Related papers: On some stochastic hyperbolic equations with sympl…
We study the homogenization of a Hamilton-Jacobi equation forced by rapidly oscillating noise that is colored in space and white in time. It is shown that the homogenized equation is deterministic, and, in general, the noise has an…
This note deals with existence and uniqueness of (variational) solutions to the following type of stochastic partial differential equations on a Hilbert space H dX(t) = A(t,X(t))dt + B(t,X(t))dW(t) + h(t) dG(t) where A and B are random…
Increasingly larger data sets of processes in space and time ask for statistical models and methods that can cope with such data. We show that the solution of a stochastic advection-diffusion partial differential equation provides a…
We consider an evolution operator for a discrete Langevin equation with a strongly hyperbolic classical dynamics and noise with finite moments. Using a perturbative expansion of the evolution operator we calculate high order corrections to…
We study the "periodic homogenization" for a class of nonlocal partial differential equations of parabolic-type with rapidly oscillating coefficients, related to stochastic differential equations driven by multiplicative isotropic…
Consider a random process s solution of the stochastic partial differential equation Ls = w with L a homogeneous operator and w a multidimensional L\'evy white noise. In this paper, we study the asymptotic effect of zooming in or zooming…
Weak noise smooths out fractals in a chaotic state space and introduces a maximum attainable resolution to its structure. The balance of noise and deterministic stretching/contraction in each neighborhood introduces local invariants of the…
We consider slow-fast systems of differential equations, in which both the slow and fast variables are perturbed by noise. When the deterministic system admits a uniformly asymptotically stable slow manifold, we show that the sample paths…
In this work, we shall consider the existence and uniqueness of stationary solutions to stochastic partial functional differential equations with additive noise in which a neutral type of delay is explicitly presented. We are especially…
This paper aims to investigate the numerical approximation of a general second order parabolic stochastic partial differential equation(SPDE) driven by multiplicative and additive noise. Our main interest is on such SPDEs where the…
We use an effective Markovian description to study the long-time behaviour of a nonlinear second order Langevin equation with Gaussian noise. When dissipation is neglected, the energy of the system grows as with time a power-law with an…
In the present paper, the effect of noise intensity on stochastic parabolic equations is discussed. We focus on the effect of noise on the energy solutions of the stochastic parabolic equations. By utilising It\^o's formula and the energy…
In this article, we consider the hyperbolic and parabolic Anderson models in arbitrary space dimension $d$, with constant initial condition, driven by a Gaussian noise which is white in time. We consider two spatial covariance structures:…
In this paper, we establish lower and upper Gaussian bounds for the probability density of the mild solution to the stochastic heat equation with multiplicative noise and in any space dimension. The driving perturbation is a Gaussian noise…
We establish a general criterion which ensures exponential mixing of parabolic Stochastic Partial Differential Equations (SPDE) driven by a non additive noise which is white in time and smooth in space. We apply this criterion on two…
Consider the following stochastic heat equation, \begin{align*} \frac{\partial u_t(x)}{\partial t}=-\nu(-\Delta)^{\alpha/2} u_t(x)+\sigma(u_t(x))\dot{F}(t,\,x), \quad t>0, \; x \in R^d. \end{align*} Here $-\nu(-\Delta)^{\alpha/2}$ is the…
We develop an asymptotic limit theory for nonparametric estimation of the noise covariance kernel in linear parabolic stochastic partial differential equations (SPDEs) with additive colored noise, using space-time infill asymptotics. The…
We study the existence and uniqueness of solutions to stochastic differential equations with Volterra processes driven by L\'evy noise. For this purpose, we study in detail smoothness properties of these processes. Special attention is…
This paper investigates the parabolic scaling limit of a damped stochastic wave map from the real line into the two-dimensional sphere, perturbed by multiplicative Gaussian noise of co-normal type. We prove that under this rescaling, the…
We study the projection of an element of fractional Gaussian noise onto its neighbouring elements. We prove some analytic results for the coefficients of this projection, in particular, we obtain recurrence relations for them. We also make…