Existence and stability of mild solutions to parabolic stochastic partial differential equations driven by Lévy space-time noise

This paper is concerned with the well-posedness and stability of parabolic stochastic partial differential equations. Firstly, we obtain some sufficient conditions ensuring the existence and uniqueness of mild solutions, and some H-stability criteria for a class of parabolic stochastic partial differential equations driven by Lévy space-time noise under the local/non-Lipschitz condition. Secondly, we establish some existenceuniqueness theorems and present sufficient conditions ensuring the H′-stability of mild solutions for a class of parabolic stochastic partial functional differential equations driven by Lévy space-time noise under the local/non-Lipschitz condition. These theoretical results generalize and improve some existing results. Finally, two examples are given to illustrate the effectiveness of our main results.


Introduction
It is well known that stochastic partial differential equations (SPDEs) are appropriate mathematical models for many multiscale systems with uncertain and fluctuating influences, which are playing an increasingly important role in accurately describing complex phenomena in physics, geophysics, biology, etc.In recent years, the theoretical research of SPDEs has attracted a large number of research workers, and has already achieved fruitful results.There are many interesting problems, such as well-posed problem, blow-up problem, stability, invariant measures and other properties, which have been extensively investigated for different kinds of SPDEs.We refer the reader to [4,9,20,[23][24][25][26]29] for more details and some new developments.There are many results in which the coefficients satisfy the global Lipschitz condition and the linear growth condition [8,14].However, the global Lipschitz condition, even the local Lipschitz condition, is seemed to be considerably strong in discussing variable applications in the real world.Obviously, we need to find some weaker or more general conditions ensuring the existence and uniqueness of solutions of SPDEs.
Xie [27] investigated the following stochastic heat equation driven by space-time white noise ∂ 2 u(t, x) ∂x 2 + b t, x, u(t, x) + σ t, x, u(t, x) where {W(t, x), t ≥ 0, x ∈ R} is a two-sided Brownian sheet and the coefficients b, σ : [0, ∞) × R × R → R are continuous nonlinear functions.Such an equation arises in many fields, such as population biology, quantum field, statistical physics, neurophysiology, and so on, see [6,12,19].By using the successive approximation argument, Xie [27] studied the existence of mild solutions to equation (1.1) under some conditions weaker than the Lipschitz condition.
It is worth pointing out that the work of [27] focuses on the SPDE driven by Browian motion whose path is continuous.However, many abrupt changes such as environmental shocks for the population, sudden earthquakes, hurricanes or epidemics may lead to the discontinuity of the sample paths.Therefore, SPDEs driven by Brownian motions are not appropriate to model some real situations where large external and/or internal fluctuations with possible large jumps might exist.But Lévy noise can produce large jumps or exhibit long heavy tails of the distribution which makes the sample paths discontinuous in time, so SPDEs driven by Lévy noise are more suitable for the actual situation (see [1,18,21]).In [1], Albeverio, Wu and Zhang established the existence and uniqueness of mild solutions for a class of stochastic heat equations driven by compensated Poisson random measures.In [21], Shi and Wang discussed the mild solutions to SPDEs driven by Lévy space-time white noise under the uniform Lipschitz condition.So far as we know, however, there has been no mathematical treatment about the pathwise uniqueness to parabolic SPDEs driven by Lévy noise under some kinds of conditions weaker than the Lipschitz condition.Inspired by the work of Xu, Pei and Guo [28], in this paper we shall promote the work of Xie [27] and investigate the following parabolic SPDE ∂ 2 u(t, x) ∂x 2 + b t, x, u(t, x) + σ t, x, u(t, x) L(t, x), t ≥ t 0 , x ∈ R, u(t 0 , x) = u 0 (x), (1.2) where t 0 ≥ 0, (t, x) ∈ [t 0 , +∞) × R, L(t, x) is Lévy space-time white noise, and the coeffi- cients b, σ : [t 0 , +∞) × R × R → R are usually continuous nonlinear functions.Here, our primary task is to investigate the existence and uniqueness of mild solutions to (1.2) under the local/non-Lipschitz condition which includes the Lipschitz condition as a special case.Furthermore, in order to obtain the dynamical properties of solutions to SPDEs driven by Lévy noise, we shall seek for some H-stability conditions under which the mild solutions of (1.2) are H-stable.
We also notice that the above mentioned results are based on the fact that the future of systems is independent of the past states and is determined solely by the present.However, in realistic models many dynamical systems depend on not only the present but also the past states and even the future states of the systems.Stochastic functional differential equations (SFDEs) give a mathematical formulation for such models.Recently, the investigation of stochastic partial functional differential equations (SPFDEs) has attracted the considerable attentions of researchers and many qualitative theories of SPFDEs have been obtained in literature [3,7,8,10,11,15,16,22].There are a lot of substantial results on the existence and uniqueness of solutions to SPFDEs.For example, Taniguchi [22] and Luo [15] employed the Banach fixed point theorem and the successive approximation method to study the existence and uniqueness of mild solutions for SPFDEs under the global Lipschitz condition and the linear growth condition.By using the stochastic convolution, Govindan [8] investigated the existence, uniqueness and almost sure exponential stability of neutral SPFDEs under the global Lipschitz condition and the linear growth condition.Luo and Guo [17] studied the existence and uniqueness of mild solutions for parabolic SPFDEs driven by Winner space-time white noise under the non-Lipschitz condition.To the best of our knowledge, there is few work about the well-posedness and stability for mild solutions to parabolic SPFDEs driven by Lévy space-time white noise.
Motivated by the previous problems, in this paper we further investigate the following parabolic SPFDE driven by Lévy space-time noise are Borel measurable functions and are perhaps not Lipschitz, and L(t, x) is Lévy space-time white noise.Therefore, the other two tasks of this paper are to discuss the well-posedness of mild solutions to (1.3) under the local/non-Lipschitz condition, and to obtain some sufficient conditions ensuring the H -stability of mild solutions to (1.3).
The rest of the paper is organized as follows.After presenting some preliminaries in the next section, we establish some existence-uniqueness theorems under the local/non-Lipschitz condition and provide some sufficient conditions ensuring the H-stability of mild solutions to SPDE (1.2) in Section 3. Section 4 is devoted to the well-posedness and H -stability of mild solutions to SPFDE (1.3) under the local/non-Lipschitz condition.Two examples are provided in Section 5 to illustrate our main results.
Throughout this paper, the letters C and C represent some positive constants which may change occasionally their values from line to line.If C and C are essential to depending on some parameters, e.g.T etc, which will be written as C T and C T , respectively.

