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We establish the existence and uniqueness of strong solutions to stochastic porous media equations driven by Lévy noise on a $σ$-finite measure space $(E,\mathcal{B}(E),μ)$, and with the Laplacian replaced by a negative definite self-adjoint operator. The coefficient $Ψ$ is only assumed to satisfy the increasing Lipschitz nonlinearity assumption, without the restriction $rΨ(r)\rightarrow\infty$ as $r\rightarrow\infty$ for $L^2(μ)$-initial data. We also extend the state space, which avoids the transience assumption on $L$ or the boundedness of $L^{-1}$ in $L^{r+1}(E,\mathcal{B}(E),μ)$ for some $r\geq1$. Examples of the negative definite self-adjoint operators include fractional powers of the Laplacian, i.e. $L=-(-Δ)^α,\ α\in(0,1]$, generalized Schrödinger operators, i.e. $L=Δ+2\frac{\nabla ρ}ρ\cdot\nabla$, and Laplacians on fractals., Revised version, 24 pages, proof of Claim 4.1 is revised by using the added assumption (H4). The statements of Theorem 3.1 and Proposition 4.1 are revised

In this paper, we establish the existence and uniqueness of both mild(/variational) solutions and weak (in the sense of PDE) solutions of coupled system of 2D stochastic Chemotaxis-Navier-Stokes equations. The mild/variational solution is obtained through introducing a new method of cutting off the stochastic system and using a fixed point argument in a carefully constructed Banach space. To get the weak solution we first prove the existence of a martingale weak solution and then we show that the pathwise uniqueness holds for the martingale solution.

In this paper, we establish the existence and uniqueness of solutions of stochastic nonlinear Schrödinger equations with additive jump noise in $L^2(\mathbb{R}^d)$. Our results cover all either focusing or defocusing nonlinearity in the full subcritical range of exponents as in the deterministic case.

In this paper, we consider stochastic second grade fluids driven by Levy noise on a bounded domain of R 2 . Global existence and uniqueness of strong probabilistic solutions are established, which improves the existing corresponding results for martingale solutions in literature.

In this paper, we establish a large deviation principle for stochastic Burgers type equation with reflection perturbed by the small multiplicative noise. The main difficulties come from the highly non-linear coefficient and the singularity caused by the reflection. Here, we adopt a new sufficient condition for the weak convergence criteria, which is proposed by Matoussi, Sabbagh and Zhang [14].

A Freidlin-Wentzell-type large deviation principle is established for \begin{document}$ 2D $\end{document} stochastic Navier-Stokes equations on time-dependent domains driven by Brownian motion, which captures situations where the regions of the fluid change with time.

In this paper, we establish a moderate deviation principle for stochastic models of two-dimensional second-grade fluids driven by Levy noise. We will adopt the weak convergence approach. Because of the appearance of jumps, this result is significantly different from that in Gaussian case.

In this paper, we consider the averaging principle for one dimensional stochastic Burgers equation with slow and fast time-scales. Under some suitable conditions, we show that the slow component strongly converges to the solution of the corresponding averaged equation. Meanwhile, when there is no noise in the slow component equation, we also prove that the slow component weakly converges to the solution of the corresponding averaged equation with the order of convergence 1 − r , for any 0 r 1 .

For a heat equation with memory driven by a L\'evy-type noise we establish the existence of a unique solution. The main part of the article focuses on the Freidlin-Wentzell large deviation principle of the solutions of heat equation with memory driven by a L\'evy-type noise. For this purpose, we exploit the recently introduced weak convergence approach.

In this paper, we establish the large deviation principle for 3D stochastic primitive equations with small perturbation multiplicative noise. The proof is mainly based on the weak convergence approach., Comment: arXiv admin note: text overlap with arXiv:1204.6417 by other authors

In this paper, we establish the large deviation principles for stochastic porous media equations driven by time-dependent multiplicative noise on $\sigma$-finite measure space $(E,\mathcal{B}(E),\mu)$, and the Laplacian replaced by a negative definite self-adjoint operator. The coefficient is only assumed to satisfy the increasing Lipschitz nonlinearity assumption without the restrictions to its monotone behavior at infinity for $L^2(\mu)$-initial data or compact embeddings in the associated Gelfand triple. Applications include fractional powers of the Laplacian, i.e. $L=-(-\Delta)^\alpha,\ \alpha\in(0,1]$, generalized $\rm Schr\ddot{o}dinger$ operators, i.e. $L=\Delta+2\frac{\nabla \rho}{\rho}\cdot\nabla$, and Laplacians on fractals., Comment: A revised version, 27 pages, the proof of Claim 4.1 is revised by using the added assumption (H2)(iii). The statements of Theorem 3.2, Theorem 4.1 and Lemma 4.1 are revised

