Related papers: It\^o formula for integral processes related to sp…
We study the convergence of a Zakharov system driven by a time white noise, colored in space, to a multiplicative stochastic nonlinear Schr{\"o}dinger equation, as the ion-sound speed tends to infinity. In the absence of noise, the…
A description in terms of phase and amplitude variables is given, for nonlinear oscillators subject to white Gaussian noise described by It\^o stochastic differential equations. The stochastic differential equations derived for the…
In this paper we develop a white noise framework for the study of stochastic partial differential equations driven by a d-parameter (pure jump) Levy white noise. As an example we use this theory to solve the stochastic Poisson equation with…
This paper is concerned with the strong approximation of a semi-linear stochastic wave equation with strong damping, driven by additive noise. Based on a spatial discretization performed by a spectral Galerkin method, we introduce a kind of…
High order and fractional PDEs have become prominent in theory and in modeling many phenomena. Here, we focus on the regularizing effect of a large class of memoryful high-order or time-fractional PDEs---through their fundamental…
This paper investigates the structure preservation and convergence analysis of a class of fully discrete finite difference schemes for the stochastic heat equation driven by L\'evy space-time white noise. The novelty lies in the…
An Ito-Skorokhod bi-linear equation driven by infinitely many independent colored noises is considered in a normal triple of Hilbert spaces. The special feature of the equation is the appearance of the Wick product in the definition of the…
In this article, we study the asymptotic behavior of the spatial integral of the solution to the hyperbolic Anderson model in dimension $d\leq 2$, as the domain of the integral gets large (for fixed time $t$). This equation is driven by a…
It is a well established result that, in classical dynamical systems with sufficient time-scale separation, the fast chaotic degrees of freedom are well modeled by (Gaussian) white noise. In this paper, we present the stochastic dynamical…
In this article, we study the stochastic wave equation in all dimensions $d\leq 3$, driven by a Gaussian noise $\dot{W}$ which does not depend on time. We assume that either the noise is white, or the covariance function of the noise…
In this article, we consider the following stochastic fractional diffusion equation \begin{equation*} \left(\partial^{\beta}+\dfrac{\nu}{2}\left(-\Delta\right)^{\alpha / 2}\right) u(t, x)= \lambda\: I_{0_+}^{\gamma}\left[u(t, x) \dot{W}(t,…
It is possible to construct L\'evy white noises as generalized random processes in the sense of Gel'fand and Vilenkin, or as an independently scattered random measures introduced by Rajput and Rosinski. In this article, we unify those two…
Several versions of It\^{o}'s formula have been obtained in the context of the functional stochastic calculus. Here, we revisit this topic in two ways. First, by defining a notion of derivative along a functional, we extend the setting of…
In this paper we consider the two-dimensional stochastic Gross-Pitaevskii equation, which is a model to describe Bose-Einstein condensation at positive temperature. The equation is a complex Ginzburg-Landau equation with a harmonic…
We introduce a general distributional framework that results in a unifying description and characterization of a rich variety of continuous-time stochastic processes. The cornerstone of our approach is an innovation model that is driven by…
Using tools from the theory of random fields with stationary increments, we introduce a new class of processes which can be used as a model for the noise perturbing an SPDE. This type of noise (called harmonizable) is not necessarily…
In this article, we consider a stochastic partial differential equation (SPDE) driven by a L\'evy white noise, with Lipschitz multiplicative term $\sigma$. We prove that under some conditions, this equation has a unique random field…
In this paper we consider a general class of second order stochastic partial differential equations on $\mathbb{R}^d$ driven by a Gaussian noise which is white in time and it has a homogeneous spatial covariance. Using the techniques of…
We extend the It\=o formula \cite{MR1837298}*{Theorem 2.3} for semimartingales with rcll paths. We also comment on Local time process of such semimartingales. We apply the It\=o formula to L\'evy processes to obtain existence of solutions…
Motivated by the traditional Lotka-Volterra competitive models, this paper proposes and analyzes a class of stochastic reaction-diffusion partial differential equations. In contrast to the models in the literature, the new formulation…