The stochastic Cauchy problem driven by a cylindrical Levy process

In this work, we derive sufficient and necessary conditions for the existence of a weak and mild solution of an abstract stochastic Cauchy problem driven by an arbitrary cylindrical Levy process. Our approach requires to establish a stochastic Fubini result for stochastic integrals with respect to cylindrical Levy processes. This approach enables us to conclude that the solution process has almost surely scalarly square integrable paths. Further properties of the solution such as the Markov property and stochastic continuity are derived.


Introduction
Cylindrical Lévy processes naturally extend the class of cylindrical Brownian motions and cover many examples of Lévy-type noise considered in the literature. A general framework of cylindrical Lévy processes in Banach spaces has been recently introduced by Applebaum and Riedle in [3]. Stochastic integration of deterministic operator-valued integrands with respect to cylindrical Lévy processes is developed in [18]. Based on this integration theory, the authors of the present article have developed a general theory of weak and mild solutions for the stochastic Cauchy problem driven by an arbitrary cylindrical Lévy process in [13].
More specifically, the stochastic Cauchy problem is a linear evolution equation driven by an additive noise of the form for all t ∈ [0, T ]. (1.1) Here, L is a cylindrical Lévy process on a separable Hilbert space U , the coefficient A is the generator of a strongly continuous semigroup (T (t)) t 0 on a separable Hilbert space V and B is a linear, bounded operator from U to V . In this general setting, we present sufficient conditions for the existence of a stationary solution of (1.1) and show that these conditions are also necessary if the semigroup is stable, in which case the invariant measure is unique. If the semigroup has a spectral decomposition, we significantly simplify these conditions without assuming any further restrictions on the driving cylindrical Lévy process. For finite dimensional Lévy processes the existence of invariant measures and its relation to operator self-decomposibilty has been studied by Jurek [11], Jurek and Vervaat [12], Sato and Yamatado [20], [21], Wolfe [23], and Zabczyk [24]. The case of an infinite dimensional Lévy process in a Hilbert space was studied by Chojnowska-Michalik in [8] and [9]. To the best of our knowledge, the case of a cylindrical Lévy process was only considered for a specific example of a cylindrical Lévy process and under further assumptions on the semigroup in [16]. The assumptions in [16] enable the authors to reduce the problem of the existence of an invariant measure to the analogue problem in one dimension. The general setting in the present paper clearly excludes this approach. Our results in the general framework can easily be applied to the example considered in [16], and we are not only able to cover these results but even improve them; see Example 4.5.
In our general framework, having in hand the integration theory developed in [18] and the probabilistic description of cylindrical Lévy processes by their characteristics introduced in [17], we are able to generalise the conditions from the case of a genuine Lévy process in [8] to the cylindrical setting. The fact, that cylindrical processes are generalised processes not attaining values in the underlying Hilbert space, prevents us from directly adopting the methods from the classical case. Instead, we exploit some of the methods developed in [13] and [18] such as tightness of finite-dimensional approximations. As in the classical setting, the derived conditions are rather difficult to verify in the general case but can be significantly simplified in typical cases such as the heat equation; see [9] for the classical case. Again, the fact that cylindrical processes are generalised processes not attaining values in the underlying Hilbert space requires some more advanced arguments.
Our article begins with Section 2 where we fix most of our notations and introduce cylindrical Lévy processes and their stochastic integral. In section 3 we briefly demonstrate the equivalence of the existence of a stationary solution of (1.1) and of an invariant measure for the corresponding Mehler semigroup. Our first main result of this article is presented in this section, which provides sufficient conditions for the existence of a stationary solution in terms of the characteristics of the driving cylindrical Lévy process. Our second main result is in the final Section 4, where we significantly simplify the conditions from Section 3 if the semigroup has a spectral decomposition. We finish the article by demonstrating our results in some examples.

