NONLINEAR STOCHASTIC PARTIAL DIFFERENTIAL EQUATIONS OF HYPERBOLIC TYPE DRIVEN BY L´EVY-TYPE NOISES

. We study a class of abstract nonlinear stochastic equations of hyperbolic type driven by jump noises, which covers both beam equations with nonlocal, nonlinear terms and nonlinear wave equations. We derive an Itˆo formula for the local mild solution which plays an important role in the proof of our main results. Under appropriate conditions, we prove the non-explosion and the asymptotic stability of the mild solution.

1. Introduction. In this paper we will study two nonlinear hyperbolic second order in time stochastic partial differential equations (SPDEs) driven by Lévy-type noises: the beam and the wave equations (with emphasis on the former). We propose a framework that covers both types of equations. Let us briefly describe the motivation for these equations.
Due to its widespread applications in structural and mechanical engineering, the subject of nonlinear vibrations of Euler-Bernoulli beam has been intensively studied by many authors (see [33] for the free oscillations of a damped beam with immovable ends under an axial force, [5] for the effect of a compressive axial load, [14] for the effect of an axial periodic load on the motion of a hinged beam and [24] for a model with a nonlinear friction force, to name just a few). A type of nonlinear stochastic beam equation describing nonlinear vibrations of an elastic panel subjected to random fluctuations (state dependent white noises) was investigated by Chow and Menaldi in [12]. By means of a stochastic energy equation, they proved the existence, uniqueness and regularity of solutions. Recently the second named author, Maslowski and Seidler in [8] studied a wide class of abstract stochastic nonlinear beam equations perturbed by a white noise in a Hilbert space which is applicable to the equation treated in [12]. With the help of Lyapunov functions, they proved non-explosion of the mild solution and established the asymptotic stability of the solution. Most of the previous studies on hyperbolic SPDEs considered Gaussianity perturbations. However a huge amount of experimental evidences have demonstrated that Lévy-type models possess properties which describe the physical, biological and financial phenomena more accurately than the pure diffusion-based models, see cf. [1,4,10,13,28]. For instance, the real asset prices move essentially by jumps and large, sudden movements may occur over the time scales. Also the growth and recruitment of planktonic fish larvae and the magnetization in ferromagnetic materials are modelled by means of jump processes. The problem of vibration of a road bridge under running vehicles is usually modeled as a simply supported beam structure subjected to moving loads. Because vehicle loads are random in nature and traffic flows may suddenly increase or decrease, this leads to discontinuous gaps between vehicle flows. Those features are unattainable by diffusion-based models but appear as prevailing in models with jumps. Lévy-type models are more sophisticated and not easily amenable to mathematical analysis. Many results achieved in diffusion models are thus deserve to be re-investigated when jumps are included and our paper could be seen as a modest contribution to theoretical underpinning of such applied research.
In our paper, we investigate the existence and uniqueness of global mild solutions to a stochastic model arising in the nonlinear theory of structural dynamics and aeroelasticity by focusing on the following abstract second order stochastic differential equation perturbed by jump noises u tt + A 2 u + ∇ u Φ(u) + f (t, u, u t ) = Z g(t, u, u t , z) N (t, dz), (1) where u is an H-valued stochastic process with H being a Hilbert space, A is a self-adjoint operator on H such that A ≥ µI for some µ > 0, Φ : D(A) → [0, ∞) is an "energy" function (with the gradient ∇Φ understood with respect to the Hilbert space structure on H) and N is a compensated Poisson random measure. The two motivating examples of the function Φ are those corresponding to the beam and nonlinear wave equations. In the former case Φ is of the form where m : [0, ∞) → ∞) is a C 1 class increasing function such that m(0) = 0 and its derivative is a locally Lipschitz continuous function, B is a self-adjoint operator such that BA −1 ∈ L(H). In the latter, more concrete, case, H = L 2 (D) for some domain D ⊂ R d , d ∈ N, A 2 being the − Laplacian with the Dirichlet boundary conditions, and, for suitable p ≥ 2.
