SPDEs driven by standard symmetric $\alpha$-stable cylindrical L\'evy processes: existence, Lyapunov functionals and It\^{o} formula

We investigate several aspects of solutions to stochastic evolution equations in Hilbert spaces driven by a standard symmetric $\alpha$-stable cylindrical noise. Similarly to cylindrical Brownian motion or Gaussian white noise, standard symmetric $\alpha$-stable noise exists only in a generalised sense in Hilbert spaces. The main results of this work are the existence of a mild solution, long-term regularity of the solutions via Lyapunov functional approach, and an It\^{o} formula for mild solutions to evolution equations under consideration. The main tools for establishing these results are Yosida approximations and an It\^{o} formula for Hilbert space-valued semi-martingales where the martingale part is represented as an integral driven by cylindrical $\alpha$-stable noise. While these tools are standard in stochastic analysis, due to the cylindrical nature of our noise, their application requires completely novel arguments and techniques.


Introduction
Standard symmetric α-stable distributions are the natural generalisations of Gaussian distributions for modelling random perturbations of finite dimensional dynamical systems.They often meet various empirical requests, such as heavy tails, self-similarity and infinite variance, but are at the same time analytically tractable and well-understood.The importance of these models is reflected by the available vast literature on dynamical systems perturbed by random noises with α-stable distributions in various areas such as economics, biology etc.
In the infinite dimensional setting of modelling random perturbations of partial differential equations, much fewer results are known for systems perturbed by α-stable distributions.In fact, only in the random field approach, based on the seminal work by Walsh, one can find several publications on stochastic partial differential equations (SPDEs) driven by multiplicative α-stable noise, e.g.Mueller [27], Mytnik [28], and more recently Chong [7] and Chong et.al. [8].However, in the semigroup approach, following the spirit of Da Prato and Zabczyk, one can find several results for equations only with additive driving noise distributed according to an α-stable law; see e.g.Brzeźniak and Zabczyk [6] and Riedle [33].The only publication in the semigroup approach for multiplicative α-stable perturbation is Kosmala and Riedle [23], where however the assumptions are rather restrictive and do not correspond to the natural Lipschitz continuity and linear growth conditions.The lack of results in the semigroup approach is due to the fact that a random noise with a standard symmetric α-stable distribution does not exist as an ordinary Hilbert space-valued process but only in the generalised sense of Gel'fand and Vilenkin [14] or Segal [37].
In this work, we investigate several aspects of solutions to equations of the form dX(t) = AX(t) + F (X(t)) dt + G(X(t−)) dL(t), (1.1) where A is the generator of a C 0 -semigroup in a separable Hilbert space H, the coefficients F : H → H and G : H → L 2 (U, H) are mappings with U being a separable Hilbert space, and L is a standard symmetric α-stable cylindrical process in U for α ∈ (1, 2).Analogously to the standard normal distribution, standard symmetric α-stable distributions in R d can only be generalised to infinite dimensional spaces as cylindrical distributions.In particular, this means that the driving noise L in (1.1) exists only in the generalised sense; see Schwartz [36].Since such processes do not attain values in the underlying Hilbert space, standard results for stochastic processes in infinite dimensional spaces are not applicable.Most notably, complications arise from the fact that while these processes are cylindrical semi-martingales, see Jakubowski et.al. [18], they do not enjoy a semi-martingale decomposition in a cylindrical sense, since semi-martingale decompositions are not invariant under linear transformations, see Jakubowski and Riedle [19,Re. 2.2].Nevertheless, the problem of stochastic integration with respect to cylindrical Lévy processes was solved in Jakubowski and Riedle [19] by arguments avoiding the usual Lévy-Itô decomposition.This approach has been further developed for standard symmetric α-stable cylindrical process by two of us in Bodó and Riedle [5], which enables us to integrate predictable integrands and to derive a dominated convergence theorem for stochastic integrals.
This work comprises of 3 main results: the existence of a mild solution to Equation (1.1), a Lypunov functional approach for long-term regularity for solutions to Equation (1.1), and an Itô formula for mild solutions to Equation (1.1).The main tools for establishing these results are an Itô formula for Hilbert space-valued semi-martingales driven by standard symmetric α-stable cylindrical Lévy noise and a Yosida approximation of solutions to Equation (1.1).
While these tools are standard in stochastic analysis, due to the cylindrical nature of our noise, their application in our setting requires completely novel arguments and techniques, which we highlight in the following.
A classical Itô formula for semi-martingales in Hilbert spaces is well known and easy to derive; see e.g.Metivier [26,Th. 27.2].However, applying this formula often requires the identification of the martingale and bounded variation components of the process, which in the classical situation of a semi-martingale driven by an ordinary Hilbert space-valued process can easily be obtained via the semi-martingale decomposition of the driving process.Since in our case, the driving cylindrical process does not enjoy a semi-martingale decomposition, one needs to identify the martingale part of the stochastic integral process by carrying out a deep analysis of its jump structure.
The second major tool in our work is a Yosida approximation, which is an often-utilised device in the classical situation with an ordinary Hilbert space-valued process as driving noise; see e.g.Peszat and Zabczyk [30].Convergence of the Yosida approximation is established by tightness arguments in the space C([0, T ], L p (Ω, H)) of p-th mean continuous Hilbert spacevalued processes for any p < α.It turns out that the space C([0, T ], L p (Ω, H)) is tailor-made for analysing equations driven by a standard symmetric α-stable cylindrical process.The observation that the solution is continuous in the above sense, despite having discontinuous paths, lies at the heart of this paper.To the best of our knowledge, we are the first to use this in the context of SPDEs driven by cylindrical stable noise.