Preliminaries
Let U and V be real separable Hilbert spaces with norms · and inner products ·, · . Let (e k ) k∈AE and (h k ) k∈AE be the orthonormal bases of U and V , respectively. We identify the dual of a Hilbert space by the space itself. The space of all linear, bounded operators from U to V is denoted by L(U, V ), equipped with the operator norm · op . By B U , we denote the open unit ball in U , that is, B U := {u ∈ U : u < 1}. The Borel σ-algebra of U is denoted by B(U ) and the space of Radon probability measures on B(U ) is denoted by M(U ) and is equipped with the Prokhorov metric. We fix a filtered probability space (Ω, F, {F t } t 0 , P ), where the filtration {F t } t 0 satisfies the usual conditions of right continuity and completeness. By L 0 P (Ω; U ), we denote the space of all equivalence classes of measurable functions g : Ω → U and it is equipped with the topology of convergence in probability. The space of all regulated functions g : [0, T ] → U is denoted by R([0, T ]; U ) and it is a Banach space when equipped with the supremum norm. Recall that a function g : [0, T ] → U is called regulated if it can be uniformly approximated by step functions. In particular, a regulated function has only countable number of discontinuities; see [6, Ch.II.1.3] for this and other properties we will use.
The set of all these cylindrical sets is denoted by Z(U, Γ) and it is a σ-algebra if Γ is finite and otherwise an algebra. We write Z(U ) for Z(U, U ). A function µ : Z(U ) → [0, 1] is called a cylindrical measure, if for each finite subset Γ ⊆ U the restriction of µ on the σ-algebra Z(U, Γ) is a measure. A cylindrical measure µ is only finitely additive and is said to extend to a measure ν on B(U ) if µ = ν on Z(U ). A cylindrical measure is called finite if µ(U ) < ∞ and a cylindrical probability measure if µ(U ) = 1. A cylindrical random variable Z in U is defined as a linear and continuous map Z : U → L 0 P (Ω; Ê). Given a cylindrical random variable Z, we can define a cylindrical probability measure λ by for cylindrical sets Z = C(u 1 , ..., u n ; B). The cylindrical probability measure λ is called the cylindrical distribution of Z. The characteristic function of a cylindrical random variable Z is defined by and it uniquely determines the cylindrical distribution of Z.
A family (Z(t) : t 0) of cylindrical random variables is called a cylindrical process. By a cylindrical Lévy process we mean a cylindrical process (L(t) : t 0) such that for all u 1 , ..., u n ∈ U and n ∈ AE, the stochastic process ((L(t)u 1 , ..., L(t)u n ) : t 0) is a Lévy process in Ê n with respect to the filtration {F t } t 0 . The characteristic function of L(t) for all t 0 is given by where Ψ : U → is called the (cylindrical) symbol of L, and is given by where a : U → Ê is a continuous mapping with a(0) = 0, the mapping Q : U → U is a positive, symmetric operator and µ is a cylindrical Lévy measure on Z(U ), that is it is a cylindrical measure on Z(U ) satisfying We call (a, Q, µ) the (cylindrical) characteristics of L. Cylindrical Lévy processes are introduced in [3] and its characteristics further studied in [17].
For a function f : [0, T ] → L(U, V ) such that the map f * (·)v : [0, T ] → U is a regulated function for each v ∈ V , one can define the stochastic integral In this way, one obtains a cylindrical random variable Z A : V → L 0 P (Ω; Ê). The function f is called stochastically integrable with respect to L if for each Borel set A ∈ B([0, T ]), the cylindrical random variable Z A extends to a genuine V -valued random variable I A , that is This stochastic integration theory is developed in [18] and applied in [13] to study the weak solution of abstract stochastic Cauchy problem driven by a cylindrical Lévy process.

Invariant measure
The main aim of this paper is to study the conditions for the existence of an invariant measure for the solution of the stochastic Cauchy problem where A is the generator of a C 0 -semigroup (T (t)) t 0 on a separable Hilbert space V , the driving noise L is a cylindrical Lévy process on a separable Hilbert space U and B : is called a weak solution of (3.1) on [0, T ] if it satisfies the following: (1) Y is progressively measurable; in probability as n → ∞; (3) for every v ∈ D(A * ) and t ∈ [0, T ], P -almost surely, we have It remains to establish the following: Proof. Choose M ∈ AE such that S/M T and define the cylindrical random variable By [18,Lemma 5.4] and the semigroup property, we obtain for each v ∈ V that Theorem IV.2.5 in [22] implies that Z is induced by a genuine V -valued random variable. Hence s → T (s)B is stochastically integrable in [0, S] which completes the proof by Theorem 4.3 in [13].