Note that in the former case ∇ u Φ(u) = m (|B 1 2 u| 2 )Bu, u ∈ D(A) while in the latter ∇ u Φ(u) = |u| p−1 u, u ∈ D(A). We will first prove, see also [34] where the case of globally Lipschitz coefficients is carefully investigated, that under some natural local Lipschitz continuity conditions on the coefficients f and g, Equation (1) has a unique maximal local mild solution given by (11). More importantly, we will prove, see Theorem 2.1, that the maximal local solution is a global one and establish in Theorem 2.2 the ultimate boundedness and stability of such solutions. Let us stress that this is not a simple issue due to the presence of a nonlinear term involving the function m. The main ingredient of the proofs of these two theorems is a general version of the Itô Lemma for mild solutions. Note that the results about maximal local mild solution (Proposition 1) and the Itô Lemma (Lemma 4) are proved in a more general form for equation (10) than what is needed to establish the non-explosion of the mild solutions, i.e. for functions F and G not being of the special forms (7) or (9). On comparing with the method used in the case of stochastic beam equation driven by Wiener process, the factorization method used in showing the uniform L p -convergence of the Yosida approximation for stochastic convolutions w.r.t. the Wiener noise, is not applicable in our case. Instead of considering the Yosida approximation A n , we follow the approximation procedure introduced in [29] and [30]. We first apply the ordinary Itô formula to D(A)-valued solution processes, then investigate its limit and obtain an Itô-type formula for the mild solution when the D(A)-valued solution processes converges to the mild solution, here D(A) is the domain of the generator A. As a result, the Itô formula we established for mild solutions is sufficient to cover the two different cases: the existence of a global solution and its asymptotic boundedness. Compared with [8], where the two results were proved separately, we give a more efficient direct proof of the main results. As our major motivation, we also show that under some standard assumptions the results we have proved for problem (1) can be applied to a wider class of models including stochastic nonlinear wave equations and stochastic nonlinear beam equations subject to either the periodic boundary condition or mixed hinged/clamped boundary conditions. Stochastic PDE driven by discontinuous noise is a very new subject. So far mainly problems with Lipschitz coefficients have been investigated, see the recent monograph [25] by Peszat and Zabczyk and/or papers [18] by Hauseblas and [27] by Riedle. A type of stochastic PDEs with monotone and coercive coefficients, which is weaker than the usual Lipschitz and linear growth assumptions, driven by some discontinuous perturbations were studied by Gyöngy and Krylov in [16] for the finite-dimensional case and extended by Gyöngy to infinite-dimensional spaces in [17]. Recently, the authors and Liu in [3] established the existence and uniqueness of strong solutions for a large class of SPDEs with coercive and locally monotone coefficients driven by Lévy processes. Stochastic reaction diffusion equations driven by Lévy noises have been a subject of a recent paper [9], where also some comments on the existing literature can be found. The approach of the current paper is different as it does not use any compactness methods but instead follow a more natural route of constructing a maximal local solution and then proving that its lifespan is equal to infinity, see [8] and [6] for the gaussian noise case. To our best knowledge the present paper is the first one in which this approach is applied to SPDEs with non-Lipschitz coefficients and non-gaussian noise. Note however, that this method has been used and further developed in a joint paper by the current authours and Liu in [3]. It is a natural and interesting question to combine the results obtained in this paper together with those from [7] and prove the existence of an invariant measure for (11). However, contrary to the finite dimensional case, see e.g. [2], this is still an open problem and its resolution is postponed till the next publication.
The rest of the paper is organized as follows. Section 2 gives a detailed description of the problem, the main results and its hypotheses. Section 3 is devoted to proving a basic auxiliary result about stopped stochastic convolutions. Section 4 proves the existence and uniqueness of maximal local mild solution, while Section 5 proves the crucial Lemma, Itô Lemma. The proofs of the main theorems 2.1 and 2.2, are given in Section 6 and Section 7.
For the convenience of a reader let us describe the structure of the proof of Theorem 2.1 about the existence and uniqueness of a global solution to Problem (10). The existence and uniqueness of a local maximal solution is formulated in Proposition 1, which in turn is proved in section 4. In section 5 we formulate an infinite dimensional Itô Lemma for processes which are not semimartingales but are defined certain stochastic convolutions. This result is general enough to be applicable in the proof of both main theorems. In section 3 we formulate Lemma 2 about equality of two possible definitions of stopped stochastic convolution processes. To conclude the proof of Theorem 2.1 it is enough to show that the lifetime of the local maximal solution is equal to infinity. The proof of that fact is contained in section 2.1. For this we define auxiliary stopping times τ n , by (51), and auxiliary processes f n , g n , F n and G n defined by (52, 53, 54, 55) and consider processes v n which are global solutions of the auxiliary linear SPDE (56). We apply to v n the Itô formula from our fundamental section 5 and, since by Lemma 2, v n and u are equal up to τ n we conclude the proof. The proof of the first part of Theorem 2.2 relies on the use of Itô formula from section 5 applied a modified energy function (58).