Preliminaries
In this section, let us recall some basic definitions and introduce some notations and assumptions.Assume that (Ω, F , {F t } t≥t 0 , P) is a complete probability space with the filtration {F t } t≥t 0 , which satisfies the usual condition, i.e., {F t } is a right continuous, increasing family of sub σ-algebras of F and F t 0 contains all P-null sets of F .Let H be the family of all random fields {X(t, x), t ≥ t 0 , x ∈ R} defined on (Ω, F , {F t } t≥t 0 , P) such that where r > 0. Following from Borel-Cantell's lemma [5], H equipped with the norm • H is a Banach space.

Gaussian kennel and its properties
Let the Gaussian kernel G(t, x) denote the fundamental solution of the Cauchy problem where δ 0 stands for the Dirac function.By Fourier transform, we obtain Let g(t, x, z) = G(t, x − z) for all t > 0, x, z ∈ R, then the heat kernel g(t, x, z) have the following properties (see [27] for details).
(i) For each r ∈ R and T > 0, there exists a constant C depending only on r, t 0 and T such that (ii) If 0 < p < 3, then there exists a positive constant C such that (iii) If 3 2 < p < 3, then there exists a positive constant C such that for all t ∈ [t 0 , T], (iv) If 1 < p < 3, then there exists a positive constant C such that for all t and t (t ) (2.8)

Bihari's lemma
We give two lemmas without proofs (see [27] for the proofs), which will be used many times in the following analysis.

Existence and stability of mild solutions to SPDE (1.2)
This section is devoted to the existence and uniqueness of mild solutions to (1.2) under the non-Lipschitz condition.Equation (1.2) can be given by the following integral equation In view of the definition of Lévy space-time white noise, we obtain for all (t, x) ∈ [t 0 , T] × R, with the mappings ψ(s, z) and h(t, z, y) defined by where we suppose that all integrals on the right-hand side of (3.1) exist, and I U 0 denotes the indicator function of the set U 0 .