In the rest of this article we assume that the map s → T (s)B is stochastically integrable with respect to L in [0, T ] for some (and hence each) T > 0. In this case  If (a, Q, µ) denotes the cylindrical characteristics of L, then the usual characteristics (c t , S t , ξ t ) of ν t is given by where T t ν denotes the forward measure ν • (T (t)) −1 . Equivalently, a measure satisfying (3.9) is also called an operator self-decomposable measure.
A stationary measure can also be defined as the invariant measure for the generalised Mehler semigroup of the process Y . The concept of a generalised Mehler semigroup has been studied in detail in [5] for the Gaussian case and [10] for the non-Gaussian case. First, we need to know that the family (ν t : t 0) defines a skew-convolution semi-group: Proof. Let ϕ Ttνs * νt : V → denotes the characteristic function of the probability measure T t ν s * ν t . For each v ∈ V and s, t 0, we obtain, which establishes (3.10).
The generalised Mehler semigroup (P t : t 0) for the family (ν t : t 0) is defined by The following equivalence result is from [1, Theorem 2.1], whose proof identically applies to the cylindrical case. A natural candidate for a stationary measure is the limit of ν t in M(V ) as t → ∞. The following result relates the limit to the stochastic integral: Lemma 3.4. The following conditions are equivalent: In this case, the probability distribution of the limit in (b) coincides with the limit in (a).
Proof. Note that the process t 0 T (s)B dL(s) : t 0 has independent increments which follows from the definition of the stochastic integral as a limit of stochastic integrals of simple integrands. Consequently, Lemma A.2.1 in [12] guarantees that convergence in probability and weak convergence coincide.
Lemma 3.5. If (ν t : t 0) converges to ν in M(V ) as t → ∞, then it follows that: (a) the limit ν is a stationary measure for the process (3.3); (b) any stationary measure λ for (3.3) has the form λ = β * ν, where β is a probability measure satisfying β = T t β for all t 0.
Proof. Lemma 3.2 guarantees ν t+s = T t ν s * ν t for any s, t 0. By taking limit as which proves (a). To establish (b), we follow the arguments in [8,Prop. 3.2]. Let λ be an invariant measure for (3.3) and (t n ) n∈AE ⊆ Ê + a sequence converging to ∞.
By the definition of the invariant measure, we have (3.12) Since (ν tn : n ∈ AE) is relatively compact in M(V ), and {λ} is trivially relatively compact, Theorem III.2.1 in [15] guarantees that the sequence (T tn λ : n ∈ AE) is relatively compact in M(V ). As a consequence of infinite divisibility of distributions ν and ν t , we obtain ϕ ν (v) = 0 and ϕ νt (v) = 0 for all v ∈ V . It follows by (3.12) that, Hence, since (t n ) n∈AE is an arbitrary sequence, Lemma VI.2.1 in [15] implies that (T t λ : t 0) converges weakly to some probability measure β, and thus we obtain λ = β * ν by (3.12). Using that both λ and ν are stationary measures for (3.3), we conclude By Lemma 3.5, if the sequence (ν t : t 0) converges in M(V ) then its limit is a stationary measure. Thus, conditions for the convergence of (ν t : t 0) provide conditions for the existence of a stationary measure. Later in the case of stable semigroups, we will see that the converse implication is also true, i.e. the existence of a stationary measure implies convergence of (ν t : t 0). Theorem 3.6. The sequence (ν t : t 0) converges in M(V ) as t → ∞ if and only if the characteristics of ν t defined in (3.6) -(3.8) satisfy the following conditions: (3.14) For the proof of Theorem 3.6 we need some results on the cylindrical measure ξ ∞ defined by The canonical projection is denoted by π n , i.e.
where (h k ) k∈AE is an orthonormal basis of V .
