Representations of mild solutions of

The main object of this paper is to give a representation of the covariance operator associated to the mild solutions of time-varying, linear, stochastic equations in Hilbert spaces. We use this representation to obtain a characterization of the uniform exponential stability of linear stochastic equations with periodic coecien ts.


Preliminaries
Let H, V be separable real Hilbert spaces and let L(H, V ) be the Banach space of all bounded linear operators from H into V (If H = V then L(H, V ) not = L(H)).We write ., .for the inner product and .for norms of elements and operators.We denote by a ⊗ b, a, b ∈ H the bounded linear operator of L(H) defined by a ⊗ b(h) = h, b a for all h ∈ H.The operator A ∈ L(H) is said to be nonnegative and we write A ≥ 0, if A is self-adjoint and Ax, x ≥ 0 for all x ∈ H.We denote by E the Banach subspace of L(H) formed by all self-adjoint operators, by L + (H) the cone of all nonnegative operators of E and by I the identity operator on H. EJQTDE, 2004 No. 4, p. 1 Let P ∈ L + (H) and A ∈ L(H).We denote by P 1/2 the square root of P and by |A| the operator (A * A) 1/2 .We put A 1 = T r(|A|) ≤ ∞ and we denote by C 1 (H) the set {A ∈ L(H)/ A 1 < ∞} (the trace class of operators).
If A ∈ C 1 (H) we say that A is nuclear and it is not difficult to see that A is compact.
The definition of nuclear operators introduced above is equivalent with that given in [6] and [9].
It is known (see [6]) that C 1 (H) is a Banach space endowed with the norm . 1 and for all A ∈ L(H) and B ∈ C 1 (H) we have AB, BA ∈ C 1 (H).
C 2 (H) is a Hilbert space with the inner product A, B 2 = T rA * B ( [5]).We denote by H 2 the subspace of C 2 (H) of all self-adjoint operators.Since H 2 is closed in C 2 (H) with respect to . 2 we deduce that it is a Hilbert space, too.It is known (see [9]) that for all A ∈ C 1 (H) we have For each interval J ⊂ R + (R + = [0, ∞)) we denote by C s (J, L(H)) the space of all mappings G(t) : J → L(H) that are strongly continuous.
If E is a Banach space we also denote by C(J, E) the space of all mappings G(t) : J → E that are continuous.
In the subsequent considerations we assume that the families of operators {A(t)} t∈R + and {G i (t)} t∈R + , i = 1, ..., m satisfied the following hypotheses: P1 : a) A(t), t ∈ [0, ∞) is a closed linear operator on H with constant domain D dense in H.
If we denote by U n (t, s) the evolution operator generates by A n (t), then it is known (see [4]) that for each x ∈ H, one has lim n→∞ U n (t, s)x = U (t, s)x uniformly on any bounded subset of {(t, s); t ≥ s ≥ 0}.
Let (Ω, F, F t , t ∈ [0, ∞), P ) be a stochastic basis and L 2 s (H) = L 2 (Ω, F s , P, H).We consider the stochastic equation where the coefficients A(t) and G i (t) satisfy the hypothesis P1, P2 and w i 's are independent real Wiener processes relative to F t .Let us consider T > 0. It is known (see [2]) that (2) has a unique mild solution in C([s, T ]; L 2 (Ω; H)) that is adapted to F t ; namely the solution of We associate to (2) the approximating system: where A n (t), n ∈ N are the Yosida approximations of A(t).
By convenience, we denote by y(t, s; ξ) (resp.y n (t, s; ξ)) the solution of (2) (resp.( 4)) with the initial condition y(s) = ξ (resp.Now we consider the following Lyapunov equation: According with [4], we say that Q is a mild solution on an interval If A n (t), n ∈ N are the Yosida approximations of A(t) then we introduce the approximating equation: Then there exists a unique mild (resp.classical) solution They are given by and for each x ∈ H, Q n (s)x → Q(s)x uniformly on any bounded subset of [0, T ].Moreover, if we denote these solutions by Q(T, s; R) and respectively Q n (T, s; R) then they are monotone in the sense that EJQTDE, 2004 No. 4, p. 4 For all n ∈ N and t ≥ 0 we consider the mapping L n (t) : It is easy to verify that L n (t) ∈ L(H 2 ) and the adjoint operator L * n (t) is the linear and bounded operator on H 2 given by for all t ≥ 0, P ∈ H 2 .
Lemma 3 Now we use P1 (the statements b) and c)) and we deduce that there exist δ < 0, α ∈ (0, 1), M > 0 and N > 0 such that we have ).We only have to prove that From Lemma 1 [7] and since 2 ) for all P ∈ H 2 and i = 1, ..., m.For s ≥ 0, P ∈ H 2 fixed and for every i ∈ {1, ..., m} we have As t → s, we obtain lim If s = 0 we only have the limit from the right.
If E is a Banach space and L ∈ C s ([0, ∞), L(E)), we consider the initial value problem continuously differentiable on [s, T ] and satisfies (12).The following results have a standard proof (see [11]).Lemma 4 For every x ∈ E the initial value problem (12) has a unique classical solution v.
We define the "solution operator" of the initial value problem (12) by where v is the solution of (12).
Let us denote by I the identity operator on E.
Proposition 5 For all 0 ≤ s ≤ t ≤ T , V (t, s) is a bounded linear operator and I in the uniform operator topology for all 0 ≤ s ≤ t ≤ T.
The operator V (t, s) is called the evolution operator generated by the family L. Let us consider the equation on H 2 , where L n is given by (10).From Lemma 3, Lemma 4 and the above proposition it follows that the unique classical solution of ( 13) is where U n (t, s) ∈ L(H 2 ) is the evolution operator generated by L n and Integrating from s to t, we have Let Q n (t, s; R) be the unique classical solution of ( 7) such as Q n (t) = R, R ≥ 0. We have 14) and ( 15) By the Uniform Boundedness Principle there exists l T > 0 such that L * n (t)P ≤ l T P for all t ∈ [0, T ], P ∈ L(H) and we obtain Now we use Gronwall's inequality and we get From Proposition 5 and (1) we deduce that for all R ∈ H 2 the map for all α, β ∈ R + and R, S ∈ H 2 , R, S ≥ 0.
3 The covariance operator of the mild solutions of linear stochastic differential equations and the Lyapunov equations We denote by E(ξ ⊗ ξ) the bounded and linear operator which act on H given by E(ξ ⊗ ξ)(x) = E( x, ξ ξ).
The operator E(ξ ⊗ ξ) is called the covariance operator of ξ.