These two tools, the Itô formula for semi-martingales driven by a standard symmetric αstable cylindrical process and convergence of the Yosida approximation, enable us to establish the 3 main results of our work.For the existence result, we use tightness of the Yosida approximation to establish existence of a mild solution to Equation (1.1).In our setting, standard methods for establishing existence of a solution, such as fix point arguments or Grönwall's lemma are not applicable, since the integral operator with a standard symmetric α-stable integrator maps to a larger space than its domain; see Kosmala and Riedle [23] or Rosinski and Woyczynski [34].
By following the classical approach of Ichikawa in [16], we demonstrate the power of the established tools by investigating the long-term regularity of the mild solution to Equation (1.1) via the functional Lyapunov approach.The functional Lyapunov approach can be used to establish various regularity properties; in this work, we focus on exponential ultimate boundedness, but other quantitative properties can be investigated similarly.As the mild solution is not a semi-martingale, the derived Itô formula for semi-martingales cannot be applied directly.However, we successfully show that the Yosida approximations are semimartingales, and thus the Itô formula can be applied to these, which immediately shows their exponential ultimate boundedness.It remains only to show that this boundedness property carries over to the limit, for which we establish convergence of the Markov generators in a suitable sense.
Mild solutions of SPDEs are not semi-martingales, and thus the classical Itô formula cannot be applied.This lack of a powerful tool is often circumvented by a specific Itô formula for mild solutions of SPDEs.One of the first versions of such an Itô formula for mild solutions can be found in Ichikawa [16] for the Gaussian case, and more recent versions in Da Prato et.al. [9] for the Gaussian case and in Alberverio et.al. [1] for the case of ordinary Lévy processes.In the last part of our work, we derive such an Itô formula for mild solutions of equation (1.1) driven by a standard symmetric α-stable cylindrical process.
We outline the structure of the paper.Selected preliminaries on standard symmetric α-stable cylindrical processes, integration with respect to them and underlying results on equations as well as the theory of predictable compensators are collected in Section 2. In Sections 3 and 4, we identify the predictable compensator and quadratic variation of the integral process.These observations lead us directly to the Itô formula for semi-martingales driven by a standard symmetric α-stable cylindrical process in Section 5.In Section 6, we prove existence of a mild solution under Lipschitz and boundedness conditions in the space of continuous functions, where the main result is formulated in Theorem 6.5.In Section 7, we establish conditions for exponential ultimate boundedness in Theorem 7.1.Finally, in Section 8, an Itô formula for mild solutions is proved.
Such sets are called cylindrical sets with respect to S and the collection of all such cylindrical sets is denoted by Z(U, S).It is a σ-algebra if S is finite and otherwise an algebra.We write shortly Z(U ) for Z(U, U ).A set function µ : Z(U ) → [0, ∞] is called a cylindrical measure on Z(U ) if for each finite subset S ⊆ U , the restriction of µ to the σ-algebra Z(U, S) is a σ-additive measure.A cylindrical measure is said to be a cylindrical probability measure if µ(U ) = 1.
Let (Ω, Σ, P ) be a complete probability space.We will denote by L 0 P (Ω, U ) the space of equivalence classes of measurable functions Y : Ω → U equipped with the topology of convergence in probability.A cylindrical random variable X in U is a linear and continuous mapping X : U → L 0 P (Ω, R).It defines a cylindrical probability measure µ X by for cylindrical sets Z = C(u 1 , ..., u n ; A).The cylindrical probability measure µ X is called the cylindrical distribution of X.We define the characteristic function of the cylindrical random variable X by ϕ Let T : U → H be a linear and continuous operator.By defining we obtain a cylindrical random variable on H.In the special case when T is a Hilbert-Schmidt operator and hence 0-Radonifying by [39, Th.VI.5.2], it follows from [39, Pr.VI.5.3] that the cylindrical random variable T X is induced by a genuine random variable Y : Ω → H, that is (T X)h = Y, h for all h ∈ H.
A family (L(t) : t ≥ 0) of cylindrical random variables L(t) : U → L 0 P (Ω, R) is called a cylindrical (F t )-Lévy process if for each n ∈ N and u 1 , ..., u n ∈ U , the stochastic process L(t)u 1 , ..., L(t)u n : t ≥ 0 is an (F t )-Lévy process in R n and the filtration (F t ) t≥0 satisfies the usual conditions.We denote by Z * (U ) the collection of cylindrical sets, which forms an algebra of subsets of U .For fixed u 1 , ..., u n ∈ U , let λ u 1 ,...,un be the Lévy measure of (L(t)u 1 , ..., L(t)u n ) : It is shown in [3] that λ is well defined.The set function λ is called the cylindrical Lévy measure of L.
In this paper, we restrict our attention to standard symmetric α-stable cylindrical Lévy processes for α ∈ (1, 2), which we simply call α-stable cylindrical Lévy processes in the sequel.These are cylindrical Lévy processes with characteristic function ϕ L(t) (u) = exp(−t u α ) for each t ≥ 0 and u ∈ U .Let (e k ) k∈N be an orthonormal basis of U .
Proof.Let m ∈ N be fixed.We approximate the integrand of the first integral in (2.2) by see [24,Th. 6.2.7].This enables us to conclude for each n ∈ N that 2 n ] (r) αr −(α+1) dr.
The monotone convergence theorem implies Since another application of the monotone convergence theorem shows we obtain from (2.1) that Applying similar arguments for the integrand in the second integral in (2.2) yields for all Φ ∈ L 2 (U, H), which completes the proof as in (2.2) we can set If L is an α-stable cylindrical Lévy process and T : H → U a Hilbert-Schmidt operator, then the cylindrical random variable T L(1) is induced by a genuine stable random variable on U with Lévy measure λ • T −1 .This Lévy measure depends continuously on T in the following way: Lemma 2.2.Let λ be the cylindrical Lévy measure of an α-stable cylindrical Lévy process for α ∈ (1, 2).Then for each r > 0, the mapping U, H) to the space of Borel measures on B H (r) c equipped with the weak topology.
Proof.If µ is a cylindrical probability measure and From this, the assertion follows from [29, Th. 5.5].

Stochastic integration
We briefly recall some facts on stochastic integration with respect to an α-stable cylindrical Lévy process L as introduced in [5].A process G : Ω × [0, T ] → L 2 (U, H) is called adapted and simple if it is of the form where N ∈ N, 0 = t 0 < t 1 < • • • < t N = T and Φ i is an F t i−1 -measurable and L 2 (U, H)valued random variable taking finitely many values.We denote by S HS adp the class of all adapted, simple processes.The integral process • 0 G dL is defined as the sum of the Radonified increments (2.6) Let S In this case, • 0 G dL is defined as the limit of It is shown in [5], that a predictable process G is stochastically integrable if and only it is an element of L 0 P Ω, L α Leb [0, T ], L 2 (G, H) .It follows from [23,Co. 3] that for every 0 < p < α we have for every stochastically integrable predictable process G, where e p,α = α α−p e p/α 2,α for some e 2,α ∈ (0, ∞) that depends only on α.Lemma 2.3.If G is a predictable stochastic process stochastically integrable with respect to the α-stable cylindrical Lévy process L for some α ∈ (1, 2) then Proof.Define the predictable stopping times τ n = inf t > 0 : H) ds > n for n ∈ N. It follows from Proposition 4.22(ii) and Lemma 1.3 in [10] that for each n ∈ N there exists a sequence of adapted, simple processes (G n,k ) k∈N such that (2.8) Since inequality (2.7) guarantees for each k, n ∈ N that the same arguments as in [32,Th. I:51] show that the processes • 0 G n,k dL are martingales.Equation (2.8) shows that • 0 G½ [0,τn] dL is a limit of martingales in L 1 (Ω, H) by (2.7), and thus a martingale.Since standard arguments, e.g.[32, Th.I.12], establish for the stopped integral process, the proof is completed.
We denote by P (resp.Õ) the predictable (resp.optional) σ-algebra on Ω × [0, T ] × H and call a function If µ is a random measure and W is optional we define A random measure µ is called predictable (resp.optional) if ( t 0 H W (s, h) µ(ds, dh) : t ∈ [0, T ]) is predictable (resp.optional) for every predictable (resp.optional) function W .An optional random measure µ is called σ-finite if there exists a sequence (A n ) n∈N ⊂ P with For each σ-finite, optional measure µ on [0, T ] × H there exists a predictable random measure ν on B([0, T ]) ⊗ B(H) such that is an optional and σ-finite random measure on B([0, T ]) ⊗ B(H).Thus, its compensator exists which we denote by ν Y .
Example 2.6.Let L be a genuine H-valued Lévy process with Lévy measure λ.Then the compensator ν L of the jump measure µ L is given as the extension of µ In the sequel, we will make use of another characterisation of compensators of jumpmeasures.We denote by C + (H) the class of non-negative, continuous functions k : H → R bounded on H and vanishing inside a neighbourhood of 0.
Proposition 2.7.The compensator ν Y of the jump-measure µ Y of an H-valued càdlàg semimartingale Y is characterised by being predictable and satisfying either of the following: is locally integrable, then so is Proof.The equivalence between (i) and (ii) follows by the same argument as in the proof of [17,

Predictable compensator
For an α-stable cylindrical Lévy process L for some α ∈ (1, 2) and a stochastically integrable predictable process G, we define the integral process X = • 0 GdL and The main result of this section is that ν extends to a random measure on B([0, T ]) ⊗ B(H) and that the extension is the predictable compensator of the jump measure of X.We will derive this result by a couple of Lemmata.
Step 2: We show that f is predictable for all B ∈ B(H \ {0}), which will immediately imply that (3.1) is almost surely well defined and predictable as it is then just an integral of a non-negative predictable process.We define and claim that D is a λ-system.Continuity of measures implies that H \ {0} ∈ D since, for all t ∈ (0, T ] and ω ∈ Ω, we have where the right hand side is the limit of processes that are predictable by Step 1.
The collection D is closed under union of increasing sequences, which follows as above from continuity of measures and predictability of the pointwise limit.This concludes the proof of the claim that D is a λ-system.We define the π-system.
and the right-hand side is predictable by Step 1.The Dynkin π-λ theorem for sets, see e.g.