Well-posedness of mild solutions to (1.2)
Now, we recall Shi and Wang's recent work [21] on the existence of mild solutions of the following SPDE with Lévy space-time white noise, which is induced by the λ-fractional differential operator, where is Lévy space-time white noise, ∆ λ is λ-fractional differential operator and is defined via Fourier transform by We introduce the following assumptions: (P1) b, σ are uniform Lipschitz, i.e. there exists a constant where the functions ψ(t, •) and h(t, •, y) are specified in (3.1), and the norm • p is defined as Under the assumptions (P1)-(P3), Shi and Wang [21] obtained the following result by the Banach's fixed point theorem.
However, it is easy to find that the condition (P1) is very stringent in Proposition 3.1 and Corollary 3.3.A natural question arises: Whether or not can the condition (P1) be relaxed to the local Lipschitz case or the non-Lipschitz case?In what follows, we shall investigate the existence and uniqueness of SPDE (1.2) under the following assumptions (i.e., the local Lipschitz condition and the non-Lipschitz condition).
(S1) b, σ are local Lipschitz, i.e. there exists a positive constant K n such that for all (t, x) ∈ (S'1) If there exist a strictly positive, nondecreasing function λ(t) defined on [t 0 , T] and a nondecreasing, continuous function φ(u) defined on R + such that for all (t, x) where λ(t) is a locally integrable function, φ(u) or φ 2 (u)/u is a concave function with (S2) The functions ψ(t, z) and h(t, z, y) satisfy the following conditions sup (S3) b(t, x, 0) and σ(t, x, 0) are locally integrable functions with respect to t and x.
Remark 3.4.The condition (S'1) is so-called non-Lipschitz condition.In particular, when λ(t) = k is a positive constant and φ(u) = u, then the condition (S'1) can be reduced to the Lipschitz condition.
Utilizing the method of [21], we obtain the following conclusion.
Before stating our main results, we give the following auxiliary conclusion.
Lemma 3.6.Suppose that the conditions (S1) and (S3) or (S'1) and (S3) hold, then there exists a positive constant K such that for all (t, x, u) Proof.Here, we shall only prove the conclusion under the conditions (S'1) and (S3).For the conditions (S1) and (S3), it can be shown by the same techniques.Since φ(u) or φ 2 (u)/u is a concave and non-negative function satisfying φ(0) = 0, we can choose two appropriate positive constants k 1 and k 2 such that Therefore, by utilizing (3.3), we have Thus the proof of Lemma 3.6 is completed.
Remark 3.7.Lemma 3.6 tells us that the linear growth condition can be obtained by utilizing the local Lipschitz condition or the non-Lipschitz condition.This result plays an important role in proving the following important conclusion.Now, we shall present a well-posed result for the mild solutions of (1.2) under the local Lipschitz condition.Theorem 3.8.Under the conditions (S1)-(S4), there exists an H-valued solution u(t, x) to SPDE (1.2) with the initial value u 0 (x).
Proof.Here we only outline the proof by utilizing a truncation procedure.For every integer n ≥ 1, we define truncation functions b n (t, x, u(t, x)) and σ n (t, x, u(t, x)) by where we set Then b n and σ n satisfy the uniform Lipschitz condition (P1).Therefore, by Corollary 3.5, there exists a unique H-valued solution u n (t, x) to the following equation g(t − s, x, z)σ n s, z, u n (s, z) h(s, z, y) Ṅ(dy, dz, ds). ( Introduce the stopping time where we set inf φ = ∞ if possible.It is easy to see that and η n ↑ T as n → ∞.Therefore, there exists an integer n 0 such that η n = T when n ≥ n 0 .Let , which combining with (3.5), yields that From the definition of truncation functions b n (t, x, u(t, x)) and σ n (t, x, u(t, x)), it follows that Let n → ∞ we observe that u(t, x) satisfies equation (3.1), which implies that u(t, x) is the solution of equation (1.2).Therefore, we complete the proof of the theorem.
The following Lemma is a corollary of Bihari's lemma which can be found in [27] and will provide some help in the forthcoming proof.Lemma 3.9.Assume that λ(t) and φ(u) satisfy the condition (S'1).If for some α ∈ (0, 1  2 ], there exists a nonnegative measurable function z(t) satisfying z(0) = 0 and In what follows, we study the existence of mild solutions for SPDE (1.2) with initial value u 0 (x) under the non-Lipschitz condition.Theorem 3.10.Under the conditions (S'1), (S2)-(S4), there exists a unique H-valued solution u(t, x) to SPDE (1.2) with the initial value u 0 (x).
We prepare a lemma in order to prove this theorem.Lemma 3.11.Under the conditions (S'1) and (S3), the solution u(t, x) of SPDE (1.2) with the initial value u 0 (x) satisfies where C i (i = 0, 1, . . ., 4) are positive constants specified in the following proof.In particular, u(t, x) is an element of the Banach space H.
Proof of Theorem 3.10.The proof will be divided into two steps.
Step 1.We firstly show the existence of mild solutions to SPDE (1.2) by the successive approximation scheme.Define u 0 (t, x) = R g(t, x, z)u(t 0 , z)dz, then for n = 1, 2, . . ., we set Here, we can show that {u n (t, x)} ∞ n=0 is a uniformly bounded sequence in H by induction.From (S4), it is easy to see u(t 0 , x) ∈ H. Assume that u n−1 (t, x) ∈ H, we will prove that u n (t, x) ∈ H. Using the similar arguments as above, we have By Lemma 3.11, we get where √ T−t 0 .Therefore, {u n (t, x)} ∞ n=0 is a uniformly bounded sequence in H.We shall prove that {u n (t, x)} ∞ n=0 is a Cauchy sequence of the Banach space H. Suppose that m, n are any two integers, we have Using the similar arguments as above, we have which implies that Therefore, by Lemma 3.9, we deduce that lim m,n→∞ u n − u m H = 0, which implies that {u n (t, x)} ∞ n=0 is a Cauchy sequence of the Banach space H. Let u(t, x) denote its limit.We pass to limits both sides of the equation (3.13) to prove that u(t, x), t ∈ [t 0 , T], x ∈ R satisfies (3.1), P a.s., which means that u(t, x) is a solution of (1.2).
Step 2. In what follows, let us show the uniqueness of mild solutions to SPDE (1.2).We suppose that u (1) (t, x) and u (2) (t, x) are two solutions of equation (1.2).From Lemma 3.11, it follows that both of them belong to the Banach space H.Moreover, we have − σ s, z, u (2) (s, z) h(s, z, y) Ṅ(dy, dz, ds) Using the similar arguments as above, we have Using Jensen's inequality and (2.4), we obtain From Lemma 3.9, it follows that u (1) − u (2) H = 0 for all t ∈ [t 0 , T] and x ∈ R, which means u (1) (t, x) = u (2) (t, x) for all t ∈ [t 0 , T], x ∈ R, P a.s.Thus the proof of Theorem 3.10 is completed.
Remark 3.12.If h 1 (t, z, y) = h 2 (t, z, y) = 0, then the Lévy space-time white noise will be reduced to Wiener space-time white noise, and equation (1.2) will be converted to the form of (1.1).Therefore, Theorem 2.1 in [27] can be regarded as a special case of Theorem 3.10.
Remark 3.13.Here, we utilize the successive approximation argument to prove Theorem 3.10.
Of course, we can use the same method to deduce Theorem 3.1 of [21].Under the conditions (S'1), (S2)-(S4), we can also employ the same approach to obtain some relevant results about (3.2), which will promote the work of [21].Indeed, our results under the non-Lipschitz condition can be considered as a generalization of those of Theorem 3.1 of [21] and Theorem 2.2 of [1].