Motivated by the remarkable works of Busse and his collaborators in the 1980s on turbulent convection in a rotating layer, we explore the long-run behavior of stochastic Lotka-Volterra (LV) systems both in pull-back trajectories and in stationary measures. A decomposition formula is established to describe the relationship between the solutions of stochastic and deterministic LV systems and the stochastic logistic equation. By virtue of this formula, it can be verified that every pull-back omega limit set is an omega limit set of the deterministic LV system multiplied by the random equilibrium of the stochastic logistic equation. The formula is also used to derive the existence of a stationary measure, its support and ergodicity. We prove the tightness of stationary measures and that their weak limits are invariant with respect to the corresponding deterministic system and supported on the Birkhoff center. The developed theory is successfully utilized to completely classify three dimensional competitive stochastic LV systems into 37 classes. The expected occupation measures weakly converge to a strongly mixing measure and all stationary measures are obtained for each class except class 27 c). Among them there are two classes possessing a continuum of random closed orbits and strongly mixing measures supported on the cone surfaces, which weakly converge to the Haar measures of periodic orbits as the noise intensity vanishes. The class 27 c) is an exception, almost every pull-back trajectory cyclically oscillates around the boundary of the stochastic carrying simplex characterized by three unstable stationary solutions. The limit of the expected occupation measures is neither unique nor ergodic. These are consistent with symptoms of turbulence. (C) 2019 Elsevier Masson SAS. All rights reserved.

In this paper, we establish a Freidlin–Wentzell-type large deviation principle for stochastic models of two-dimensional second grade fluids driven by Lévy noise. The weak convergence method introduced by Budhiraja, Dupuis and Maroulas plays a key role.

In this paper, we establish the exponential mixing property of stochastic model for the incompressible second grade fluids. The general criterion established by Odasso (2008) plays an important role.

We consider the dynamics of the stochastic shadow Gierer–Meinhardt system with one-dimensional standard Brownian motion. We establish the global existence and uniqueness of solutions. We also prove a large deviation result.

In this paper, we establish a large deviation principle for a type of stochastic partial differential equations (SPDEs) with locally monotone coefficients driven by L\'evy noise. The weak convergence method plays an important role., Comment: arXiv admin note: text overlap with arXiv:1203.4020 by other authors

In this paper we first obtain a necessary condition for \(H\)-cadlag modification and \(H\)-weakly cadlag modification of generalized Ornstein–Uhlenbeck processes with Levy noises in Hilbert spaces \(H\). Then we give a necessary and sufficient condition for the \(H\)-cadlag modification and \(H\)-weakly cadlag modification of Ornstein–Uhlenbeck processes driven by cylindrical \(\alpha \)-semistable processes. Secondly, we investigate the properties of cylindrical cadlag modification and \(V\)-cylindrical cadlag modification. Applying the obtained results to diagonal Ornstein–Uhlenbeck processes with \(\alpha \)-stable noises, we show a necessary and sufficient condition for cylindrical cadlag modification and \(V\)-cylindrical cadlag modification in the symmetric case for \(\alpha \in (0,1)\) and give a sufficient condition in the general case for \(\alpha \in (0,2)\). Some examples illustrate the relations among the concepts of various cadlag modifications.