2. Framework and main results. Throughout the whole paper we assume that (Ω, F, P) is a probability space, equipped with a filtration F = (F t ) t≥0 satisfying the usual hypotheses 1 , and that (Z, Z, ν) is a σ-finite measure space. We denote by N the compensated Poisson random measure on [0, T ] × Ω × Z with the intensity measure ν, so that We use P to denote the predictable σ-field on R + × Ω, that is the σ-field generated by all left continuous and F-adapted real-valued processes. We write BF for the σ-field of all F-progressively measurable sets on R + × Ω, i.e.
The two motivating stochastic evolution equations are the wave equation in R d , for p > 1 and the beam equation in O ⊂ R d , where m ∈ C 1 (R + , R + ).
Of course the above equations have to be supplemented by initial and boundary conditions. A common feature of these two problems is that they can be written as where E is the energy defined by for the wave equation, and in the case of the beam equation, and ∇ u E is the gradient of E with respect to the Hilbert space L 2 (R d ) or L 2 (D). In fact, one consider the following generalised energy for the nonlinear beam equation In such a case, the generalised energy for the nonlinear beam equation will take the following form With an appropriate choice of the linear operators A and B, these problems can be written in a unified way As a byproduct of this approach we will see that Equation (3) with the energy function (5) has at least two non-equivalent formulations depending on the choice of the operator A (and hence the boundary conditions) while B is the Laplace operator with fixed boundary conditions.
Suppose that H is a real separable Hilbert space with an inner product ·, · and a corresponding norm | · | H . By B(H) we denote the Borel σ-field on H. Let A and B be self-adjoint operators in H. Suppose that B ≥ 0 and A ≥ µI for some µ > 0. We also assume that BA −1 ∈ L(H) and functions which is a generator of a C 0 -unitary group e tA , t ∈ R, on H, see [19], and functions M : Then Equation (6) can be rewritten as a system of first order equations for the unknown function u(t) = (u(t), u t (t)) with respect to the time variable in the following form, with u 0 = (u 0 , u 1 ) ∈ H, It is useful to notice that the above equation is more general than the problem (6). Indeed, the functions F and G do not need to be of the special form (7) and (9). We will formulate our local existence result, see Proposition 1, and the Itô Lemma, see Lemma 4, in this more general way, i.e. for general functions F and G. From that we will deduce a corresponding result for special functions F and G as in (7) and (9), however, with general functions f , m and g (of course satisfying some natural assumptions).
2 As before, one could add, to avoid a slight chance of ambiguity, "H-valued"-càdlàg.
Let us recall, see for instance [6], that a stopping time τ is called to be accessible if there exists an increasing sequence {τ n } n∈N of stopping times such that τ n < τ and lim n→∞ τ n = τ a.s. and this sequence {τ n } n∈N will be called an approximating sequence for τ .
Using the above introduced notion we can state the main definition in this paper. One can fully understand it only after becoming familiar with the results presented in section 3. (10) is an H-valued, F-adapted, càdlàg local process u = (u(t)) 0≤t<τ , where τ is an accessible stopping time with an approximating sequence {τ n } n∈N , such that for any n ∈ N, the stopped process

Definition 2.3. A local mild solution to Equation
where I τn (G(u)) is a process defined by For the future reference we specifically state the following important observation.

Remark 2. If the local mild solution of Equation
In order to show the existence and uniqueness of a global mild solution to Equation (10), it is sufficient to impose local Lipschitz continuity and some natural growth conditions on the functions F and G.
There exists a constant L F > 0 such that for all t ∈ [0, T ], Let us now formulate a basic result about the existence of a local maximal solution, see Definition 2.3. For the existence of global solutions we will additionally need the following classical assumptions on F and G.

Condition (C.3).
Function G is of linear growth on H locally uniformly w.r.t. t ∈ R + , i.e. for every T > 0, there exists constants
where, m : [0, ∞) → ∞) is a C 1 class increasing function such that m(0) = 0 and the derivative m is locally Lipschitz continuous.  Let us now formulate two main results of our paper. The first one is about the existence of global solutions while the second is concerned with stability of solutions.
Theorem 2.1. Suppose that conditions (C.1)-(C.5) are satisfied. Then for every F 0 -measurable H-valued intial data u 0 , there exists a unique global mild solution to Equation (10).
Our last result is about asymptotic behaviour of solutions. This result is proved under more stringent conditions than the previous results. Let us formulate the relevant conditions.

Condition (C.6).
The function F is related to a function f : R + × D(A) × H → H via formula (7) and the latter is of the following form. There exists δ > 0 such that

Condition (C.7).
There exists α > 0 such that the function m from (C.5) satisfies Note that in this case, the function M defined by (8) is Lipschitz on balls.
For the stability of the global solutions we will need the following stronger version of condition (C.3).

Condition (C.8).
Function G is of linear growth on H uniformly w.r.t. t ∈ R + , i.e. there exists constants K G , R G ≥ 0 such that for all t ≥ 0, Theorem 2.2. Assume (C.2), (C.8) and (C.5)-(C.7). Assume furthermore that then there exist constants C > 0 and λ > 0 such that Note that Proposition 1 holds true for problem (10) in the general form, i.e. for the functions F and G satisfying conditions (C.1) and (C.2), and the M satisfying the locally Lipschitz condition. Theorem 2.1 holds true for problem (10) in the special form since the functions M, F and G are of special form (8), (7) and (9) respectively, and the corresponding functions m, f and g satisfy conditions (C.5), (C.4) and (C.3). Finally, Theorem 2.2 holds for problem (10) in the above case when also functions M, F and G are special form (8), (7) and (9) respectively, for functions m, f and g satisfying also conditions (C.5), (C.7), (C.6) and (C.8).
Since the stochastic nonlinear beam equation with either the hinged or the clamped boundary conditions can be treated in almost the same way as in [8], as only the the stochastic term will be different, we will discuss applications to the stochastic nonlinear beam equation with periodic boundary conditions. However, in order that a reader can easily spot the differences and similarities between our Example and section 4 from [8] we have decided to keep as much as possible the notation from that paper.
Definition 2.5. Suppose that Λ is a topological space and X 1 , X 2 and Y are normed vector spaces. A Borel function This definition can be given of various natural generalizations which we simply do not state.
Example 1 (Stochastic nonlinear beam equation with the periodic boundary condition). Assume that L > 0 and let T n be a n-dimensional torus of length L, i.e.
It is well known that T n is a riemannian manifold (without boundary) and that functions defined on T n can be identified with functions defined on R n which are L-periodic in each coordinate (i.e. L n -periodic). Partial differential equations on a torus are often used as the simplest model of PDES on manifolds where there is no need to introduce deep theory from Differential Geometry. A boundary or initial value problem with periodic boundary conditions is often identified as a corresponding problem on the torus. Somehow incorrectly, one can identify T n with the product on n intervals [0, L) each equipped with a metric d(s, t) = min{|t − s|, |t − L − s|}. One can also identify (correctly) that T n with a cartesian product of copies of a circle of length L. In this way, one can see that T n is endowed with the Lebesgue measure m on the σ-fields of its Borel sets or of Lebesgue measurable sets. This gives rise to the family of Lebesgue spaces L p (T n ) for each p ∈ [1, ∞]. With a bit of extra work one can define the scales of the Sobolev spaces H k,p (T n ) for k ∈ N and p ∈ [1, ∞]. Every function u ∈ H k,p (T n ) can be uniquely lifted to an L nperiodic function U ∈ H k,p loc (R n ) so that u satisfies naturally periodicity conditions. These spaces satisfy the classical Gagliardo-Nirenberg inequalities, for instance if k − n p > 0, then u ∈ H k,p (T n ) has a unique representativeũ which belongs to C(T n ). Note also that the liftŨ ofũ belongs to C(R n ).
Define a linear operator B in H = L 2 (T n ) by It is well known that B is a self-adjoint operator in H. Next let us now define an operator P by It is well know that P is a strictly position self-adjoint operator on the Hilbert space H = L 2 (T n ) and Let A 1 be the unique positive square root of P . It satisfies Obviously, since D(P 1/2 ) is equal to the complex interpolation space of order with the periodic boundary conditions (written for simplicity in the case n = 2) u(0, ·) = u(L, ·), u(·, 0) = u(·, L) Moreover we assume that the following growth conditions are satisfied.
1. There exists a constant K Υ such that for all t ∈ [0, T ], x ∈ D, a ∈ R, b ∈ R n and c ∈ R, 2. There exists a constant L Π such that for all t ∈ [0, T ], x ∈ D, a ∈ R, b ∈ R n and c ∈ R, If n = 1 or n ≤ 3 and the functions Υ and ψ depend only on the first and the second variables (i.e. on x and u), then there exists a unique maximal global mild solution to Equation (16). P ϕ = ∆ 2 ϕ, for ϕ ∈ D(P ). It is well known that both B and P are self-adjoint, B is positive and P is nonnegative. As in [8] we can check that it is also positive. Indeed, if ϕ ∈ D(P ) then by the Stokes Theorem and by [15,Lemma 9.17] where the constant K > 0 is independent of ϕ. Set A = P Thus, as in the previous Example, we have the following Assertion: if n = 1 or n ≤ 3 and the functions Υ and ψ depend only on the first and the second variables (i.e. on x and u), then there exists a unique maximal global mild solution to Equation (16) with the mixed clamped/hinged boundary conditions (19).
Example 3 (Nonlinear stochastic wave equations). We consider the following stochastic equation on R d where γ > 0 and Γ and Λ are nonlinear terms. Let H = L 2 (R d ). We consider two cases d = 1 and d ≥ 2. We suppose that: respect to the 3rd, 4th and 5th variables, locally uniformly with respect to the 1st and uniformly with respect to the 2nd.
x, u, ∇u, u t )) is locally Lipschitz is with respect to the 3rd variable and globally Lipschitz with respect to 4th and 5th, locally uniformly with respect to the 1st and uniformly with respect to the 2nd.
x, u, ∇u, u t )) is globally Lipschitz is with respect to the 3rd, 4th and 5th variables, locally uniformly with respect to the 1st and uniformly with respect to the 2nd.
Then Equation (20) has a unique local mild solution. Moreover, if there exists L > 0 and L and nonnegative functions κ, κ ∈ L 1 (R d ) such that for all t ∈ [0, ∞), almost all x ∈ R d and all a, c ∈ R and b ∈ R d , then Equation (20) has a unique global mild solution.
Remark 5. As we will see below, the wave equation is a special case of the general problem with m = 0.
Proof of the Claim made in Example 3. Let A be positive square root of the strictly positive self-adjoint operator −δ 2 ∆ + γ 2 I on H. Then A ≥ γI, D(A) = W 1,2 (R d ) and |Au| 2 H = δ 2 |∇u| 2 L 2 + γ 2 |Au| 2 H . Although, we could consider a more general case, we put m = 0. Define f and g to be the Nemytski maps corresponding to functions Γ and Λ respectively, i.e.
Put C = 1 min{δ 2 ,γ 2 } . Consider the case d = 1 as the other one is trivial. Let ψ 1 , ψ 2 ∈ D(A) with their norms being bounded by a fixed R > 0. Since by the Sobolev embedding Theorem, W 1,2 (R) ⊂ C b (R) ⊂ L ∞ (R), we can find N > 0 such that |ψ i | L ∞ ≤ N . Denoting by L N the Lipschitz constant of the function Γ on ball of radius N with respect to the 3rd variable and by L the Lipschitz constant of the function Γ with respect to the 4th and 5th variables we get, for all t ∈ [0, ∞), Similarly, in the general case, it's easy to verify that Remark 6. As in [8] it is worth pointing out that our results are valid for bounded and unbounded domains.
3. An auxiliary Lemma. Let (e tA ) t∈R be a C 0 -semigroup on a Hilbert space H. Assume that τ is an accessible stopping time. Let ϕ = ϕ(t)), t ≥ 0 be an H-valued process belonging to M 2 loc (P ⊗ Z, H). Set Note that by the choice of process ϕ and the assumption about (e tA ) t∈R , the stochastic convolution process I(t), t ≥ 0, is well defined. Also for any stopping time τ , the process 1 [0,τ ] (t, ω) is predictable. In fact, the predictable σ-field is generated by the family of closed stochastic intervals {[0, T ] : T is a stopping time}, see [22]. This together with the predictability of ϕ implies that integrand of I τ (t) is predictable. Thus the stochastic convolution I τ (t) is well defined as well. Moreover, one can always assume that the stochastic convolution process I(t) and I τ (t), t ≥ 0 are H-valued càdlàg, see [35]. The following lemma verifies (11) in Definition 2.3 of a local mild solution. The proof below is mainly based on [8] (which in turn was provided by Martin Ondreját, see [23]). Let us point out a minor but important difference with [8]. We consider here closed random intervals [0, τ ], while in the case of a Wiener process [8] we considered open random intervals [0, τ ). Our formula (21) is also more general than formula [8, (A.4)] as we allow an additional time parameter r.

Lemma 2.