To show that the random measure ν characterised by (3.1) is the compensator of the jump-measure µ X of the integral process X, we first consider the case when the integrand is an adapted, simple process.Lemma 3.2.Suppose that G is an adapted, simple process in S HS adp .Then the random measure ν obtained in Lemma 3.1 is the predictable compensator of µ X .Proof.Since Lemma 3.1 guarantees that ν is predictable, it remains to show (2.10), which by the functional monotone class theorem reduces to proving Let G be of the form (2.5), and assume that the points of the partition contain s and t; otherwise these can be added.Then X takes the form (2.6), and it follows and the compensator of the jump measure of the Lévy process Before we show that the result of Lemma 3.2 can be extended to general integrands, we need to prove some technical Lemmata.Recall the class of functions C + (H) used in Proposition 2.7 (and defined just before) to determine the compensator.
The proof is complete.
Leb ([0, T ], L 2 (U, H)) and pointwise for almost every s ∈ [0, T ].Then we obtain for each Proof.Lemma 2.2 implies for almost each s ∈ [0, T ] and every n ∈ N that Since k is bounded and vanishes in a neighbourhood of 0, we conclude from inequality (2.1) the generalised Lebesgue's dominated convergence theorem, see e.g.[35,Th. 4.19], implies As the functions are continuous monotone and converge pointwise to a continuous limit on [0, T ], the convergence is uniform by [31, p. 81/127] (or deuxième théorème de Dini).
Now we can prove the main result of this section.
Theorem 3.5.Let L be an α-stable cylindrical Lévy process L for some α ∈ (1, 2) and G a stochastically integrable predictable process.Then the predictable compensator ν X of the jump measure µ X of X := • 0 GdL is characterised by (3.1).Proof.In light of Proposition 2.7, it suffices to show that the process M k defined by is a local martingale for any k ∈ C + (H).Lemma 4.3 in [5] guarantees that there exists a sequence (G n ) n∈N of adapted, simple processes in S HS adp converging both in L α Leb ([0, T ], L 2 (U, H)) a.s. and P ⊗ Leb| [0,T ] − a.e. to G. Letting X n := • 0 G n dL and denoting the jump-measure of X n by µ Xn , we define for each k ∈ C + (H) and n ∈ N a process M k n by Proposition 2.7 and Lemma 3.2 imply that M k n is a local martingale for all n ∈ N. Since for each n ∈ N and t ∈ [0, T ] we have that µ Xn ({t} × H) ≤ 1 almost surely, it follows that s. for all n ∈ N. Almost sure uniform convergence of X n and Lemma 3.3 guarantee that there exists an Ω 1 ⊆ Ω with P (Ω 1 ) = 1 such that, for all ω ∈ Ω 1 , we have In the same way, by convergence of G n both in L α Leb ([0, T ], L 2 (U, H)) a.s. and P ⊗Leb| [0,T ] −a.e. and Lemma 3.4 there exists an Ω 2 ⊆ Ω with P (Ω 2 ) = 1 such that, for all ω ∈ Ω 2 , we have

Quadratic variation of the integral process
The quadratic covariation of two real-valued càdlàg semimartingales V 1 and V 2 starting from zero is the process where ⊗ denotes the tensor product and Z i (t) = Z(t), f i for t ∈ [0, T ] are the projection processes of Z; see [26,Se. 21.2] for brief introduction.The process [[Z]] does not depend on the choice of the orthonormal basis (f i ) i∈N .The process [[Z]] is called the tensor quadratic variation of Z and its continuous part We say that Z is purely discontinuous if Proposition 4.1.Let L be an α-stable cylindrical Lévy process for some α ∈ (1, 2) and G a stochastically integrable predictable process with values in L 2 (U, H).Then the integral process X := • 0 G dL is purely discontinuous.
Proof.We proceed in three steps.
Step 1: Assume H = R and U = R d for some d ∈ N. In this case, L is a U -valued standard symmetric α-stable Lévy process, and therefore purely discontinuous; see e.g.[32, p. 71 Step 2: Assume H = R, but without any further restrictions on U .In that case, by the identification Fix an orthonormal basis (f k ) k∈N in U and define for each n ∈ N the projection Since the projection π n is a Hilbert-Schmidt operator, there exists a U -valued Lévy process L n with the property L n , u = L(π * n u) for all u ∈ U .We define the approximations Since L n attains values in a finite-dimensional subspace and is a symmetric α-stable process by [33, Le. 2.4], it follows that X n is purely discontinuous by Step 1.
Let M be a real-valued, continuous martingale and define for k ∈ N the stopping times It follows that τ k → T as k → ∞ by (4.2).Since X n is purely discontinuous, it follows from [17, Le.I.4.14] that (X n M ) τ k is a local martingale for each k, n ∈ N. Since applying inequality (2.7) and equality (2.9) shows we obtain that (X n M ) τ k is a martingale by [32, Th.I:51].Noting ) and equality (2.9) establish for each t ≥ 0 that It follows that the process (XM ) τ k as a limit of martingales is itself a martingale.Since X is a local martingale according to Lemma 2.3 and M is an arbitrary real-valued continuous martingale, it follows from [17, Le.I.4.14] that X is purely discontinuous.
Step 3: For the general case, we fix an orthonormal basis (e i ) i∈N in H and choose any i, j ∈ N. Since X(t), e i = t 0 G * e i dL for every t ≥ 0, the polarisation formula for realvalued covariation shows Linearity of the integral and binomial formula enable us to conclude

Strong Itô formula
In this section, we establish an Itô formula for processes that are given by a differential driven by a standard symmetric α-stable cylindrical Lévy process L for α ∈ (1, 2) and are of the form where We denote by C 2 b (H) the space of continuous functions f : H → R having bounded first and second Fréchet derivatives, which are denoted by Df and D 2 f , respectively.Theorem 5.1.Let X be a stochastic process of the form (5.1).It follows for each where Lemma 5.2.Let λ be the cylindrical Lévy measure of an α-stable cylindrical Lévy process for α ∈ (1, 2).Then we have for each where d 1 α is a constant depending only on α as defined in Inequality (2.2).
Proof.Taylor's remainder theorem in the integral form, see [2, Th. 5.8], and Inequality (2.2) imply Similarly, Taylor's remainder theorem in the integral form and Inequality 2.2 show Another application of Inequality 2.2 shows Combining inequalities (5.3) to (5.5) completes the proof.
Proof of Theorem 5.1.The stochastic process X given by (5.1) is purely discontinuous as it is the sum of a finite-variation process and a purely discontinuous process according to Proposition 4.1.The Itô formula in [26,Th. 27.2] takes for all t ∈ [0, T ] the form One can show by approximating with simple integrands that where both integrals are well defined since (5.2) guarantees The definition of the compensator ν X and Lemma 5.2 imply (5.7) The stopping times τ n := inf t > 0 : H) ds ≥ n ∧ T satisfy τ n → T as n → ∞ by (5.2).Since inequality (5.7) guarantees for all n ∈ N that Proposition 2.7 shows that M f is a local martingale.This concludes the proof, since the claimed formula is just a different form of (5.6).

Mild solutions for stochastic evolution equations
We recall that U and H are separable Hilbert spaces with norms • and L is a standard symmetric α-stable cylindrical (F t )-Lévy process in U with α ∈ (1, 2).In this section we consider the mild solution of the stochastic evolution equation: where A is a generator of a C 0 -semigroup (S(t)) t≥0 in H, x 0 is an F 0 -measurable H-valued random variable, F : H → H and G : H → L 2 (U, H) are measurable mappings and T > 0.
Definition 6.1.An H-valued predictable process X is a mild solution to (6.1) if We work under the following assumptions: (A1) The C 0 -semigroup (S(t)) t≥0 is compact, analytic and a semigroup of contractions and 0 is an element of the resolvent set of A.
(A2) The mapping F is Lipschitz and bounded, i.e. there exists for every h 1 , h 2 , h ∈ H.
(A3) The mapping G is Lipschitz and bounded, i.e. there exists for every h 1 , h 2 , h ∈ H.
We will obtain the solution to (6.1) by using the Yosida approximations.For this purpose, we define R n = n (nI − A) −1 for n ∈ N and denote by X n the mild solution to Existence of the mild solution X n to (6.7) with cádlág paths is guaranteed by [23,Th. 12].
Remark 6.6.We recall that under Assumption (A1) we have for all δ ∈ [0, 1] that This follows from the fact, that if an operator commutes with A then it commutes with A γ , see e.g.[15, Pr. 3.1.1],which enables us to conclude for every n ∈ N that Since (S(t)) t≥0 is a contraction semigroup, Theorem 3.5 in [13] guarantees R n L(H) ≤ 1 for all n ∈ N.
The solution to (6.1) will be constructed as a limit of X n in C([0, T ], L p (Ω, H)) for an arbitrary but fixed p < α.In the first three Lemmata, we establish relative compactness of the Yosida approximation {X n : n ∈ N} in the space C([0, T ], L p (Ω, H)).Lemma 6.7.The set {X n (t) : n ∈ N} is tight in H for every t ∈ [0, T ].
Proof.The case t = 0 follows immediately from the strong convergence of R n .For the case t ∈ (0, T ] we first prove that for every 1 ≤ q < α, and 0 ≤ δ < 1/α we have Applying Hölder's inequality and inequality (2.7) shows for every n ∈ N that Commutativity of S and R n , Remark 6.3 and Remark 6.6 verify Assumption (A2) on boundedness of F together with Remark 6.3 and Remark 6.6 yield Similarly, Assumption (A3) on boundedness of G implies Combining the above estimates establishes (6.8), which in turn gives the statement of the Lemma.Indeed, choose any δ ∈ (0, 1/α) and use Markov's inequality and (6.8) for q = 1 to obtain for each N > 0 that sup for some constant c ∈ (0, ∞).Since the embedding D δ ֒→ H is compact according to Remark 6.2, we obtain tightness of {X n (t) : n ∈ N} by Prokhorov's theorem.
Raising both sides of (6.10) to the power of α/p shows for each t ∈ [0, T ] that where we use It follows from (6.11)  then, for each ǫ ∈ (0, 1), we can find some p * ∈ (1, α) such that 2 2 , and some N ∈ N such that for all m, n ≥ N 2 This completes the proof since we obtain for any m, n ≥ N that It remains to establish (6.13) and (6.14), which we first do under the additional assumption (6.9): Argument for (6.13):Since e p,α = α α−p e p/α 2,α for a constant e 2,α ∈ (0, ∞) independent of α, we obtain T α−1 ≤ e α p p,α for p sufficiently close to α.In this case, we obtain Since we assume (6.9), applying L'Hospital's rule verifies (6.13).
Argument for (6. = 0, (6.17) which establishes (6.14).Thus, we have proved the lemma under the additional assumption (6.9).For establishing the general case, we fix a time Since our choice of T 0 implies that {S(t−T 0 )X n (T 0 ), n ∈ N} is relatively compact in L 0 P (Ω, H), and that (t − T 0 ) 12 2 e 2,α (K α F + K α G ) < 1, we can use the same argument as before to obtain relative compactness of (X n (t)) n∈AE in L 0 P (Ω, H) for each t ∈ (T 0 , 2T 0 ].Using a standard induction argument, we can now cover intervals of arbitrary length.This concludes the proof of the general case. We now step from relative compactness of {X n (t) : n ∈ N} in L 0 P (Ω, H) for fixed time t to relative compactness of the processes {X n : n ∈ N} using the Arzelà-Ascoli Theorem.Lemma 6.9.The collection {X n : n ∈ N} is relatively compact in C([0, T ], L p (Ω, H)) for any 0 < p < α.
Proof.We consider the case 1 < p < α as the case p ≤ 1 follows from the fact that relative compactness in C([0, T ], L p (Ω, H)) implies relative compactness in C([0, T ], L p ′ (Ω, H)) for p > p ′ .In light of the Arzelà-Ascoli Theorem, cf.e.g.[21,Th. 7.17]), it suffices to show that The claim in (a) follows from [12,Cor. 3.3] by Lemmata 6.7, 6.8 and the fact that Equation (6.8) with δ = 0 and any q ∈ (p, α) implies via the Vallee-Poussin Theorem [11,Th. II.22] that the collection {X n (t) : n ∈ N} is p-uniformly integrable and bounded in L p (Ω, H).Hence, it remains only to prove (b).To that end, we take t ∈ [0, T ] and h ∈ (0, T − t], and estimate Commutativity of R n and S and contractivity of S implies Applying Hölder's inequality, boundedness of F in Assumption (A2) and contractivity of S we get We conclude from Inequality (2.7) by using boundedness of G in Assumption (A3) and contractivity of S that It follows from Lemma 6. Proof of Theorem 6.5.It is enough to consider the case p ≥ αβ where β is the Hölder exponent from (6.6).Lemma 6.9 guarantees that there is a subsequence for some Z ∈ C([0, T ], L p (Ω, H)).The proof will be complete if we show that Z is a mild solution to (6.1).We conclude for each k ∈ N and t ∈ [0, T ] from Lipschitz continuity of F and Hölder continuity of G in (6.2) and (6.6) and contractivity of S by applying Hölder's inequality and Inequality (2.7) that As the last line tends to 0 as k → ∞ by (6.24), it follows that Z is a mild solution to (6.1).It remains to establish that Z has càdlàg paths, but this follows immediately from the following corollary as X n has càdlàg paths.
At the end of this section, we present a stronger convergence result for Yosida approximations that not only completes the proof of Theorem 6.5 but also turns out to be useful in applications as will be seen in the following sections.Corollary 6.10.For all 0 < p < α there exists a subsequence (X n k ) k∈N of the Yosida approximations, which converges to a solution to (6.1) both in C([0, T ], L p (Ω, H)) and uniformly on [0, T ] almost surely.
Proof.Lemma 6.9 enables us to choose a subsequence (X n ) n∈N of the Yosida approximations which converges in C([0, T ], L p (Ω, H)) to the mild solution X.To prove almost sure convergence, we fix an arbitrary r > 0 and estimate For the following arguments, we define m := sup t∈[0,T ] S(t) L(H) .As I − R n converges to zero strongly as n → ∞ we obtain For estimating the second term in (6.25), we apply Markov's inequality and Lipschitz continuity of F in (A2) to obtain We conclude from the last inequality by Lebesgue's dominated convergence theorem and convergence of To estimate the last term in (6.25), we apply the dilation theorem for contraction semigroups, see [38, Th.I.8.1]): there exists a C 0 -group ( Ŝ(t)) t∈R of unitary operators Ŝ(t) on a larger Hilbert space Ĥ in which H is continuously embedded satisfying S(t) = π Ŝ(t)i for all t ≥ 0, where π is the projection from Ĥ to H and i is the continuous embedding of H into Ĥ.Thus, if we denote m = sup t∈[0,T ] π Ŝ(t) , we may estimate using Markov's inequality, Inequality (2.7) and Hölder continuity of G in (6.6) We conclude from the last inequality by Lebesgue's dominated convergence, strong convergence of R n to I, boundedness G in (6.6) and convergence of We have shown that all the terms on the right hand side of (6.25) converge to zero as n tends to infinity which gives uniform convergence of X n to X in probability on [0, T ].This concludes the proof, since uniform convergence in probability implies the existence of a desired subsequence.

Moment boundedness for evolution equations
In this section, we investigate stability properties of the solution for the stochastic evolution equation (6.1) by applying the Itô's formula derived in Theorem 5.1.More precisely, we shall derive conditions on the coefficients such that the mild solution X is ultimately exponentially bounded in the p-th moment, that is there exist constants m 1 , m 2 , m 3 > 0 such that Recall that C 2 b (H) denotes the space of continuous real-valued functions defined on H with bounded first and second Fréchet derivatives.In what follows, our goal is to derive a Lyapunov-type criterion using the following operator on C 2 b (H): for f ∈ C 2 b (H).Note that the right hand side of (7.1) is well defined by Lemma 5.2.We can now state the main result of this section, the following general moment boundedness criterion.Theorem 7.1.Let p ∈ (0, 1) be fixed and V be a function in C 2 b (H) satisfying for some constants β 1 , β 2 , β 3 , k 1 , k 3 > 0 the inequalities Then the solution X to (6.1) is exponentially ultimately bounded in the p-th moment: Before we prove Theorem 7.1 we demonstrate its application by deriving conditions for moment boundedness in terms of the coefficients of Equation (6.1).Corollary 7.2.Suppose that there exists ǫ > 0 such that Ah + F (h), h ≤ −ǫ ||h|| 2 for all h ∈ D 1 , then the solution to (6.1) is exponentially ultimately bounded in the p-th moment for every p ∈ (0, 1).
Proof.Fix p ∈ (0, 1) and let ζ be a function in C 2 ([0, ∞)) satisfying ζ(x) = x p/2 for x ≥ 1 and ζ(x) ≤ 1 for x < 1.By defining V (h) = ζ( h 2 ) for all h ∈ H, we obtain V ∈ C 2 b (H) and It follows that (7.2) holds with β 1 = β 2 = k 1 = 1.We show that (7.3) also holds.By the definition of V , it follows for each h For h ∈ D 1 ∩ B H , one obtains by boundedness of F in Assumption (A2) that Since Lemma 5.2 together with boundedness of G in Assumption (A3) implies for each h ∈ H that we have verified Condition (7.3).
In the remaining of this section, we prove Theorem 7.1 using the Yosida approximations established in the previous sections.For this purpose, let X n denote the mild solution to the approximating equations (6.7) for each n ∈ N. We may assume due to Corollary 6.10, by passing to a subsequence if necessary, that X n converges to the solution X of (6.1) uniformly almost surely on [0, T ].In what follows, we will routinely pass on to a subsequence without changing the indices.
Proposition 7.3.The mild solution X n of (6.7) is a strong solution attaining values in D 1 , that is, for each t ∈ [0, T ], it satisfies Proof.Our argument will follow closely the proof of [1,Th.2].As mild solution, X n satisfies The process X n is (F t )-measurable with càdlàg paths and attains values in D 1 .First, we obtain from (7.4) by interchanging integrals and A ∈ L(D 1 ) for t ≥ 0 that Each term on the right hand side of (7.5) is almost surely Bochner integrable.Indeed, integrability of the first term is immediate from the uniform boundedness principle.For the second term, boundedness of F in Condition (A2) and commutativity of S and R n implies Almost sure Bochner integrability of the stochastic integral in (7.5) follows from boundedness of G in Assumption(A3), commutativity of S and R n , and Theorem 2.4 via the estimate Integrating both sides of (7.5) results in the equality Applying Fubini's theorems, see Theorem 2.4 for the stochastic version, and the equality which verifies X n as a strong solution to (6.1).
We denote by L n the usual generator associated with the Yosida approximations X n , n ∈ N, defined for f ∈ C 2 b (H) and h ∈ D 1 by The right hand side of (7.6) is well defined by Lemma 5.2.Recall that the counterpart to L n for the mild solution X denoted by L was introduced in (7.1).The generators L n and L are related by the following crucial convergence result.
Lemma 7.4.Let (X n ) n∈N be solutions of (6.7) which a.s.converges uniformly to the solution of (6.1).It follows for each Proof.We obtain for each h ∈ D 1 that Taylor's remainder theorem in the integral form implies In the same way, we obtain and also Applying the last three estimates to (7.7) and taking expectation on both sides, it follows from Inequality (2.2) and for each n ∈ N that Since the boundedness conditions in (A2) and (A3) guarantee an application of Lebesgue's dominated convergence theorem verifies (7.8) and (7.9), which completes the proof.
Proof of Theorem 7.1.Let (X n ) n∈N be the solutions of (6.7).Because of Corollary 6.10, we can assume that (X n ) n∈N converges uniformly to the solution of (6.1) a.s.Proposition 7.3 enables us to apply the Itô formula in Theorem 5.1 to X n , which results in almost surely for all t ≥ 0. Applying the product formula to the real-valued semi-martingale V (X n (•)) and the function t → e β 3 t and taking expectations on both sides of (7.10) shows Here, we used the fact that the last two integrals in (7.10) define martingales, and thus have expectation zero.This follows from the observation that they are local martingales according to Lemma 2.3 and Theorem 5.1 and are uniformly bounded.The latter is guaranteed by the boundedness of G in (A3), since similarly, by using Lemma 5.2, The first term on the right hand side in (7.11) is finite since The same holds for the second term, which can be shown using the same arguments as in the proof of Lemma 7.4.By applying Inequality (7.3) to (7.11), we conclude Lemma 7.4 together with Fatou's lemma implies Applying Assumption (7.2) completes the proof.

Mild Itô formula
In this section, we prove an Itô formula for mild solutions of Equation (6.1) and mappings f ∈ C 2 b (H) such that the second derivative D 2 f is not only continuous but satisfies The subspace of all these functions is denoted by C 2 b,u (H).
Theorem 8.1 (Itô formula for mild solutions).A mild solution X of (6.1) satisfies for each where the limit is taken in L 0 P (Ω, R).
Remark 8.2.Note that while X may not be a semimartingale, the compensated measure µ X − ν X in (8.2) still exists as X is both adapted and càdlàg; see [17, Chap.II].
Remark 8.3.Unlike in similar situation with the driving noise being Gaussian e.g. in [25] we do not identify the limit in (8.2) as then the imposed assumptions on f are very restrictive.In many applications (see e.g.[1]), it is enough to identify some bound on which leads to natural assumptions on the generator A.