In this paper we prove uniform Hausdorff and packing dimension results for the images of a large family of Markov processes. The main tools are the two covering principles of Xiao (second author). As applications, uniform Hausdorff and packing dimension results for certain classes of L\'evy processes, stable jump diffusion and non-symmetric stable-like processes are obtained., Comment: We revised the assumptions (A2) and (A3) in the previous version, the new assumptions are more general and expected to cover more examples

The limiting behavior of stochastic evolution processes with small noise intensity $\epsilon$ is investigated in distribution-based approach. Let $\mu^{\epsilon}$ be stationary measure for stochastic process $X^{\epsilon}$ with small $\epsilon$ and $X^{0}$ be a semiflow on a Polish space. Assume that $\{\mu^{\epsilon}: 0

In this paper, we establish a moderate deviation principle for two-dimensional stochastic Navier-Stokes equations driven by multiplicative $L\acute{e}vy$ noises. The weak convergence method introduced by Budhiraja, Dupuis and Ganguly in arXiv:1401.73v1 plays a key role., Comment: arXiv admin note: text overlap with arXiv:1401.7316, arXiv:1203.4020 by other authors

A necessary and sufficient condition of cadlag modification of Ornstein–Uhlenbeck process with cylindrical stable noise in a Hilbert space is given in this Note. Applying this result, some questions in Time irregularity of generalized Ornstein–Uhlenbeck processes [C. R. Acad. Sci. Paris, Ser. I 348 (2010) 273–276] and Structural properties of semilinear SPDEs driven by cylindrical stable process [Probab. Theory Related Fields 149 (2011) 97–137] are answered.

In this paper, we study the existence of martingale solutions of stochastic 3D Navier–Stokes equations with jump, and following Flandoli and Romito (2008) [7] and Goldys et al. (2009) [8] , we prove the existence of Markov selections for the martingale solutions.

In this paper, we establish a large deviation principle for stochastic models of incompressible second grade fluids. The weak convergence method introduced by Budhiraja and Dupuis (Probab Math Statist 20:39–61, 2000) plays an important role.

In this paper, we establish a large deviation principle for two-dimensional stochastic Navier–Stokes equations driven by multiplicative Lévy noises. The weak convergence method introduced by Budhiraja, Dupuis and Maroulas [ Ann. Inst. Henri Poincaré Probab. Stat. 47 (2011) 725–747] plays a key role.

We study SPDEs with two reflecting walls Λ 1 Λ 2 and two singular drifts c 1 ( X − Λ 1 ) ϑ , c 2 ( Λ 2 − X ) ϑ , driven by space–time white noise. First, we establish the existence and uniqueness of the solutions X for ϑ ≥ 0 , which satisfy Λ 1 ≤ X ≤ Λ 2 . Second, we obtain the following pathwise properties of the solutions X. If ϑ > 3 , then a.s. Λ 1 X Λ 2 for all t ≥ 0 ; If 0 ϑ 3 , then X hits Λ 1 or Λ 2 with positive probability in finite time. Thus ϑ = 3 is the critical parameter for X to hit reflecting walls.

In this paper, we established a large deviation principle for stochastic models of incompressible second grade fluids. The weak convergence method introduced by \cite{Budhiraja-Dupuis} plays an important role., Comment: arXiv admin note: substantial text overlap with arXiv:1505.03940

In this paper, we prove a central limit theorem and establish a moderate deviation principle for 2D stochastic hydrodynamical type systems with multiplicative noise in unbounded domains, which covers 2D Navier–Stokes equations, 2D MHD models and the 2D magnetic Bénard problem and also shell models of turbulence. The weak convergence method plays an important role in obtaining the moderate deviation principle.

In this paper, we study an ergodic theorem of a parabolic Andersen model driven by Levy noise. Under the assumption that A = (a(i, j))i,j∈S is symmetric with respect to a σ-finite measure gp, we obtain the long-time convergence to an invariant probability measure νh starting from a bounded nonnegative A-harmonic function h based on self-duality property. Furthermore, under some mild conditions, we obtain the one to one correspondence between the bounded nonnegative A-harmonic functions and the extremal invariant probability measures with finite second moment of the nonnegative solution of the parabolic Anderson model driven by Levy noise, which is an extension of the result of Y. Liu and F. X. Yang.

In this paper, we establish a large deviation principle for a fully non-linear stochastic evolution equation driven by both Brownian motions and Poisson random measures on a given Hilbert space $H$. The weak convergence method plays an important role., Comment: arXiv admin note: substantial text overlap with arXiv:1203.4020 by other authors

We first study the existence of stationary weak solutions of stochastic 3D Navier–Stokes equations involving jumps, and the associated Galerkin stationary probability measures for this case. Then we present a comparison between the Galerkin stationary probability measures for the case driven by Lévy noise and the one driven by Wiener processes.