Lemma 3.7. If (3.15) holds, then it follows that Proof. For any v ∈ V and n ∈ AE we obtain by the Cauchy-Schwarz inequality that [17,Lemma 4.4], it follows that
Proof. We first note that by monotone convergence theorem and (3.15), it follows that Let v ∈ V be fixed and define for each n ∈ AE the function It follows that Define the bounded and continuous function Lemma 4.4 in [17] implies that For each n ∈ AE and s 0, we obtain by the Cauchy-Schwarz inequality that . Then ρ N extends to a measure as π N is Hilbert-Schmidt, and satisfies Consequently, ρ N is a genuine Lévy measure on B(V ). The Lévy-Khinchine Theorem implies that there exists an infinitely divisible probability measure θ N on B(V ) with characteristic function By an application of the inequality 1 − cos β By denoting the density of the standard normal distribution on B(Ê m ) by g m , we obtain for every m, n ∈ AE with m n and N ∈ AE that Furthermore, for fixed n ∈ AE and for each N ∈ AE, define the function For each β = (β 1 , . . . , β n ) ∈ Ê n we have For fixed n ∈ AE, Lebesgue's theorem on dominated convergence implies → 0 as |β| → 0 and uniformly in N ∈ AE.
It follows from (3.24) that the family (ψ N : N ∈ AE) is equicontinuous at the origin.
It follows by [15, Lemma VI.2.1] that (θ N ) N ∈AE converges weakly to an infinitely divisible probability measure θ and the characteristic function ϕ θ of θ is given by Consequently, ξ ∞ + ξ − ∞ extends to the Lévy measure of θ. Since Theorem 3.4 in [18] implies that ξ ∞ extends to a Lévy measure on B(V ), which completes the proof.
Proof of Theorem 3.6. Sufficiency: suppose that (3.13)-(3.16) hold. We first show that the family (ν t : t 0) of infinitely divisible probability measures with characteristics (c t , S t , ξ t ) is relatively compact in M(V ), for which we use the compactness criterion for infinitely divisible probability measures as given in [15, Th. VI.5.3]. We only need to show that the set (ξ t : t 0) restricted to the complement of any neighbourhood of the origin is relatively compact and the operators R t : V → V defined by For any cylindrical set C ∈ Z(V ) and t 0, we have Since B(V ) is the σ-algebra generated by Z(V ) and Z(V ) is a π-system, we obtain ξ t ξ ∞ on B(V ) for all t 0. Let ξ c t and ξ c ∞ denote the restrictions of the measures ξ t and ξ ∞ to the complement of a neighbourhood V 1 ⊆ V of origin. By Lemma 3.9 and [14, Prop 1.1.3], the finite measure ξ c ∞ is a Radon measure and therefore, for each ε > 0 there exists a compact set K ⊆ V c 1 such that ξ c ∞ (K c ) ε. As a consequence, which implies that (ξ t : t 0) restricted to the complement of any neighbourhood of the origin is relatively compact. Furthermore, Lebesgue's theorem on dominated convergence and (3.14) imply The limits in (3.29) and (3.30) show that Condition (3.27) is satisfied. Condition (3.26) can be proved analogously using (3.15), and thus Theorem VI.5.3 in [15] imply that (ν t : t 0) is relatively compact. Since t →tr(S t ) is increasing, Condition (3.14) implies that the operator is well-defined and as t → ∞. It follows from (3.13), (3.31) and (3.32) that the characteristic function ϕ νt of ν t converges to the characteristic function ϕ ν of an infinitely divisible measure ν with characteristics (c ∞ , S ∞ , ξ ∞ ). Together with relative compactness of (ν t : t 0), Lemma VI.2.1 in [15] guarantees that (ν t : t 0) converges in M(V ). Necessity: if (ν t : t 0) converges weakly as t → ∞ then (3.13)-(3.16) follow by the compactness criterion of infinitely divisible probability measures in Hilbert spaces as applied before.
Example 3.10. In this example, we assume that U = V and B = Id in equation (3.1). Let L be the canonical α-stable cylindrical Lévy process for α ∈ (0, 2), which is defined in [19] by requiring that its characteristic function is of the form for all u ∈ U, t 0.
Assume that there exists an orthonormal basis (e k ) k∈AE of U and an increasing se-  [19]. For example, a sufficient assumption for the validity of (3.33) is This follows since Cauchy-Schwartz inequality implies Example 3.11. More specifically, we consider the heat equation on a bounded domain O in Ê d with smooth boundary for some d ∈ AE in the setting of the previous Example 3. 10. In this case, the generator A is given by the Laplace operator ∆ on U = L 2 (O) and L is the canonical α-stable cylindrical Lévy process for α ∈ (0, 2).

Weyl's law guarantees that the eigenvalues of
where c k ∈ [a, b] for some a, b > 0. Example 4.2 in [19] shows that there exists a solution if and only if αd < 4. We claim that the same condition is sufficient and necessary for the existence of a stationary solution. First, suppose αd < 4. By the integral test for convergence of series we obtain for each s > 0 that . On the other hand, the integral test for series implies

Consequently, Condition (3.33) is satisfied since
which results in 1 0 T (s) α HS ds = ∞ for αd 4. Remark 3.12. If L is the genuine Lévy process with (classical) characteristics (b, Q, µ), then the cylindrical characteristics of L are given by (a, Q, µ) where Then for every v ∈ V , we have by (3.6) and (3.34), As a consequence, we observe that in this case, Theorem 3.6 is equivalent to the well-known result from [8]: the sequence (ν t : t 0) converges weakly if and only if the following conditions are satisfied: (i) There exists The equivalence of (3.36) and the Conditions (3.15) and (3.16) can be obtained by noting that in this case µ is a genuine Lévy measure and consequently ξ ∞ = (leb ⊗ µ) • χ −1 [0,∞) is also a genuine measure. By Lemma 3.9, the Conditions (3.15) and (3.16) are equivalent to the Condition that ξ ∞ is a Lévy measure which is equivalent to (3.36).
It is well known that in general an invariant measure is not necessarily unique. As in the case of genuine Lévy processes, we obtain uniqueness if the semigroup Theorem 3.13. If the semigroup (T (t)) t 0 is stable, then there exists a stationary measure ν for the process (3.3) if and only if (ν t ) t 0 converges weakly; in this case the limit of (ν t ) t 0 equals the stationary measure.
Proof. Our claim in the cylindrical setting can be proved as in the classical situation; see [8,Prop. 6.1].
Combining Theorem 3.13 with Theorem 3.6, we obtain that, if the semigroup is stable, then Conditions (3.13)-(3.16) of Theorem 3.6 are necessary and sufficient for the existence of a stationary measure for the process (3.3), which in this case is unique.

The case of exponentially stable semigroups
In general the conditions of Theorem 3.6 may be difficult to verify in practice, in particular Condition (3.13) for the drift component c t . If the semigroup is exponentially stable, i.e. there exists C > 1 and λ > 0 such that T (t) Ce −λt for all t 0, and L is a genuine Lévy process, then a sufficient condition for the existence of stationary measure is that the Lévy measure µ of L satisfies the following simple condition where log + x := log x if x 1 and 0 otherwise; see [8,Th. 6.7]. This condition is also necessary if V is finite dimensional (see [2,Th. 4.3.17] and references therein) or if the semigroup (T (t)) t∈Ê is a group (see [8,Prop. 6.8]) but in general is not necessary (see [7,Ex. 3.15]). In the case of a semigroup (T (t)) t 0 with spectral decomposition T (t)e k = e −λ k e k (e.g. the heat semigroup) where the eigenvalues (λ k ) satisfy some mild conditions (see (4.5)), the following weaker condition is shown in [9] to be both necessary and sufficient for the existence of a stationary measure when L is a genuine Lévy process. In the main result of this section, we generalise this condition for the case of cylindrical Lévy processes and give some examples. Without loss of generality, we assume U = V and B = Id in the rest of this section. We assume that A is a self-adjoint strictly negative operator with compact resolvent. Consequently, A has a purely point spectrum (−λ k ) k∈AE , where 0 < λ 1 λ 2 · · · and lim k→∞ λ k = ∞, (4.2) and there is an orthonormal basis (e k ) k∈AE in V consisting of eigenvectors e k of A corresponding to the eigenvalues −λ k . Then A is a generator of the C 0 -semigroup (T (t)) t 0 of bounded linear operators on V , given by the formula: Clearly, the semigroup (T (t)) t 0 satisfies and is therefore exponentially stable.
which verifies Condition (3.14) due to our assumption(4.5). We next show that (3.15) and (3.16) are satisfied. It follows by Lemma 3.1 in [18] that for any c > 0, u, e k 2 µ(du) Using the definition of the sets C k (s), we obtain Noting that ∪ n k=m C c k (s) = u : max m k n 2 log + | u,e k | λ k > s , we have where the last equality follows from a Fubini argument; see [4,App. 2] for details. Hence substituting (4.11), (4.12) and (4.13) in (4.10) and using (4.5) and (4.7) verifies Condition (3.16). Similarly, by setting m = 1 in (4.11), (4.12) and (4.13) and using (4.5) and (4.6) verifies Condition (3.15). It remains to prove that (3.13) is satisfied, that is there exists lim t→∞ c t where for each v ∈ V , (4.14) We first prove that ∞ 0 |a(T * (s)v)| ds < ∞ for each v ∈ V . For this we will use the following equality which holds for all u * ∈ U and β > 0 as given by (3.9) in [18], Let π n : U → U be the projection operator defined by π n (v) := n k=1 v, e k e k . Then by (4.15) and assuming T * (s)π n v = 0 , we obtain Since a maps bounded sets to bounded sets, it follows by (4.4) that Integrating the second term on the right side in (4.16) results in | u, T * (s)π n v | µ(du) ds =: I 4 + I 5 . (4.18) Using (4.4) we estimate I 4 by If C k (s) denotes the set defined in (4.9), we obtain (4.20) For the integral I 6 we obtain which is finite by using (4.5). Using the same equality from [4, App. 2] as in (4.13), we obtain Applying (4.6) to (4.22) and using (4.17) -(4.22), it follows from (4.16) that there exists some C 1 > 0 such that Fatou's Lemma implies that for any δ > 0, For considering the second term in (4.14), define Then for any h, v ∈ V , , from which the integrability of f (·, v) with respect to ξ ∞ follows by using the properties of Lévy measures. Consequently, since ξ t (C) ↑ ξ ∞ (C) as t → ∞ for each C ∈ B(V ) and ξ ∞ is a Lévy measure due to Lemma 3.9, we obtain by the same arguments as in Lemma 3.3 in [10] that Together with (4.24) it follows from (4.14) that ( c t , v ) t 0 converges for each v ∈ V and lim To prove that (c t ) t 0 converges in V , it is enough to show that (c t ) t 0 is relatively compact in V , which in this case, as (c t ) t 0 is bounded, reduces to establish  Using (4.23), Cauchy-Schwarz inequality and the fact that ξ t ξ ∞ , we obtain h, e k 2 ξ ∞ (dh).
Similar application of Fubini's theorem implies for the second term in (4.26) that Using the estimates obtained in (4.27) and (4.28) in (4.26), it follows that there exists a constant C 2 > 0 such that which implies that (c t ) t 0 is relatively compact in V and hence (3.13) is satisfied.  From the proof of Theorem 4.1 it also follows that without assuming (4.5), the following conditions: together with (4.6) and (4.7) are sufficient for the existence of an invariant measure. which shows the equivalence of (4.6) and (4.33) by taking m = 1. Furthermore, Condition (4.2) implies for each u ∈ U that sup n m log + | u, e n | λ n sup n m log + u λ n log + u λ m → 0 as m → ∞.
An application of Lebesgue's theorem together with (4.33) implies (4.7). Condition (4.32) is weaker than (4.5). But it is well known (and also mentioned in above Example) that the stochastic heat equation driven by a cylindrical Brownian motion has a weak solution if and only if d = 1. Therefore, condition (4.5) is more natural for an arbitrary cylindrical Lévy process. But if L is a genuine Lévy process or if L has characteristics (0, 0, µ) where µ is symmetric, then the above proof can be easily modified by assuming (4.32) instead of (4.5).