Under the assumptions listed above, for any stopping time τ and for all r ≥ t ≥ 0, In particular, Remark 7. It is known, see [6] and [8] for the Wiener process case, that is if ξ is another process satisfying the same conditions as ϕ such that for some Ω 0 ∈ F, In particular, if Ω 0 ⊂ {ω ∈ Ω : ξ(·, ω, ·) = ϕ(·, ω, ·) on [0, τ (ω)]}, then the above equality implies that P-a.s. on Ω 0 One of the consequences of Lemma 2 is the following modification of the above assertions. Proof. Indeed, note that obviously, for t ≥ 0, P-a.s., Thus, equality follows by applying (22).

Remark 8.
An alternative approach to Lemma 2 and Corollary 1 is via the method implemented by [7] and [11], i.e. maximal inequalities for stopped stochastic convolution processes. In our case, this would require to use Theorems 4.4, 4.5 and 5.1 from [35].
Proof of Lemma 2. It is enough to proof identity (21), as the equality (22) follows from the former by replacing the deterministic times r = t by a stopping time t ∧ τ . It is also enough to proof identity (21) when r = t. We first verify it for deterministic time. Let τ = a. If t < a, then Thus equality (21) holds for any deterministic time. Now let τ be an arbitrary stopping time. Define τ n := 2 −n ([2 n τ ] + 1), for each n ∈ N. That is τ n = k+1 2 n if k 2 n ≤ τ < k+1 2 n . Then τ n τ as n → ∞ pointwise. Since equality (21) has been proved for each deterministic time k2 −n , in view of Remark 7 we infer that Since τ n τ , by the P-a.s. right-continuity of the process I, see [35], the random variable I(t ∧ τ n ) converges pointwise on Ω to I(t ∧ τ ) as n → ∞ for every t ≥ 0 P-a.s. Also, observe that converges to 0 as n → ∞. Thus we conclude that e (t−t∧τn)A I(t ∧ τ n ) converges to e (t−t∧τ )A I(t ∧ τ ), for each t ≥ 0, P-a.s. For the term I τn (t), by the Itô isometry we find out that Recall that that τ n ↓ τ as n → ∞. So 1 [0,τn] converges to 1 [0,τ ] as n → ∞.
Obviously, the integrand is bounded by |ϕ(s, z)| 2 for all n. Thus by the Lebesgue dominated convergence theorem it follows that Hence we can always find a subsequence which is convergent a.s. Finally, Letting n → ∞ in both sides of (23) yields (21). This completes the proof.

Proofs of Proposition 1.
Proof of Proposition 1.
Since functions F , M and G are Lipschitz on closed balls in H, for every n ∈ N we may find globally Lipschitz functions F n : H → H and G n : H → H such that F n = F and G n = G onB H (0, n), the closed ball in H of radius n and centered at the origin. By using a classical argument we infer that there exists a unique mild solution (u n (t)) t≥0 to problem (10) with F replaced by F n and G replaced by G n , see e.g. Theorem 4.1.10 in [34]. By the càdlàg property of the process u n , a random variable τ n defined by Let Φ(u n ) denote the right hand side of (24). Note that the value of Φ(u n ) at τ n depends only on the values of u n on [0, τ n ), i.e.
Hence we may extend the process u n from the interval [0, τ n ) to the closed interval [0, τ n ] by setting where I τn (G(u n )) is a process defined by (12) with u replaced by u n , i.e.
The proof that the maximal local solution is a global one is based on the use of the Khashminski test, see [20] and [8]. The essence of this method is to prove first the existence and uniqueness of a local maximal solution, then to find an appropriate Lyapunov function, with the help of which prove the life span of that local maximal solution is equal to ∞. This method had been earlier used in the parabolic case in [11].
In order to prove the main result we need the following auxiliary standard result whose proof can be found in [8]. The function V satisfying conditions below is called a Lyapunov function for Equation (10).

5.
An Itô Lemma. In this section we will formulate a general form of an Itô Lemma which in the next two sections will be used to prove the existence of a global solution and the asymptotic boundedness of the solutions. The novelty of our approach is that we prove one result general enough to cover the two different cases. In the case of stochastic beam equation driven by a Wiener process, see [8], the authors formulated and proved two separate results. Our proof would yield, had we considered that case here, those results as corollaries.
We begin with formulation of the general assumptions and the Itô Lemma. This will be proceeded by two examples when these general assumptions are satisfied. The section will be finished with the proof of the Itô Lemma.
and define a function V : H → R + by An obvious consequence of the above definition is that V (x) = V 0 (x) whenever π 1 x = 0 i.e.
