69 research outputs found
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 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 are unbounded operators with domains
which may be time dependent. We assume that
satisfies the conditions of Acquistapace and Terreni. The
functions and are nonlinear functions defined on certain interpolation
spaces and is the initial value. is a cylindrical Brownian
motion on a separable Hilbert space .
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 of
\eqref{eq:SEab}. For Hilbert spaces 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
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