTIAL EQUATIONS 13 = ∞∑ i=1 ∫ 0 −h dθ ∫ D ∣∣ ∫ D G(T0 + θ, x, y) ei(y)√ ρ̄(y) dy ∣∣∣2ρ(x) dx = ∫ 0 −h dθ ∫ D ρ(x) ∫ D G2(T0 + θ, x, y) ρ̄(y) dy dx ≤ ∫ 0 −h dθ ∫ D ∫ D ρ(x) ρ̄(y) C1(T0)(T0 + θ)−d exp{−2C2(T0) |x− y|2 T0 + θ } dy dx ≤ C17 ∫ 0 −h dθ (T0 + θ)d/2 ∫ Rd ∫ Rd 1 (T0 + θ)d/2 exp{−2C2(T0) |x− y|2 T0 + θ }ρ(x) ρ̄(y) dx dy. But ∫ Rd 1 (T0 + θ)d/2 exp{−2C2(T0) |x− y|2 T0 + θ }ρ(x) ρ(y) dxρ(y) ≤ C(r) ∫ Rd 1 (T0 + θ)d/2 exp { − 2C2(T0) |x− y|2 T0 + θ } (1 + |x− y|r) dxρ(y) ≤ C18(T, r)ρ(y). Thus ‖A‖2L2 ≤ C19(T, r) ∫ 0 −h dθ (T0 + θ)d/2 ∫ Rd 1 + |y|r̄ 1 + |y|r dy <∞, which completes the proof. � Corollary 3.2. Following the lines of the proof of Lemma 3.1 we can show that S(t) is a compact operator from Bρ̄0 to Bρ0 for t > 0. We now return to the proof of Theorem 2.10. Following the approach in [9, Theorem 11.29], we have u(T0) = S(T0)ϕ(0, ·) + ∫ T0 0 S(T0 − s)f(us) ds+ ∫ T0 0 S(T0 − s)σ(us) dW (s), (3.12) uT0 = u(T0 + θ) = S(T0 + θ)ϕ(0, ·) + ∫ T0+θ 0 S(T0 + θ − s)f(us) ds + ∫ T0+θ 0 S(T0 + θ − s)σ(us) dW (s). (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 (Gαϕ)(θ) = ∫ T0+θ 0 (T0 + θ − s)α−1S(T0 + θ − s)ϕ(s) ds is compact from Lp(0, T0;Bρ̄0) to C([−h, 0], Bρ0). Remark 3.4. Compactness in C([−h, 0], Bρ0) implies compactness in Bρ1 . Proof of Lemma 3.3. We denote ‖ϕ‖pLp := ∫ T0 0 ‖ϕ‖p Bρ̄0 dt 14 J. CLARK, O. MISIATS, V. MOGYLOVA, O. STANZHYTSKYI EJDE-2023/35 We will use the infinite dimensional version of Arzela-Ascoli Theorem. To this, we need to show (i) For any fixed θ ∈ [−h, 0] the set {Gα(ϕ)(θ), ‖ϕ‖Lp ≤ 1} is compact in Bρ0 ; (ii) for any ε > 0 there exists δ > 0 such that if ‖ϕ‖Lp ≤ 1 such that if ‖ϕ‖Lp ≤ 1 and for all θ1, θ2 with |θ1 − θ2| ≤ δ we have ‖Gα(ϕ)(θ1)−Gα(ϕ)(θ2)‖Bρ0 < ε. To check (i), for fixed θ ∈ [−h, 0] and 0 < ε < T0 + θ, introduce Gεαϕ := ∫ T0+θ−ε 0 (T0 + θ − s)α−1S(T0 + θ − s)ϕ(s) ds = S(ε) ∫ T0+θ−ε 0 (T0 + θ − s)α−1S(T0 + θ − s− ε)ϕ(s) ds Clearly ∫ T0+θ−ε 0 (T0 + θ − s)α−1S(T0 + θ − s− ε)ϕ(s) ds is in Bρ̄0 . Using Corollary 3.2, S(ε) is a compact operator from Bρ̄0 to Bρ0 . Then, following [9, p.227], Gεα converges to Gα strongly as ε→ 0, hence Gα is compact and (i) follows. To prove (ii), fix θ and r such that −h ≤ θ ≤ θ + r ≤ 0, and ‖ϕ‖Lp ≤ 1. Then ‖(Gαϕ)(θ + r)− (Gαϕ)(θ)‖Bρ0 = ∥∥∫ T0+θ+r 0 (T0 + θ + r − s)α−1S(T0 + θ + r − s)ϕ(s) ds − ∫ T0+θ 0 (T0 + θ − s)α−1S(T0 + θ − s)ϕ(s) ds ∥∥ Bρ0 ≤ ∫ T0+θ 0 ∥∥(T0 + θ + r − s)(α−1)S(T0 + θ + r − s) − (T0 + θ − s)(α−1)S(T0 + θ − s)‖‖ϕ(s) ∥∥ ds + ∫ T0+θ+r T0+θ ‖(T0 + θ + r − s)(α−1)S(T0 + θ + r − s)ϕ(s)‖ ds ≤ (∫ T0 0 ‖(r + s)α−1S(s+ r)− sα−1S(s)‖q ds )1/q ‖ϕ‖Lp + C20 (∫ T0 0 s(α−1)q ds )1/q ‖ϕ‖Lp := J1 + J2. Direct calculations yield J2 = C20 rα− 1 p ((α− 1)q + 1)1/q ‖ϕ‖Lp → 0 as r → 0. We now proceed with estimating J1. Since S(t) is compact, then S(t) is strongly continuous for t > 0 (see [24, Theorem 3.27]), hence ‖S(s + r) − S(s)‖ → 0 as r → 0, for any s > 0. Furthermore, the integrand in J1 is bounded by 2C20s (α−1)q. Hence, by Dominated Convergence Theorem, J1 → 0 as r → 0, which concludes the proof of the Lemma. � We now complete the proof of Theorem 2.10. For any r > 0 introduce K(r) := {(µ, ν), µ ∈ Bρ0 , ν ∈ B ρ 1} EJDE-2023/35 STOCHASTIC FUNCTIONAL DIFFERENTIAL EQUATIONS 15 such that µ := S(T0)v + (G1ϕ)(0) + (Gαh)(0), ν := S(T0 + θ)v + (G1ϕ)(0) + (Gαh)(0) with ‖v‖Bρ0 ≤ r, ‖ϕ‖Lp(0,T0,B ρ̄ 0 ) ≤ r and ‖h‖Lp(0,T0,B ρ̄ 0 ) ≤ r. It follows from Lemma 3.1, Corollary 3.2 and Lemma 3.3 that K(r) is compact in Bρ. Lemma 3.5. Under the conditions of Theorem 2.3, there is C > 0 such that for arbitrary r > 0 and y = (x, z) ∈ Bρ̄ such that ‖y‖Bρ̄ ≤ r we have P{(u(T0, x, z), uT0(x, z)) ∈ K(r)} ≥ 1− cr−p(1 + ‖y‖pBρ̄), (3.14) where u(0, x, z) = x ∈ Bρ̄0 and u0(x, z) = z ∈ Bρ̄1 . Proof. From the factorization formula [10, Thm. 5.2.5], we have u(T0, y) = S(T0)x+ (G1f(us))(0) + sin(απ) π (GαY (s))(0), (3.15) uT0 (y) = S(T0 + θ)x+ (G1f(us))(θ) + sin(απ) π (GαY (s))(θ), (3.16) Y (s) = ∫ s 0 (s− τ)−αS(s− τ)σ(uτ ) dW (τ). (3.17) Using Lemma 7.2 [9], we obtain E ∫ T0 0 ‖Y (s)‖p Bρ̄0 ds = E ∫ T0 0 ‖ ∫ s 0 (s− τ)−αS(s− τ)σ(uτ ) dW (τ)‖p Bρ̄0 ds ≤ Cp,T0 E ∫ T0 0 (∫ s 0 (s− τ)−2α‖S(s− τ)σ(uτ ) ◦Q1/2‖L2(H,Bρ̄0 ) )p/2 ds ≤ C21E ∫ T0 0 (∫ s 0 (s− τ)−2α‖σ(uτ )‖2 Bρ̄0 dτ )p/2 ds. (3.18) Using Hausdorff-Young’s inequality and (2.11), we have E ∫ T0 0 ‖Y (s)‖p Bρ̄0 ≤ C21 (∫ T0 0 t−2α dt )p/2 ∫ T0 0 E‖σ(ut)‖pBρ̄0 dt ≤ C22 ∫ T0 0 (1 + E‖ut‖pBρ̄1 ) dt ≤ C23(1 + ‖y‖pBρ̄). (3.19) In a similar way, E ∫ T0 0 ‖f(us)‖pBρ̄0 ds ≤ C23(1 + ‖y‖pBρ̄). (3.20) Hence, if ‖y‖Bρ̄ ≤ r, ‖f(us)‖Lp(0,T0,B ρ̄ 0 ) ≤ r, and ‖σ(us)‖Lp(0,T0,B ρ̄ 0 ) ≤ πr sin(απ) , then from the definition of K(r) we have (u(T0, y), uT0 (y)) ∈ K(r). Assume ‖y‖Bρ̄ ≤ r. Then P{(u(T0, y), uT0 (y)) /∈ K(r)} ≤ P{‖f(us)‖Lp(0,T0;Bρ̄0 ) > r}+ P{‖Y (s)‖Lp(0,T0,B ρ̄ 0 )} 16 J. CLARK, O. MISIATS, V. MOGYLOVA, O. STANZHYTSKYI EJDE-2023/35 ≤ 2rpC23(1 + ‖y‖pBρ̄) where we used (3.19) and (3.20). The proof is complete. � 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 E‖y(t)‖2Bρ = E ∫ Rd |u(t, x)|2ρ(x) dx+ E ∫ 0 −h dθ ∫ Rd |u(t+ θ, x)|2ρ(x) dx. (3.21) By