Stability of mild solutions to (1.2)
In this subsection, we mainly investigate the stability of mild solutions to (1.2).Now let us give the definition of H-stability.
Definition 3.14.A solution u(t, x) of SPDE (1.2) with initial value u(t 0 , x) is said to be H-stable if for all > 0 there exist δ > 0 such that whenever sup x∈R E |u(t 0 , x) − u(t 0 , x)| 2 < δ, where u(t, x) is another solution of SPDE (1.2) with initial value u(t 0 , x).
Remark 3.15.The H-stability is defined by utilizing the norm of space H, which can be regarded as a promotion of ordinary stochastic stability, and can be applied to describe the dynamic behaviors of non-trivial solutions of SPDEs in the Banach space H.
Proof.The existence has been shown in Theorem 3.10, we now only need to prove the Hstability of the mild solution.Let u(t, x) and u(t, x) be two solutions of equation (1.2) with initial value u(t 0 , x) and u(t 0 , x), respectively.Then, we have Using the similar arguments as above, it is not difficult to get where C and C are two positive constants.We now set ϕ(m) = C λ(s) From assumption (S'1), it follows that ϕ(m) is obviously a positive, continuous and nondecreasing concave function for any fixed s, which satisfies ϕ(0) = 0 and 0 + 1 ϕ(m) dm = ∞.Therefore, for any > 0, set 1 2 , we have lim γ→0 Therefore, by utilizing Lemma 2.2 and Lemma 2.3, we can obtain for any t ∈ [t 0 , T], the estimate χ(t) = u − u H < holds, which implies the stability of the mild solution.Thus, we complete the proof of Theorem 3.16.