However, we also have two important equalities, for x ∈ H and y = (0, y 2 ) ∈ H: V (x)(y) = Qx, y H Note that since we assume that m is a C 1 -class function with locally Lipschitz derivative, the function V is also of C 1 -class on H, with the Fréchet derivative of V being Lipschitz on balls of H. See also Lemma 5 for further important consequences of this regularity assumption. But it seems that in order to be able to apply our Itô Lemma, it would be sufficient to assume that the derivative functions m is only locally Hölder continuous. Let us also note the following inequalities comparing the behaviour of V (x) and |x| H .
In order to prove (34) let us note that in view of (35) it is sufficient to prove the second inequality in (34) for V 1 . Since by assumptions: the function m is increasing and, for some C > 0, |By| ≤ C Let us note that it follows from (34) that V is a bounded function on bounded subsets of H and that it satisfies condition (29) of the Khasminski Lemma 3. Now we are ready to formulate the announced Itô Lemma, the main result in this section.
Lemma 4. Assume that the operator Q satisfies Assumption 1 and the function m satisfies Assumption (C.5). Let the function V be defined by formula (31). Assume that u = (u, v) be a global mild solution of the problem (10) and let σ be a bounded stopping time such that the processes u(r), F (r, u(r)) and G(r, u(r)) are uniformly bounded on [0, σ) × Ω. Then we have P-a.s., V (u(σ))e λσ − V (u(s))e λs = σ s e λr λV (u(r)) + Γ(u(r), u(r)) + 2m (|B Note that Q 0 = 2I. The following properties of Q β have been established in the last cited paper. The operator Q β is an isomorphism of H and (39) and hence the quadratic form has a unique extension from D(A) to a bounded quadratic form on the whole H and thus the quadratic form Γ β satisfies In particular, Q β satisfies Assumption 1. In sections 6 and resp. 7 we will use Lemma 4 with Q = Q β for β = 0, resp. β > 0. To our best of our knowledge, the operator Q appeared for the first time in the paper [26] in connection with a stability analysis of linear deterministic hyperbolic equations and was applied to stochastic hyperbolic problems in [21]. The above result was also used in [8] for similar purposes as in the current paper.
Remark 9. As noted in Example 4 above, if β = 0 then Q = 2I, i.e. Q 0 = 2I. Then, by (40), Γ 0 = 0 and Q 0 x, y H = 2 x 2 , y 2 for all x = (x 1 , x 2 ) ∈ H and y = (0, y 2 ) ∈ H. Thus, invoking the definitions (7), (8) and (9) of maps F , M and G in terms of maps f , m and g respectively, we infer that the following is thus a special case of our equality (36), where we also put λ = 0, Indeed, when Q = Q β , by (32) and (33), we have for x = (x 1 , x 2 ) ∈ H and y = (0, y 2 ) ∈ H Note however, that in our proof, contrary to the corresponding proof in [8], we will not prove the above special formula (41) but the general one (36) Note that every Hilbert space is 2-smooth. The following result is an immediate consequence of [31, Lemma 2.1 and Lemma 2.2].
Lemma 5. For every r > 0, there exists C = C(r) > 0 such that Indeed, function V is of C 2 class and it's first Frechét derivative is Lipschitz on balls.
Proof of Lemma 4. We start the proof with constructing a sequence of global strong solutions which converges to the global mild solution uniformly. To do this, let us set, see [30], for l ∈ N, t ∈ R + , ω ∈ Ω and z ∈ Z, , ω)) and We will apply the standard version of Itô formula for each fixed l and then a limit when l → ∞ will be taken.
Let us introduce the following two canonical linear projections: Let us also observe that Our approach here differs from the one used in [8], where instead the Yosida approximation of the operator A was used.
The following result can be applied to the above approximations with Y equal to H and S equal to either (0, T ) × Ω or (0, T ) × Ω × Z.
Let A be the infinitesimal generator of a contraction C 0 -semigroup on Y . Then Proof. The proof is straightforward, since by the Hille-Yosida Theorem, we know |(lI − A) −1 | ≤ 1 l and for every s ∈ S, (lI − A) −1 ξ(s) → ξ(s) pointwisely.
Since by definition, the processes F l and G l take values in D(A), we infer F l ∈ M 2 loc (BF ⊗ Z; D(A)) and G l ∈ M 2 loc (P ⊗ Z; D(A)), for l ∈ N. Therefore, the equation has a unique global strong solution u l given by, for t ≥ 0, u l (t) = u l (0) + t 0 Au l (r) + F l (r, u(r)) dr + t 0 Z G l (r, u(r−), z) N (dr, dz). Now we can apply Itô formula to the process u l and the function e λt V (x) to get V (u l (σ))e λσ − V (u l (s))e λs = σ s e λr λV (u l (r)) + V (u l (r) Au l (r) + F l (r, u(r)) dr (45) We next prove the following auxiliary result.
Lemma 7. For every T > 0 and every n ∈ N, Proof. Let us fix T > 0. Then we have The result follows by applying the Cauchy-Schwarz inequality, Lemma 6 to processes F and G and the Davis inequality for stochastic convolution processes, see [35].
Therefore, by taking a subsequence we can deduce that P-a.s., for every T > 0, Since H is a Hilbert space, the Frechét derivative V of function V can be identified with the gradient DV of V which satisfy (we continue to use notation x = (x 1 , x 2 ) and y = (y 1 , y 2 )) V (x)(y) = Qx, y H + 2m (|B , see [8]. By taking into account (44) and Assumption 1 we infer that Since m ∈ C 1 and B ∈ L(D(A), H), we infer that P-a.s. uniformly in [0, σ(ω)], m (|B and Bπ 1 u l (s) → Bπ 1 u(s), as l → ∞.
Moreover, as the maps Γ; H × H → R and Q : H → H are continuous, we infer that Γ(u l (r), u l (r)) → Γ(u(r), u(r)) and Qu l (r), F l (r) H → Qu(r), F (r, u(r)) + M(u(r)) H . Since the process u l is càdlàg, we infer that for every l ∈ N, the set {u l (t, ω) : t ∈ [0, T ]} is relatively compact subset of R for almost all ω and the sequence {u l } l∈N converges uniformly to v, P-a.s., we infer that the set {u l (s), s ∈ [0, T ], l ∈ N} is bounded in H, P-a.s. Therefore, P-a.s., s ∈ [0, T ], Hence, by the Lebesgue DCT, we conclude that σ s Qu l (r), F l (r) H dr → σ s Qu(r), F (r, u(r)) + M(u(r)) H dr P-a.s.
For the last term on the right side of inequality (45), we observe that By applying the Itô isometry property of the stochastic integral, see [34] and then the the Lebesgue DCT we get Therefore, by passing a subsequence in (45), it is evident to see that the Itô formula (36) holds.
6. Proof of Theorem 2.1. We will now prove Theorem 2.1 with the help of the two lemmas from the preceding sections. In particular, as announced earlier, we will use Lemma 4 with the operator Q = Q 0 = 2I and the parameter β = 0, see Example 4 and Remark 9.
Proof of Theorem 2.1. Let u(t), 0 ≤ t < τ ∞ , be a maximal local mild solution to problem (10). Our aim is to prove that τ ∞ = ∞. Before we continue with the proof let us observe that it is sufficient to prove that for any T > 0, τ ∞ ≥ T . This is of particular importance because our time dependent coefficients satisfy Conditions (C.3) and (C.4). For this purpose, we fix a positive time T > 0 and we replace the stopping time τ ∞ by τ ∞ ∧ T . We will prove that τ ∞ ∧ T ≥ T . Whenever we will speak in this section about the infimum of an empty set we will define it to be equal to T . Define a sequence of stopping times by As in the proof of Proposition 1, we can show that {τ n } n∈N is an approximating sequence of the accessible stopping time τ ∞ ∧ T . Let us set β = 0. We have Q = 2I. Then V (x) = |x| 2 H + m(|B Now it remains to verify condition (30) from Lemma 3. Notice that the solution u to Equation (10) is a process with possibly finite lifespan. We therefore introduce a sequence of globally defined processes F n and G n , n ∈ N, such that, roughly speaking, up to the stopping time τ n , the solution u agrees with a solution v n of a certain linear stochastic evolution equation. The Itô formula is applied to the process v n and then a limit when n → ∞ is taken. Let us now show details of this program. We begin with fixing n ∈ N. Then we introduce the following processes, for t ∈ [0, T ], f n (t) = 1 [0,τn) (t)f (t, u(t ∧ τ n )), g n (t, z) = 1 [0,τn] (t)g(t, u(t ∧ τ n −), z), for z ∈ Z, G n (t, z) = 0, g n (t, z) , for z ∈ Z.
One can see that both processes F n and G n are bounded. Consider the following linear equation dv n (t) = Av n (t)dt + F n (t)dt + Z G n (t, z) N (dt, dz), t ∈ [0, T ], v n (0) = u(0). Finally, according to the fact that u coincides with v P-a.s. up to time t ∧ τ n , by applying conditions (13) and (14), we get, for t ∈ [0, T ], This implies inequality (30) with C 1 = 2K f,T + K G,T and C 2 = 2R f,T + R G,T .