We divide the proof of the above theorem in some technical lemmas.To simplify the notation, we introduce the function Proof.Taylor's remainder theorem in the integral form and Inequality (2.2) imply A similar argument yields Proof.Theorem 3.5 guarantees for each n ∈ N that T f (X(s−), h) µ Xn (ds, dh) Since the last line is independent of n ∈ N and d m α → 0 as m → ∞ according to Inequality (2.2), Markov's inequality implies that there exists m 1 ∈ N such that for all m ≥ m 1 and all n ∈ N P T f (X(s−), ∆X n (s)) − T f (X(s−), ∆X(s)) ½ B H (1/m) c (∆X(s)). (8.10) For estimating the first term in the last line, we use the equality ½ A (x)−½ A (y) = ½ A (x)½ A c (y)− ½ A c (x)½ A (y).For the first term, resulting from the application of this identity, we conclude from Taylor's remainder theorem in the integral form that  (8.11)   where we used the notation c f := D 2 f ∞ .Applying Theorem 3. .
Since X has only finitely many jumps in B H (1/2m) c on [0, t] and ∆X n (s) converges to ∆X(s) for all s ∈ [0, t], there exists n 2 such that for all n ≥ n 2 Applying this together with (8.12) to Inequality (8.11) proves that for m ≥ m 2 there exists n 2 such that for all n ≥ n 2 T f (X(s−), ∆X n (s)) − T f (X(s−), ∆X(s)) ½ B H (1/m) c (∆X(s)) ≥ Applying this together with (8.14)   Continuity of f shows that f (X n (t)) → f (X(t)) and f (R n x 0 ) → f (x 0 ) a.s.Inequality (2.7) implies for the first integral in (8.16) that

• 0 G
n dL in the topology of uniform convergence in probability on [0, T ].

Lemma 3 . 1 .
The set function ν defined in (3.1) is well defined and extends to a predictable random measure on B([0, T ]) ⊗ B(H).This extension is unique among the class of σ-finite random measures on B([0, T ]) ⊗ B(H) that assign 0 mass to the origin.Proof.Step 1: We show that for all open sets B ⊆ H with 0 / ∈ B the process

0≤s≤t∆∆ s 0 G * e i dL s 0 G * e j dL = 0≤s≤t 1 2 ∆ s 0 G 2 − ∆ s 0 G * e i dL 2 − ∆ s 0 G * e j dL 2 . 2 • 0 G• 0 G * e i dL c − • 0 G * e j dL c .
X(s), e i X(s), e j = 0≤s≤t * (e i + e j ) dL The very definition (4.1) of the continuous part leads us to [[X]] c , e i ⊗ e j = [[X]] , e i ⊗ e j − 0≤s≤t ∆ X(s), e i X(s), e j = 1 * (e i + e j ) dL c − Since Step 2 guarantees that the processes • 0 G * (e i + e j ) dL, • 0 G * e i dL and • 0 G * e i dL are purely discontinuous, it follows that [[X]] c , e i ⊗ e j = 0 for all i, j ∈ N which completes the proof.

Lemma 8 . 4 .
Let λ be the cylindrial Lévy measure of L. It follows for every
1,op adp denote the class of adapted, simple L(H)-valued processes bounded in the operator norm by 1 on [0, T ].An arbitrary predictable process G : Ω × [0, T ] → L 2 (U, H) is stochastically integrable if there exists a sequence of adapted simple processes (G n ) n∈N ⊂ S HS [17, and any non-negative predictable function W .The measure ν is determined uniquely up to a set of probability zero by (2.10) and is called the compensator of µ; see[17,  th.II.1.8].If Y is an H-valued, adapted càdlàg process then the integer-valued random measure µ Y characterised by Th. II.2.21.].The fact that (ii) is an equivalent definition of the compensator is proved in [17, Th.II.1.8.].Proposition 2.7 justifies the following standard notation: if W is predictable and (2.11) is locally integrable, we define the following local martingale we say that V is purely discontinuous; see e.g.[32, Se.II.6].The concept of quadratic variation is generalised for a càdlàg semimartingale Z with values in the separable Hilbert space H in [26, Se. 26].Let (f i ) i∈N denote an orthonormal basis of H.There exists a unique stochastic process [[Z]] with values in the Hilbert-Schmidt tensor product of H satisfying .16)where the assumptions of Lemma 9.1 are satisfied by boundedness of F , see (6.2), and tightness of {F (X n (s)), m ∈ N} implied by Lemma 6.7 and continuity of F .
Lemma 8.5.Let (X n ) n∈N be a sequence of càdlàg processes in H which converges to a process X both in C([0, T ], L p (Ω, H)) and uniformly on [0, T ] almost surely.Then it follows for any f ∈ C 2 b,u (H) and t ∈ [0, T ] that to(8.10)showsT f (X(s−), h) µ Xn (ds, dh) − µ X (ds, dh) ≥ ǫ 3 ≤ ǫ ′ .(8.15)By applying Equations (8.8),(8.9)and (8.15) to (8.7), the proof is now complete.Proof of Theorem 8.1.Let (X n ) n∈N be the solutions to (6.7).According to Corollary 6.10, we can assume that (X n ) n∈N converges both in C([0, T ], L p (Ω, H)) and uniformly on [0, T ] almost surely to the mild solution X, which has càdlàg paths.Since X n is a strong solution to (6.7) according to Proposition 7.3, the Itô formula in Theorem 5.1 implies for all t ≥ 0 and n ∈ N that