Existence and stability of mild solutions to SPFDE (1.3)
In this section, we mainly study the well-posedness and stability of mild solutions to SPFDE (1.3).As before, we still work on the given complete probability space (Ω, F , {F t } t≥t 0 , P) with the filtration {F t } t≥t 0 satisfying the usual conditions, and L(t, x) is Lévy space-time white noise.Let τ > 0, C([−τ, 0] × R, R) be the space of all continuous functions ϕ(θ, x) defined on [−τ, 0] × R with the norm ϕ = sup −τ≤θ≤0,x∈R |ϕ(θ, x)|.We also assume that u t (x) Let H be the family of all random fields {X(t, x), t ≥ t 0 − τ, x ∈ R} defined on (Ω, F , {F t } t≥t 0 , P) such that where r > 0. Then the space H with the norm • H will be a Banach space.Therefore, for all (t, x) ∈ [t 0 , T] × R, the mild solution of (1.3) can be written as follow where we assume that all integrals on the right-hand side of (4.1) exist, and I stands for the indicator function.

Well-posedness of mild solutions to (1.3)
In this subsection, our main task is to study the existence of mild solutions to (1.3) under the non-Lipschitz condition by the successive approximation method.Before stating our main results, we impose the following assumptions on the coefficients b and σ.
(N1) b, σ are local Lipschitz, i.e. there exists a positive constant K n such that for all (t, x) ∈ [t 0 , T] × R and u 1 , (N'1) If there exist a strictly positive, nondecreasing function λ(t) defined on [t 0 , T] and a nondecreasing, continuous function φ(u) defined on R + such that for all (t, x) ∈ [t 0 , T] × R and u 1 , where λ(t) is a locally integrable function, φ(u) or φ 2 (u)/u is a concave function with φ(0) = 0 satisfying 0 + 1 φ(u) du = ∞.Remark 4.1.Here, we can obtain the similar conclusion as Lemma 3.6 by the same argument.Suppose that the conditions (N1) and (S3) or (N'1) and (S3) hold, then there exists a positive constant K such that for all (t, x, u) Now we provide a result on the existence and uniqueness of (1.3) under the non-Lipschitz condition (N'1).For the reader's convenience, we give a lemma in order to prove this theorem.Lemma 4.3.Under the conditions (N'1) and (S3), the solution u(t, x) of SPFDE (1.3) with the initial value ξ(θ, x) satisfies where the constants C 0 and C 1 will be specified later.In particular, u(t, x) is an element of the Banach space H .
Proof.Here the proof is very similar to Lemma 3.11, thus we only sketch the proof.For any integer n ≥ 1, we define the stopping time . By (4.1), then for t ∈ [t 0 , T], u n (t, x) satisfies the following equation Therefore the Gronwall inequality yields that Consequently Finally, the required inequality follows by letting n → ∞.
Proof of Theorem 4.2.The proof will be developed in two steps.
Step 1.We shall show the existence of mild solutions to (1.3) by using the successive approximation method in the Banach space H . Set u 0 (t, x) = R g(t, x, y)ξ(0, y)dy, u 0 x, y)ξ(θ, y)dy and define Similarly to the proof of Theorem 3.10, we can show that {u n (t, x)} ∞ n=0 is a uniformly bounded sequence of the Banach space H by induction.Meanwhile, for any two positive integer m and n, we can obtain Utilizing Fatou's lemma and Lemma 3.9, we derive that lim m,n→∞ which implies that {u n (t, x)} ∞ n=0 is a Cauchy sequence of the Banach space H .We assume that u(t, x) ∈ H is its limit.It is easy to see that u(t, x), t ∈ [t 0 , T], x ∈ R satisfies equation (4.1) by taking limits both sides of equation (4.4), which means that u(t, x) is the solution of system (1.3).
Remark 4.4.It is worth pointing out that when h 1 (t, z, y) = h 2 (t, z, y) = 0, the Lévy space-time noise will be reduced to Wiener space-time white noise, and equation (1.3) will be converted to the form of equation (1.3) in [17].Thus, Theorem 4.2 can be seen as a generalization of Theorem 3.4 of [17].
Example 5.2.Consider the following parabolic delayed SPDE driven by Lévy space-time noise: