Szymon Peszat

Stochastic evolution equations with a spatially homogeneous Wiener process

A semilinear parabolic equation on d with a non-additive random perturbation is studied. The noise is supposed to be a spatially homogeneous Wiener process. Conditions for the existence and uniqueness of the solution in terms of the spectral measure of the noise are given. Applications to population and geophysical models are indicated. The Freidlin-Wentzell large deviation estimates are obtained as well.

On ergodicity of some Markov processes

We formulate a criterion for the existence and uniqueness of an invariant measure for a Markov process taking values in a Polish phase space. In addition, weak- * ergodicity, that is, the weak convergence of the ergodic averages of the laws of the process starting from any initial distribution, is established. The principal assumptions are the existence of a lower bound for the ergodic averages of the transition probability function and its local uniform continuity. The latter is called the e-property. The general result is applied to solutions of some stochastic evolution equations in Hilbert spaces. As an example, we consider an evolution equation whose solution describes the Lagrangian observations of the velocity field in the passive tracer model. The weak- * mean ergodicity of the corresponding invariant measure is used to derive the law of large numbers for the trajectory of a tracer.

Large deviation principle for stochastic evolution equations

SummaryThe large deviation principle obtained by Freidlin and Wentzell for measures associated with finite-dimensional diffusions is extended to measures given by stochastic evolution equations with non-additive random perturbations. The proof of the main result is adopted from the Priouret paper concerning finite-dimensional diffusions. Exponential tail estimates for infinite-dimensional stochastic convolutions are used as main tools.

The Cauchy problem for a nonlinear stochastic wave equation in any dimension

Abstract. A nonlinear wave equation on

On the existence of a solution to stochastic Navier—Stokes equations

mathbb{R}^d

Local existence and uniqueness of strong solutions to 3-D stochastic Navier-Stokes equations

driven by a spatially homogeneous Wiener process is studied. Conditions for the existence of a function-valued solution in terms of the covariance kernel of the noise are given for an arbitrary dimension d.

Lagrangian dynamics for a passive tracer in a class of Gaussian Markovian flows

We consider stochastic Navier{Stokes equations on a possibly unbounded domain O R d , where d is equal to 2 or 3. First we prove the existence of a martingale solution for the initial value being a probability measure on the space of square integrable R d-valued functions. Then we show the existence of a spatially homogeneous solution to the equation on the whole R d , driven by a spatially homogeneous Wiener random eld. 0. Introduction Let O be a possibly unbounded open subset of R d , where d is equal to 2 or 3. We assume that O is connected with the boundary @O of class C 2. Let us x a bounded time interval 0; T]. The paper is concerned with the existence of a martingal solution to the following system of stochastic Navier{Stokes equations (0.1)

Continuity of Stochastic Convolutions

Abstract. For stochastic Navier-Stokes equations in a 3-dimensional bounded domain we first show that if the initial value is sufficiently regular, then martingale solutions are strong on a random time interval and we estimate its length. Then we prove the uniqueness of the strong solution in the class of all martingale solutions.

Hyperbolic Equations with Random Boundary Conditions

We formulate a stochastic differential equation describing the Lagrangian environment process of a passive tracer in Ornstein-Uhlenbeck velocity fields. We subsequently prove a local existence and uniqueness result when the velocity field is regular. When the Ornstein-Uhlenbeck velocity field is only spatially Holder continuous we construct and identify the probability law for the Lagranging process under a condition on the time correlation function and the Holder exponent.

Transport of a Passive Tracer by an Irregular Velocity Field

AbstractLet B be a Brownian motion, and let n