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No. 4, p. 8 Lemma 6 [10] Let V be another real, separable Hilbert space and s (H) is the classical solution of ( 4) then E[y n (t, s; ξ)⊗y n (t, s; ξ)] is the unique classical solution of the following initial value problem Proof.Let u ∈ H and T ≥ 0, fixed.We consider the function Using Ito's formula for F and y n (t, s; ξ) we obtain for all 0 Taking expectations, we have According with lemmas L.3, L.4 and the statements of the last section, the equation ( 19) has a unique classical solution U n (t, s)E (ξ ⊗ ξ) in H 2 and we have We note that U n (t, s) is the evolution operator generated by L n .Then From (20) and the last equality we obtain Since there exists l T > 0 such that L n (t) ≤ l T for all t ∈ [0, T ] and U n (t, s)E (ξ ⊗ ξ) , P n (t) ∈ E we can use the Gronwall's inequality to deduce that for all t ∈ [s, T ] .Since T is arbitrary we obtain the conclusion.
The following theorem gives a representation of the covariance operator associated to the mild solution of (2), by using the mild solution of the Lyapunov equation (5).
Theorem 8 Let V be another real separable Hilbert space and B ∈ L(H, V ).If y(t, s; ξ), ξ ∈ L 2 s (H) is the mild solution of (2) and Q(t, s, R) is the unique mild solution of ( 5) with the final value Proof.a) Let u ∈ H, ξ ∈ L 2 s (H) and y n (t, s; ξ) be the classical solution of (4).By (21) we obtain successively As n → ∞ we get the conclusion.Indeed, since Q n (t, s; u⊗u) → n→∞ Q(t, s; u⊗ u) in the strong operator topology (Lemma 2) then it is not difficult to deduce from Lemma 1 [7] that Since B p ∈ H 2 and B p ≥ 0 we deduce from (17) and the hypothesis that is continuous.On the other hand we have for all r ∈ [s, t] Thus it follows that r → φ n,s,t (r) is a Borel measurable and nonnegative function defined on [s, t] and bounded above by a continuous function, namely r → Q n (t, r; B * B)G i (r)U n (r, s)x, G i (r)U n (r, s)x .
From the Monotone Convergence Theorem we can pass to limit p → ∞ in (26) and we have where the integral is in Lebesgue sense.From (26) it follows EJQTDE, 2004 No. 4, p. 13 Since B 1 = {x ∈ H, x = 1} is separable [1], then there exists a net {y n } n∈N ⊂ B 1 which is dense in B 1 and is Lebesgue integrable.By (27) we have Using the Gronwall's inequality, we get By ( 24), ( 25) and since T r is continuous on C 1 (H) we obtain (23).As n → ∞ we obtain the conclusion.
We note that if A is time invariant (A(t) = A, for all t ≥ 0), then the condition P1 can be replaced with the hypothesis H0 : A is the infinitesimal generator of a C 0 -semigroup and arguing as above we can prove the following result.
Proposition 9 If P2 and H0 hold, then the conclusions of the above theorem stay true.Particularly, if we replace P2 with the condition G i ∈ L(H), i = 1, ..., m the statement b) becomes: EJQTDE, 2004 No. 4, p. 14 then we don't need to work with the approximating systems and all the main results of the last two sections (including this) can be reformulated ( and proved) adequately.So, we have the following proposition:

