ELECTRONICCOMMUNICATIONSinPROBABILITY INVARIANT MEASURES FOR STOCHASTIC CAUCHY PROBLEMS WITH ASYMPTOTICALLY UNSTABLE DRIFT SEMIGROUP

We investigate existence and permanence properties of invariant measures for abstract stochastic Cauchy problems of the form governed by the generator A of an asymptotically unstable C 0 -semigroup on a Banach space E . Here f ∈ E is ﬁxed, W H is a cylindrical Brownian motion over a separable real Hilbert space H , and B : H → E is a bounded operator. We show that if c 0 6⊆ E , such invariant measures fail to exist generically but may exist for a dense set of operators B . It turns out that many results on invariant measures which hold under the assumption of uniform exponential stability of S break down without this assumption.


Introduction
Let A be the infinitesimal generator of a C 0 -semigroup S = {S(t)} t 0 on a real Banach space E and let W H = {W H (t)} t 0 be a cylindrical Brownian motion over a separable real Hilbert space H. In this note we study invariant measures for the stochastic abstract Cauchy problem of the form where f ∈ E is a fixed vector and B ∈ L (H, E) is a bounded operator. We are interested in the situation where the semigroup S fails to be uniformly exponentially stable and intend to answer such questions as for 'how many' operators B an invariant measure exists and what can be said about its properties. If the problem (1.1) with initial condition U (0) = x has a weak solution U = U x , then this solution is unique up to modification and it is given explicitly by the stochastic convolution integral We refer to [11] for more details and unexplained terminology. A Radon probability measure µ on E is called an invariant measure for the problem (1.1) if for all t 0 and all bounded real-valued Borel functions ϕ ∈ C b (E) we have where P (t)ϕ ∈ C b (E) is given as As is well known [3,4,11] a unique invariant measure µ exists if a weak solution U x exists for some (all) x ∈ E and the semigroup generated by A is uniformly exponentially stable. It is obtained as the weak limit µ = lim t→∞ µ(t), where µ(t) is the distribution of U (t) := U 0 (t) as given by (1.2) with initial value x = 0. If the operators S(t) are compact for all t > 0, the existence of a nondegenerate invariant measure for the problem (1.1) with f = 0 implies that the semigroup S is uniformly exponentially stable [8,Theorem 2.6]. Recall that an invariant measure µ is said to be nondegenerate if µ(O) > 0 for every nonempty open set O ⊆ E. The following example, adapted from [4,Chapter 7], shows that in general the uniform exponential stability is by no means a necessary condition for the existence of a nondegenerate invariant measure, even if E is a Hilbert space. Although more refined examples will be presented below, this one is included because of its particular simplicity.
Example 1. Let H = E = L 2 (R + ) and let S be the rescaled left translation semigroup defined by Define, for n = 1, 2, . . . , the functions f n ∈ E by f n (x) := p n (x)e −x 2 , where the polynomials p n are chosen in such a way that (f n ) n 1 is an orthonormal basis for E. The fact that such polynomials exist can be deduced, e.g., from [7, Theorem 9.1]. Choose constants λ n > 0 such that n 1 λ 2 n < ∞. The operator B := n 1 λ n B n , where B n f := [f, f n ] E f n , is well defined, Hilbert-Schmidt, and has dense range.
For t 0 define the operators Q t ∈ L (E) by A simple computation using the orthonormality of the f n gives If we let λ n → 0 fast enough the right hand side is finite and we infer that By [3,Theorem 11.7] this implies that the stochastic Cauchy problem admits an invariant measure µ ∞ whose covariance operator is given by the strong operator limit Q ∞ = lim t→∞ Q t . Since B (and therefore also BB * ) has dense range, it follows from [6, Lemma 5.2] that the operator Q ∞ has dense range as well. By standard results on Gaussian measures this implies that µ ∞ is nondegenerate.

Linear equations with additive noise
In this section we consider the problem (1.1) for f = 0, that is, we study the linear stochastic Cauchy problem We begin our discussion with recalling some definitions from the theory of C 0 -semigroups. Our notations are standard and may be looked up in e.g. [5]. The spectral bound and growth bound of A are denoted by s(A) and ω 0 (A), respectively. The abscissa of uniform boundedness of the resolvent of A is defined as One has s(A) s 0 (A) ω 0 (A), and both inequalities may be strict. As a consequence of the Pringsheim-Landau theorem one has s(A) = s 0 (A) for positive C 0 -semigroups on Banach lattices E. The celebrated Gearhart-Herbst-Prüss theorem asserts that for C 0 -semigroups on Hilbert spaces E one has s 0 (A) = ω 0 (A). Let H be a separable real Hilbert space with orthonormal basis (h n ) n 1 . In the applications below, H will be either H or L 2 (R + ; H). Let (γ n ) n 1 be a sequence of independent standard Gaussian random variables on a probability space (Ω, P). A bounded operator R ∈ L (H , E) is called γ-radonifying if the sum n 1 γ n Rh n converges in L 2 (Ω; E). The space γ(H , E) of all γ-radonifying operators in L (H , E) is a Banach space with respect to the norm extends to a bounded operator I B ∈ γ(L 2 (R + ; H), E).
Concerning uniqueness, in [11] it was shown that if there exists a weak * -sequentially dense subspace F of E * such that weak * -lim t→∞ S * (t)x * = 0 for all x * ∈ F , then the problem (2.1) admits at most one invariant measure. In passing we mention the following application of this result: If for all B ∈ B the problem (2.1) admits an invariant measure, then for all B ∈ L (H, E) the problem (2.1) admits at most one invariant measure. In particular, for all B ∈ B the problem (2.1) then admits a unique invariant measure.
Proof. Let x * ∈ D(A * ) be arbitrary. By the result quoted above it suffices to prove that weak * -lim t→∞ S * (t)x * = 0. Let x ∈ E be arbitrary. Choose B ∈ B and h ∈ H such that Bh = x. We claim that lim t→∞ x, S * (t)x * = 0. For y * ∈ E * let f y * : R + → H be defined as f y * (t) = x, S * (t)y * . By the assumptions and Proposition 2 we have that for all y * ∈ E * , f y * ∈ L 2 (R + ) and Hence by the Cauchy-Schwartz inequality, g ∈ L 1 (R + ). From lim t,s→∞ it follows that the limit L = lim t→∞ g(t) exists. If L > 0, there exist ε > 0 and T > 0 such that for all t T we have g(t) ε, which contradicts the fact that g ∈ L 1 (R + ). We conclude that lim Since x ∈ E was arbitrary, this proves that weak * -lim t→∞ S * (t)x * = 0.
A subset of a topological space is called residual if it is the intersection of a countable family of open dense sets. By the Baire category theorem, every residual set in a complete metric space is dense.
Theorem 4. Let E be a Banach space not containing a closed subspace isomorphic to c 0 and let I be an operator ideal in L (H, E). Let A be the generator of a C 0 -semigroup on E. If s 0 (A) > 0, then the set J of all B ∈ I such that the problem (2.1) does not admit an invariant measure is residual in I . If, moreover, the finite rank operators are dense in I , then the finite rank operators of J are dense in I .
Proof. For k = 1, 2, . . . let . We shall prove that each G k is open and dense in I . The residual set G := k 1 G k is precisely the set of all B ∈ I for which I B ∈ γ(L 2 (R + ; H), E), or equivalently, for which the problem (2.1) has no invariant measure.
Both parts of the theorem apply to the operator ideal I = γ(H, E). The interest of this particular example is explained by the fact that roughly speaking there is a correspondence between operators B ∈ γ(H, E) on the one hand and E-valued Brownian motions on the other. To be more precise let W H be a cylindrical Brownian motion on a probability space (Ω, P). If (h n ) n 1 is an orthonormal basis for H, then for each B ∈ γ(H, E) and t 0 the sum W B (t) := n 1 W H (t)h n Bh n converges in L 2 (Ω; E) and the resulting process W B is an E-valued Brownian motion on (Ω, P) which is independent of the choice of (h n ) n 1 . Conversely, every E-valued Brownian motion W arises in such a way by taking for H the so-called reproducing kernel Hilbert space associated with W and for B the (γ-radonifying) inclusion mapping from H into E. Although in general the problem (1.1) may fail to have a solution even if B ∈ γ(H, E) (an example is presented in [9]), a solution always exists if in addition to B ∈ γ(H, E) we assume that either E has type 2 (in particular, if E is a Hilbert space or if E = L p for 2 p < ∞) or the semigroup generated by A is analytic. See [9,10,11] for more details. Our next aim is to exhibit an example of a C 0 -semigroup generator A on a Hilbert space E with the following properties: Its construction is based on [11, Example 4] which we recall first.

It was shown by Arendt [1] that its generator
is an arbitrary but fixed number. As is shown in [11], for every B ∈ γ(H, F ) the stochastic Cauchy problem (2.1) associated with the operator A F β := A F + β admits a unique invariant measure. Note that ω 0 (A F β ) = − 1 p + β, which is strictly positive by the choice of β. Example 6. We construct a Hilbert space semigroup with the properties (a) and (b) announced above. The idea is to embed the space F of Example 5 into a suitable weighted L 2 -space in such a way that the relevant properties of the semigroup S F β are preserved. We have contractive and dense embeddings The semigroup S F β on F defined in Example 5 extends to a C 0 -semigroup S on E. To see this, note that for f ∈ F and t 0 we have Thus S F β (t) extends to a bounded operator S(t) on E of norm S(t) e βt . In combination with the strong continuity of S on the dense subspace F of E it follows that S is a C 0 -semigroup on E. For the function f c := 1 (e t ,ce t ) with c > 1 we have Hence S(t) e βt , and we conclude that S(t) = e βt . Stated differently, the generator A of S satisfies ω 0 (A) = β. Since S is positive and E is a Hilbert space, we have s(A) = s 0 (A) = ω 0 (A) and property (a) holds. To prove that property (b) holds we make the simple observation that the dense embedding j : F → E induces a dense embedding The density of this embedding follows from the fact that the finite rank operators with values in F are dense in both spaces. Now if B ∈ γ(H, F ) is given, let µ F β denote an invariant measure of the linear stochastic Cauchy problem in F associated with A F β and B. Then the image measure µ := j(µ F β ) is an invariant measure for the linear stochastic Cauchy problem in E associated with A and jB.
Notice that in the previous example the invariant measure µ is nondegenerate whenever B has dense range. It was shown in [8] that the existence of a nondegenerate invariant measure for the problem (2.1) implies that the adjoint operator A * has no point spectrum in the closed right half-plane {Re λ 0}. If in addition we assume that the semigroup generated by A is uniformly bounded, then one has σ p (A) ∩ iR ⊆ σ p (A * ) ∩ iR and the existence of a nondegenerate invariant measure for the problem (2.1) implies that A has no point spectrum in {Re λ 0}. As was shown in [8,Theorem 4.4] this implies that there is at most one nondegenerate invariant measure for (2.1). The following example shows that for semigroups with linear growth and 0 ∈ σ p (A), a continuum of nondegenerate invariant measures may exist.
Example 7. Let 2 < p < ∞ and fix 1 p < β < 1 2 . Put and let E w := L 2 (1, ∞; w(x) dx). The space F of Example 5 is continuously and densely embedded in E w , and the semigroup S F β extends to a C 0 -semigroup S w on E w . We check that S w (t) = 1 2 t 2 + 1 + 1 2 t t 2 + 4 1/2 , t 0, so S w grows linearly. Indeed, for f ∈ E w and t 0, It is easy to compute that the function ξ → 1 + log 2 ξ 1 + (log ξ − t) 2 attains its maximal value on (e t , ∞) at the point and that the maximum equals α t = 1 2 t 2 + 1 + 1 2 t t 2 + 4. Hence S w (t)f 2 α t f 2 . For t 0 and ε > 0, let f t,ε := 1 (ξt,ξt+ε) . A straightforward computation shows that An elementary computation shows that b ∈ E w and that S w (t)b = b. It follows that b ∈ D(A w ) and A w b = 0. Since b is nonzero, this shows that 0 ∈ σ p (A w ). As in Example 6, in E w the problem (2.1) admits an invariant measure for every operator B ∈ γ(H, F ), where we identify γ(H, F ) with a dense subspace of γ(H, E w ) = L 2 (H, E w ), and a nondegenerate invariant measure exists whenever B has dense range. If µ is such a measure, then for all c ∈ R the translated measure µ c (C) := µ(C + cb) is a nondegenerate invariant measure for (2.1); here b is the function defined in (2.2). Thus, a continuum of such measures exists.

The inhomogeneous problem with additive noise
Next we consider the inhomogeneous problem (1.1), where f ∈ E is a fixed vector. Following the arguments of [3, Propositions 11.2 and 11.5] one sees that a Radon probability measure µ on E is invariant if and only if there exists a stationary solution V of (1.1) (on a possibly larger probability space) such that µ is the distribution of V (t) for all t 0. If µ has a first moment, i.e., if there exists an element m(µ) ∈ E such that for all x * ∈ E * we have x * ∈ L 1 (E, µ) and then by applying x * on both sides of the identity and taking expectations, the Hahn-Banach theorem shows that m(µ) satisfies the identity The following proposition relates invariant measures with first moments of the problem (1.1) to the invariant measure of the homogeneous problem (2.1) with f = 0. Its proof follows readily from the identity on [3, p. 122], which extends without change to the present Banach space setting. We proceed with a Hilbert space example which shows that even if s(A) > 0 it may happen that the inhomogeneous problem (1.1) has an invariant measure for all choices of f ∈ E.
Example 9. We show that Example 6 displays the stated properties. For the proof we fix B ∈ γ(H, F ), where F is the space of Example 5. Let U be a stationary solution of the problem (2.1) in E, which exists according to the facts proved in Example 6 and the observations made above. Let f ∈ E be arbitrary and define for c ∈ R, Observe that so that, by integration by parts, for all T > 1 we obtain Since b(x) := x −β belongs to E it follows that a c = cb+a 0 ∈ E for all c ∈ R. By an elementary computation we check that We infer that a c ∈ D(A) and −Aa c = f . This shows that f ∈ ran (A). Thus by Proposition 8, the inhomogeneous problem admits a stationary solution and hence an invariant measure.
More can be said in the above example. If µ is an invariant measure for the inhomogeneous problem with first moment m(µ), then necessarily m(µ) is given by the right hand-side of (3.2) for some c ∈ R. Indeed, since m(µ) satisfies (3.1) it suffices to show that the only elements b ∈ E satisfying b − S(t)b = 0 for all t 0 are given by b(x) = cx −β for some c ∈ R. Since by assumption for all t 0 we have e βt b(xe t ) = b(x) for almost all x > 1, it follows that for all τ 1 we have τ β b(xτ ) = b(x) for almost all x > 1. Since (x, τ ) → τ β b(xτ ) is measurable, Fubini's theorem yields that for almost all x > 1 the equality τ β b(xτ ) = b(x) holds for almost all τ 1. Consider a fixed x = x 0 with this property. Then, with θ := x 0 τ , we obtain b(θ) = cθ −β for almost all θ x 0 , where c = b(x 0 )x β 0 . By letting x 0 ↓ 1 we infer that c is independent of x 0 and that b(x) = cx −β for almost all x > 1. Summarizing what we proved so far, we see that for every f ∈ E the set of means of all invariant measures having a first moment is a one-parameter family parametrized by the real parameter c. Note that even for the homogeneous problem (2.1), invariant measures may exist whose first moment does not exist. Indeed, any weak limit of convex combinations of invariant measures is invariant as well. Returning to Example 6, if µ is an invariant measure of the homogeneous problem with mean zero and if we put µ n (C) := µ(C + 2 n b) and ν := n 1 2 −n µ n , where b ∈ E is defined by (2.2), then ν is an invariant measure and Let U denote the solution of (1.1) with initial condition U (0) = 0; thus If ω 0 (A) < 0, then this proposition shows that the measures µ(t) converge weakly to an invariant measure µ of (1.1). We will show next that, even in the presence of invariant measures, this convergence may fail if the semigroup has linear growth.
Example 11. We continue with Example 7 and show that for certain functions f an invariant measure for (1.1) exists, although the integrals t 0 S w (s)f ds fail to converge in E w as t → ∞. An appeal to Proposition 10 then shows that the measures µ(t) fail to converge weakly. Then |f (x)| x −β , so f ∈ E w . The function x −β x 1 ξ β−1 f (ξ) dξ = x −β log log log x for x > e e 0 for 1 < x e e is a member of E w and in the same way as in Example 9 we infer that f ∈ ran (A w ). Due to Proposition 8 and the existence of an invariant measure with first moment for the homogeneous problem, there exists an invariant measure for (1.1) with A replaced by A w . For all t e and x > 1, x −β (log(s + log x))(s + log x) ds = x −β log log(t + log x) log((e − log x) + + log x) .
From this we infer that the integrals t 0 S w (s)f ds diverge in E w as t → ∞. Acknowledgment -Proposition 3 was obtained in a discussion with Mark Veraar.