definition of a mild solution (2.11), we have ‖u(t, x)‖2Bρ0 ≤ 3(I1(t) + I2(t) + I3(t)) where I1(t) = ∫ Rd (∫ Rd G(t, x, y)ϕ(0, y) dy )2 ρ(x) dx, I2(t) = ∫ Rd (∫ t 0 ∫ Rd G(t− s, x, y)f(us(y)) dy ds )2 ρ(x) dx, I3(t) = ∫ Rd (∫ Rd G(t− s, x, y)σ(us(y)) dW (s) dy )2 ρ(x) dx. It follows from (2.4) that for all t ≥ 0, EI1 ≤ ∫ Rd (∫ Rd G(t, x, y) dy ∫ Rd G(t, x, y)ϕ2(0, y) dy ) ρ(x) dx ≤ C24E ∫ Rd (∫ Rd K(t, x− y)ϕ2(0, y) dy ) ρ(x) dx ≤ C24‖ρ‖∞E‖ϕ(0, ·)‖2Bρ0 <∞, where K is the heat kernel in Rd. The estimates for I2 and I3 can be estimated along the lines of [20, Theorem 1] using the Nash-Aranson type estimates for the kernel (2.3). To estimate the second term in (3.21), once again we consider two cases: t ∈ [0, h] and t ≥ h. If t ∈ [0, h], then E‖ut‖2Bρ1 = E ∫ 0 −h ‖u(t+ θ)‖2Bρ0 dθ ≤ E ∫ 0 −h ‖u(s)‖2Bρ0 ds+ E ∫ h 0 ‖u(s)‖2Bρ0 ds ≤ E‖ϕ(t, ·)‖2Bρ1 + h sup t≥0 E‖u(t)‖2Bρ0 <∞. Finally, if t ≥ h, then E‖ut‖2Bρ1 = E ∫ 0 −h ‖u(t+ θ)‖2Bρ0 dθ ≤ sup t≥0 E‖u(t)‖2Bρ0 <∞, which completes the proof. � EJDE-2023/35 STOCHASTIC FUNCTIONAL DIFFERENTIAL EQUATIONS 17 Example 3.6. Let f̄ : R→ R and σ̄ : R→ R be Lipschitz functions with Lipschitz constants L. We define f [ϕ] := f̄ (∫ 0 −h ϕ(θ) dθ ) , σ[ϕ] := σ̄ (∫ 0 −h ϕ(θ) dθ ) Then for all ϕ1, ϕ2 ∈ Bρ1 we have |f [ϕ1]− f [ϕ2]| ≤ L ∫ 0 −h |ϕ1(θ)− ϕ2(θ)| dθ. Hence ‖f [ϕ1]− f [ϕ2]‖2Bρ0 ≤ L 2 ∫ Rd (∫ 0 −h |ϕ1(θ)− ϕ2(θ)| dθ )2 ρ dx ≤ L2h‖ϕ1 − ϕ2‖2Bρ1 . Similarly, ‖σ[ϕ1]− σ[ϕ2]‖2Bρ0 ≤ L 2h‖ϕ1 − ϕ2‖2Bρ1 . Thus f and σ are examples of Lipschits maps from Bρ1 to Bρ0 , for which the theorems above apply. 4. Uniqueness of the invariant measure Proof of Theorem 2.14. Let B be the class of Ft measurable B0-valued processes ξ(t), such that sup t∈R E‖ξ(t)‖2B0 <∞. Since sup t∈R ‖ξ(t)‖2B ≤ (1 + h) sup t∈R E‖ξ(t)‖2B0 , we follow the procedure in [20] and define the successive approximations u(0) ≡ 0 and du(n+1) = (Au(n+1) + f(u (n) t )) dt+ σ(u (n) t ) dW (t). (4.1) Then sup t∈R E‖f(u (n) t )‖2B0 ≤ 2‖f(0)‖2B0 + 2L2h2 sup t∈R E‖u(n)(t)‖2B0 <∞. Similarly, sup t∈R E‖σ(u (n) t )‖2B0 <∞. Thus by Theorem 5 [20], equation (4.1) has the unique solution u(n+1)(t) such that sup t∈R E‖u(n+1)(t)‖2B0 <∞, and therefore, sup t∈R E‖u(n+1)(t)‖2B <∞. But sup t∈R E‖u(n)‖2B0 ≤ (1 + h) sup t∈R E‖u(n)(t)‖2B0 ≤ C + hL2 ( 4 λ2 1 + 2a λ1 ) sup t∈R E‖u(n−1)‖2B0 . Hence for hL2 ( 4 λ2 1 + 2a λ1 ) < 1 (4.2) 18 J. CLARK, O. MISIATS, V. MOGYLOVA, O. STANZHYTSKYI EJDE-2023/35 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‖un(t)− u∗(t)‖2B → 0, n→∞. Furthermore, we can argue that u∗ satisfies u∗(t) = S(t− t0)u∗(t0) + ∫ t t0 S(t− t0)f(u∗s) ds+ ∫ t t0 S(t− s)σ(u∗s) dW (s). (4.3) Consider any other solution (4.3) such that η(t0) is Ft0 -measurable, and E‖η(t0)‖2B < ∞. Here ηt0 = ϕ(θ, x) is defined on [−h, 0]. Let us show that the solution η con- verges to u∗ exponentially. Since we are interested in the behavior of the solutions for large t, suppose t > t0 + h. Then t+ θ > t0 and η(t) is defined via the formula (4.3). Hence E‖u∗(t)− η(t)‖2B0 ≤ 3e−λ1(t−t0)E‖u∗(t0)− η(t0)‖2B0 + 3 L2 λ1 ∫ t t0 e−λ1(t−s)E‖u∗s − ηs‖2B1 ds + 3L2a ∫ t t0 e−λ1(t−s)E‖u∗s − ηs‖2B1 ds = 3e−λ1(t−t0)E‖u∗(t0)− η(t0)‖2B0 + 3 (L2 λ1 + L2a )∫ t t0 e−λ1(t−s)E‖u∗s − ηs‖2B1 ds. In addition, E‖u∗t − ηt‖2B1 = ∫ 0 −h E‖u∗(t+ θ)− η(t+ θ)‖2B0 dθ + 3 ∫ 0 −h e−λ1(t+θ−t0)E‖u∗(t0)− η(t0)‖2B0 dθ + 3 ∫ 0 −h (L2 λ1 ∫ t+θ t0 e−λ1(t+θ−s)E‖u∗s − ηs‖2B1 ds ) dθ + 3 ∫ 0 −h ( L2a ∫ t+θ t0 e−λ1(t+θ−s)E‖u∗s − ηs‖2B1 ds ) dθ. However, e−λ1(t+θ−s) ≤ e−λ1(t−s) · eλ1h, thus E‖u∗t − ηt‖2B1 ≤ 3heλ1he−λ1(t−t0)E‖u∗(t0)− η(t0)‖2B0 + 3eλ1hh (L2 λ1 + L2a )∫ t t0 e−λ1(t−s)E‖u∗s − ηs‖2B1 ds. Altogether, E‖u∗(t)− η(t)‖2B ≤ (3eλ1hh+ 3)e−λ1(t−t0)E‖u∗(t0)− η(t0)‖2B EJDE-2023/35 STOCHASTIC FUNCTIONAL DIFFERENTIAL EQUATIONS 19 + (3 + 3heλ1h) (L2 λ1 + L2a )∫ t t0 e−λ1(t−s)E‖u∗(s)− η(s)‖2B ds. Therefore, if (3 + 3heλ1h) (L2 λ1 + L2a ) := γ0L 2 < λ1 (4.4) we have E‖u∗(t)− η(t)‖2B ≤ (3eλ1hh+ 3)e(γ0L 2−λ1)(t−t0)E‖u∗(t0)− η(t0)‖2B . Then the existence and uniqueness of invariant measure can be established in the same manner as in [20]. � 5. 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. Acknowledgments. The research of Oleksandr Misiats was supported by Simons Collaboration Grant for Mathematicians No. 854856. 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Jason Clark College of Engineering, Oregon State University, Corvallis, OR 97331, USA Email address: jason.clark@oregonstate.edu Oleksandr Misiats Department of Mathematics, Virginia Commonwealth University, Richmond, VA 23284, USA Email address: omisiats@vcu.edu EJDE-2023/35 STOCHASTIC FUNCTIONAL DIFFERENTIAL EQUATIONS 21 Viktoriia Mogylova Department of Physics and Mathematics, Igor Sikorsky Kyiv Polytechnic Institute, Ukraine Email address: mogylova.viktoria@gmail.com Oleksandr Stanzhytskyi Department of Mathematics, Taras Shevchenko National University of Kyiv, Ukraine Email address: ostanzh@gmail.com 1. Introduction 2. Preliminaries and main results 3. Proofs of main results 4. Uniqueness of the invariant measure 5. Conclusions Acknowledgments References