It is not difficult to see that if the coefficients of the stochastic equation (2) verify the condition
Proposition 10 If the assumption H1 holds then the statements a) and b) of the Theorem 8 are true.

The solution operators associated to the Lyapunov equations
Let Q(T, s; R), R ∈ L + (H), T ≥ s ≥ 0 be the unique mild solution of the Lyapunov equation ( 5), which satisfies the condition Using the Gronwall's inequality we deduce K(s) = 0 for all s ∈ [0, T ] and the conclusion follows.Similarly we can prove b).
The following lemma is known [13].
where I is the identity operator on H.
We introduce the mapping T (t, s) : for all t ≥ s ≥ 0. The mapping T (t, s) has the following properties: 5. For all R ∈ E and x ∈ H we have (It follows from the Theorem 8 and from the definition of T (t, s)(R).) 6. T (t, s) is a linear and bounded operator and T (t, s) = T (t, s)(I) .
From 5. we deduce that T (t, s) is linear.If R ∈ E, we use (29) and we get Thus T (t, s) is bounded.Using 4. and Lemma 12 we obtain the conclusion.
It follows from Lemma 11 and the definition of T (t, s).
Now, we use the Gronwall's inequality and we obtain the conclusion.The proof for the approximating equation ( 7) goes on similarly.5 The uniform exponential stability of linear stochastic system with periodic coefficients We need the following hypothesis: P3 There exists τ > 0 such that A(t) = A(t + τ ), G i (t) = G i (t + τ ), i = 1, ..., m for all t ≥ 0.
It is known (see [12], [3]) that if P1, P3 hold then we have Definition 14 We say that ( 2) is uniformly exponentially stable if there exist the constants M ≥ 1, ω > 0 such that E y(t, s; x) 2 ≤ M e −ω(t−s) x 2 for all t ≥ s ≥ 0 and x ∈ H.

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No. 4, p. 17 Proposition 15 If P3 holds and Q(t, s; R) is the unique mild solution of ( 5) such that Q(t) = R, R ≥ 0, then for all t ≥ s ≥ 0 and x ∈ H we have a) Proof.a) Since P3 holds we deduce from (30) and Lemma 2 that Now, we can use (8) and Gronwall's inequality to deduce the conclusion.The statement b) follows from a) and from the definition of the operator T (t, s).Using b) and the property 7. of the operator T (t, s) we obtain c).d) follows from Theorem 8 and a).
Next remark is a consequence of the Theorem 8 and of the property 6. of the operator T (t, s).

Remark 16
The following statements are equivalent: a) the equation ( 2) is uniformly exponentially stable b) there exist the constants M ≥ 1, ω > 0 such that Q(t, s; I) ≤ M e −ω(t−s) I for all t ≥ s ≥ 0, c) there exist the constants M ≥ 1, ω > 0 such that T (t, s) ≤ M e −ω(t−s) .Now we establish the main result of this section.Proof.The implication "a) ⇒ b)" is a consequence of the Definition 14.We will prove "b) ⇒ a)".Since b) holds we deduce that for all ε > 0 there exists n(ε) ∈ N such that E y(nτ, 0; x) 2 < ε for all n ≥ n(ε) and x ∈ H, x = 1.By (29) we get E y(nτ, 0; x) 2 = T (nτ, 0)(I)x, x .
For α = γ we deduce by Proposition 15 that Using Lemma 2 and Gronwall's inequality it is easy to deduce that there exists M τ > 0 such that Q(t, s; I) ≤ M τ for all 0 ≤ s ≤ t ≤ τ .
It is known that the operator A is self adjoint, Ay, y ≤ −π 2 y 2 for all y ∈ D(A) and A is the infinitesimal generator of an analytic semigroup S(t) [11], which satisfies the following inequality: We use Proposition 19 and Remark (18) to deduce that the solution of (32) is uniformly exponentially stable.EJQTDE, 2004 No. 4, p. 21
Then {B p } p∈N converges in the strong operator topology to the operator B * B ∈ L + (H).By Lemma 2 we deduce that the sequence {Q n (t, s; B p } p∈N * is increasing (Q n (t, s; B p ) ≤ Q n (t, s; B * B) for all p ∈ N * ) and consequently it converges in the strong operator topology to the operatorQ n (t, s) ∈ L + (H).If U n (t,s) is the evolution operator relative to A n (t), we have for all x ∈ H Q n (t, s; B p )x, x = B p U n (t, s)x, U n (t, s)x (26) b) Let ξ ∈ L 2 s (H) and n ∈ N. It is sufficient to prove that E By n (t, s; ξ) 2 = T rQ n (t, s; B * B)E (ξ ⊗ ξ) .(23)ByLemma6wehaveE By n (t, s; ξ) 2 = BE[y n (t, s; ξ) ⊗ y n (t, s; ξ)]B * 1 .(24)If{e i } i∈N * is an orthonormal basis in V then we deduce from (a) BE[y n (t, s; ξ) ⊗ y n (t, s; ξ)]B * 1 = ∞ i=1 E[y n (t, s; ξ) ⊗ y n (t, s; ξ)]B * e i , B * e i = ∞ i=1 T rQ n (t, s; B * e i ⊗ B * e i )E (ξ ⊗ ξ) .Since B * e i ⊗ B * e i ∈ H 2 and B * e i ⊗ B * e i ≥ 0, i ∈ N * , we have by (18) BE[ y n (t, s; ξ) ⊗ y n (t, s; ξ)]B * p→∞ T rQ n (t, s; p i=1 B * e i ⊗ e i B)E (ξ ⊗ ξ) .* e i ⊗ e i B is increasing and bounded above: