Stationary Solutions of Neutral Stochastic Partial Differential Equations with Delays in the Highest-Order Derivatives

In this work, we shall consider the existence and uniqueness of stationary solutions to stochastic partial functional differential equations with additive noise in which a neutral type of delay is explicitly presented. We are especially concerned about those delays appearing in both spatial and temporal derivative terms in which the coefficient operator under spatial variables may take the same form as the infinitesimal generator of the equation. We establish the stationary property of the neutral system under investigation by focusing on distributed delays. In the end, an illustrative example is analysed to explain the theory in this work.

Moreover, this stationary solution is unique in the sense that any two stationary solutions of (1.1) have the same finite dimensional distribution.

Strongly Continuous Semigroup
For arbitrary Banach spaces X and Y with their respective norms · X and · Y , we always denote by L (X, Y ) the space of all bounded, linear operators from X into Y . If X = Y , we simply write L (X) for L (X, X). Let V be a separable Hilbert space and a : V × V → R a bilinear form satisfying the so-called Gårding's inequalities for some constants β > 0, α > 0. In association with the form a(·, ·), let A be a linear operator defined by where V * is the dual space of V and ·, · V,V * is the dual pairing between V and V * . Then A ∈ L (V, V * ) and A generates an analytic semigroup e tA , t ≥ 0, on V * . We also introduce the standard interpolation Hilbert space H = (V, V * ) 1/2,2 , which is described by Ae tA x, Ae tA y V * dt, x, y ∈ V * .
We identity the dual H * of H with H, then it is easy to see that where the imbedding ֒→ is dense and continuous with x 2 H ≤ ν x 2 V , x ∈ V, for some constant ν > 0. Hence, x, Ay H = x, Ay V,V * for all x ∈ V and y ∈ V with Ay ∈ H. Moreover, for any T ≥ 0 it is well known (see [8]) that is the Sobolev space consisting of all functions y : [0, T ] → V * such that y and its first order distributional derivative are in L 2 ([0, T ], V * ) and C([0, T ], H) is the space of all continuous functions from [0, T ] into H, respectively. It can be also shown (see, e.g., [13]) that the semigroup e tA , t ≥ 0, is bounded and analytic on both V * and H such that e tA : V * → V for each t > 0 and for some constant M > 0, Consider an abstract evolution equation in V * as follows: 3) The proofs of the following result are referred to Theorems 2.3 and 2.4 in [3]. and 3) has a unique solution given by In addition, there exists a constant C 0 > 0 such that x ∈ C([0, T ], H) and and for any t ≥ 0, . We introduce two linear mappings D and F on C([−r, T ], V ), respectively, by and

Proof.
We only prove (2.4) since the relation (2.5) can be obtained in an analogous manner. By using Hölder's inequality and Fubini's theorem, we can obtain that for any Let H = H ×L 2 ([−r, 0], V ) and consider the following deterministic functional differential equation of neutral type in V * , (2.6) The integral form of (2.6) is given by (2.7) We say that x is a (strict) solution of (2.
Moreover, we have the relations and for some positive number M = M(T ) > 0.
Proof. We shall use a fixed point argument to the integral equation (2.7). Let t 0 ∈ (0, r) and Σ be a closed subspace of L 2 ([−r, t 0 ], V ) such that x(0) = φ 0 + Dφ 1 and x 0 = φ 1 for x ∈ Σ. Define a mapping S on Σ as follows: for x ∈ Σ, (2.10) Note that we have (2.11) To obtain a unique solution of (2.6), it suffices to show that S is a contraction from Σ into itself for sufficiently small t 0 > 0 and then implement a successive interval argument to extend the solution onto the whole interval [0, T ]. Indeed, for any x,x ∈ Σ and t ∈ [0, t 0 ], we have which, in addition to (2.11) and (2.12), immediately yields that The map S is thus a contraction in Σ and the equation (2.7) has a unique solution x on [−r, t 0 ].
To show the relation (2.8), we re-write (2.7) for t ∈ [0, t 0 ], t 0 < r, to get Then, from Theorem 2.1 (i) and (ii) and (2.11), we obtain where C 1 (t 0 ), C 2 (t 0 ) > 0 are those numbers given in Theorem 2.1. On the other hand, it is immediate that and similarly to (2.13), we have Hence, by letting t 0 be sufficiently small, we have from (2.14), (2.15) and (2.16) that In a similar manner, we can show the relation (2.9).
The following results give conditions on the initial data in order to obtain a solution which is more regular with respect to time or space variables.
then the solution x of (2.7) satisfies for each T ≥ 0.

Proof.
In correspondence with the time t 0 ∈ [0, r] in the proof of Theorem 2.2, let us consider the following closed subspace Σ 0 of W 1,2 ([−r, t 0 ], V ) such that x(0) = φ 0 + Dφ 1 and x 0 = φ 1 for x ∈ Σ 0 . Define the same mapping S as in (2.10) on Σ 0 by We shall show that S is a contraction from Σ 0 into itself for sufficiently small t 0 > 0.
and the equation (2.7) has a unique solution x such that and, similarly, Moreover, we can proceed as in the proof of Theorem 2.2 to get a solution x in [−r, T ] and the relation (2.18) or (2.19) for all T ≥ 0. The proof is easily completed.
Let x(t), t ≥ −r (and y(t), t ≥ 0) denote the unique solution of system (2.6) with initial x(0) = φ 0 + Dφ 1 and x 0 = φ 1 , φ = (φ 0 , φ 1 ) ∈ H. We define a family of operators S(t) : H → H, t ≥ 0, by (2.20) Theorem 2.4. The family t → S(t) is a strongly continuous semigroup on H, i.e., Then we have by virtue of Theorem 2.2 that for any t ∈ [0, T ], where C 1 , C 2 (T ) > 0, which shows (i). To show (ii), it is easy to see from (2.7) that for any t ≥ s ≥ 0, On the other hand, for t ≥ s, which immediately implies that for any t ≥ s ≥ 0, Thus, by the uniqueness of solutions to (2.22), it follows that Finally, to show (iii) we notice that The proof is complete.
The following theorem whose finite dimensional version was established in Ito and Tarn [6] gives a complete description of the generator A of semigroup e tA , t ≥ 0.
Proof. Let φ = (φ 0 , φ 1 ) ∈ D(A). Then it is known by Theorem 2.3 that the corresponding solution x of (2.6) with initial φ is in C 1 ([0, T ], H) for each T > 0. Hence, by the equality (2.20) we have On the other hand, as x ∈ W 1,2 ([−r, T ], V ) for each T > 0 according to Theorem 2.3, we can write for θ ∈ [−r, 0] and t > 0 that Therefore, it is easy to get from (2.25) that The conclusion follows from (2.24) and (2.26) and the proof is thus complete.

Semigroup and Resolvent
For each λ ∈ C, we define a linear operator D(e λ· ) : V → V by D(e λ· )x = D(e λ· x) for any x ∈ V.
Then it is easy to see that D(e λ· ) ∈ L (V ). Indeed, for any x ∈ V , In a similar way, one can show that F (e λ· ) ∈ L (V, V * ). For each λ ∈ C, we define a linear operator ∆(λ) : The following proposition can be used to establish useful relations between the resolvent ∆(λ, A, D, F ) −1 and resolvent (λI − A) −1 of A. and Proof. The equation (3.2) can be equivalently written as and It is easy to see that (3.7) is equivalent to (3.3). Hence, if (3.2) holds, we get φ 1 (0) ∈ V and by virtue of (3.6) and (3.3), we have which is the equality (3.4).

Spectrum and Stationary Solution
First, let us consider the following deterministic functional differential equation of neutral type in V * , (4.1) where α 1 , α 2 ∈ R and β, γ ∈ L 2 ([−r, 0]; R). By virtue of (2.20) and Theorem 2.5, the equation (4.1) can be equivalently lifted up into a deterministic equation without time delay where A is the generator given in Theorem 2.5 and Y (t) = (y(t), y t ) for all t ≥ 0. On the other hand, the characteristic operator ∆(λ) defined in (3.1) is given in this case by (4.5) By using Propositions 3.1-3.5, one can obtain the following result whose proof is similar to that one of Theorem 3.9 in [4].
(ii) σ r (A) = σ r (∆) = Γ r ; (iii) If α 1 = 0, γ(·) ≡ 0 and β(·) ≡ 0, then the equation (4.1) reduces to a simple form Let us suppose at present that A is some linear operator, e.g., Laplace operator, in conjunction with the form a(·, ·) in (2.1) to generate a compact semigroup. It was shown, however, by Di Blasio, Kunisch and Sinestrari [4] that the associated C 0 -semigroup e tA , t ≥ 0, is generally not compact or even not eventually norm continuous, as shown by Jeong [7]. A direct consequence of this fact is that we cannot use the well-known spectral mapping theorem to establish stability, based on the spectrum knowledge of A for system (4.6).
Bearing the above statement in mind, let us consider the following version of equation (4.1) with distributed delay by taking α 1 = α 2 = 0, It was shown by Liu [11] that when the weight functions γ(·), β(·) satisfy Now let us consider the following stochastic functional differential equation of neutral type with additive noise, where b ∈ H and w(·) is a standard real-valued Brownian motion. We can re-write (4.9) as a stochastic differential equation without time delay in H, where A is the generator given in Theorem 2.5, B = (b, 0) ∈ H and Y (t) = (y(t), y t ) for all t ≥ 0. For equation (4.10), if we can find conditions by showing sup{Re λ : λ ∈ σ(A)} < 0, (4.11) to secure an exponentially stable semigroup e tA , t ≥ 0, then we will obtain a unique stationary solution to the equation (4.9) (cf, e.g., Prévôt and Röckner [12]). Then there exists a unique stationary solution for the equation (4.9).
Proof. Note that from Proposition 4.1 we have σ(A) ⊂ Γ 0 ∪ Γ 1 . We shall show that under the assumptions in this proposition, there is a constant µ > 0 such that Re λ ≤ −µ for all λ ∈ Γ 0 ∪ Γ 1 and hence for all λ ∈ σ(A).
First, for elements in Γ 0 , if there exist a sequence {λ n } ⊂ C such that Re λ n ≥ 0 or Re λ n → 0 as n → ∞, then by using (4.3), (4.4) and Dominated Convergence Theorem, we have 1 ≤ lim sup which, once again, yields a contradiction. Combining the above results, we thus obtain that Re λ ≤ −µ for some µ > 0 and all λ ∈ σ(A).
Therefore, the solution semigroup e tA , t ≥ 0, is exponentially stable. This fact further implies the existence of a unique stationary solution to (4.7). The proof is complete.
We can re-write (4.13) as a stochastic neutral initial boundary problem Then for any random initial (φ 0 , φ 1 ) ∈ H, there exists a unique strong solution to (4.13) defined in [0, ∞). Furthermore, by applying the results derived in this section to (4.13), one can obtain a unique stationary solution. In fact, note that A = ∂ 2 /∂x 2 is a self-adjoint and negative operator in H and its spectrum satisfies σ(A) = σ p (A) ⊂ (−∞, −c 0 ] for some c 0 > 0. Then by Proposition 4.2 and a direct computation, we obtain that if |α| < 1 r and |κ| ≤ e rµ (1 − |α|r)/r if µ ≤ 0, (1 − |α|r)/r if µ > 0, the associated solution semigroup of (4.13) is exponentially stable. Moreover, the lift-up system (4.10) of equation (4.13) in this case has a unique stationary solution.