Regularity of Ornstein-Uhlenbeck processes driven by a L{\'e}vy white noise

The paper is concerned with spatial and time regularity of solutions to linear stochastic evolution equation perturbed by L\'evy white noise "obtained by subordination of a Gaussian white noise". Sufficient conditions for spatial continuity are derived. It is also shown that solutions do not have in general \cadlag modifications. General results are applied to equations with fractional Laplacian. Applications to Burgers stochastic equations are considered as well.Comment: This is an updated version of the same paper. In fact, it has already been publishe

Stochastic Reaction-diffusion Equations Driven by Jump Processes

We establish the existence of weak martingale solutions to a class of second order parabolic stochastic partial differential equations. The equations are driven by multiplicative jump type noise, with a non-Lipschitz multiplicative functional. The drift in the equations contains a dissipative nonlinearity of polynomial growth.Comment: See journal reference for teh final published versio

Inviscid Large deviation principle and the 2D Navier Stokes equations with a free boundary condition

Using a weak convergence approach, we prove a LPD for the solution of 2D stochastic Navier Stokes equations when the viscosity converges to 0 and the noise intensity is multiplied by the square root of the viscosity. Unlike previous results on LDP for hydrodynamical models, the weak convergence is proven by tightness properties of the distribution of the solution in appropriate functional spaces

Non-autonomous stochastic evolution equations and applications to stochastic partial differential equations

In this paper we study the following non-autonomous stochastic evolution equation on a UMD Banach space $E$ with type 2, {equation}\label{eq:SEab}\tag{SE} {{aligned} dU(t) & = (A(t)U(t) + F(t,U(t))) dt + B(t,U(t)) dW_H(t), \quad t\in [0,T], U(0) & = u_0. {aligned}. {equation} Here $(A(t))_{t\in [0,T]}$ are unbounded operators with domains $(D(A(t)))_{t\in [0,T]}$ which may be time dependent. We assume that $(A(t))_{t\in [0,T]}$ satisfies the conditions of Acquistapace and Terreni. The functions $F$ and $B$ are nonlinear functions defined on certain interpolation spaces and $u_0\in E$ is the initial value. $W_H$ is a cylindrical Brownian motion on a separable Hilbert space $H$. Under Lipschitz and linear growth conditions we show that there exists a unique mild solution of \eqref{eq:SEab}. Under assumptions on the interpolation spaces we extend the factorization method of Da Prato, Kwapie\'n, and Zabczyk, to obtain space-time regularity results for the solution $U$ of \eqref{eq:SEab}. For Hilbert spaces $E$ we obtain a maximal regularity result. The results improve several previous results from the literature. The theory is applied to a second order stochastic partial differential equation which has been studied by Sanz-Sol\'e and Vuillermot. This leads to several improvements of their result.Comment: Accepted for publication in Journal of Evolution Equation

Large deviations principle for the invariant measures of the 2D stochastic Navier-Stokes equations on a torus

We prove here the validity of a large deviation principle for the family of invariant measures associated to a two dimensional Navier-Stokes equation on a torus, perturbed by a smooth additive noise

Cylindrical fractional Brownian motion in Banach spaces

In this article we introduce cylindrical fractional Brownian motions in Banach spaces and develop the related stochastic integration theory. Here a cylindrical fractional Brownian motion is understood in the classical framework of cylindrical random variables and cylindrical measures. The developed stochastic integral for deterministic operator valued integrands is based on a series representation of the cylindrical fractional Brownian motion, which is analogous to the Karhunen-Loève expansion for genuine stochastic processes. In the last part we apply our results to study the abstract stochastic Cauchy problem in a Banach space driven by cylindrical fractional Brownian motion

On L_p- theory for stochastic parabolic integro-differential equations

The existence and uniqueness in fractional Sobolev spaces of the Cauchy problem to a stochastic parabolic integro-differential equation is investigated. A model problem with coefficients independent of space variable is considered. The equation arises, for example, in a filtering problem with a jump signal and jump observation process

Stochastic evolution equations driven by Liouville fractional Brownian motion

Let H be a Hilbert space and E a Banach space. We set up a theory of stochastic integration of L(H,E)-valued functions with respect to H-cylindrical Liouville fractional Brownian motions (fBm) with arbitrary Hurst parameter in the interval (0,1). For Hurst parameters in (0,1/2) we show that a function F:(0,T)\to L(H,E) is stochastically integrable with respect to an H-cylindrical Liouville fBm if and only if it is stochastically integrable with respect to an H-cylindrical fBm with the same Hurst parameter. As an application we show that second-order parabolic SPDEs on bounded domains in \mathbb{R}^d, driven by space-time noise which is white in space and Liouville fractional in time with Hurst parameter in (d/4,1) admit mild solution which are H\"older continuous both and space.Comment: To appear in Czech. Math.

Conservative interacting particles system with anomalous rate of ergodicity

We analyze certain conservative interacting particle system and establish ergodicity of the system for a family of invariant measures. Furthermore, we show that convergence rate to equilibrium is exponential. This result is of interest because it presents counterexample to the standard assumption of physicists that conservative system implies polynomial rate of convergence.Comment: 16 pages; In the previous version there was a mistake in the proof of uniqueness of weak Leray solution. Uniqueness had been claimed in a space of solutions which was too large (see remark 2.6 for more details). Now the mistake is corrected by introducing a new class of moderate solutions (see definition 2.10) where we have both existence and uniquenes