Stochastic partial differential equations driven by Levy space-time white noise

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 some generalized white noise process, which may be Gaussian or not (e.g., Laplace, impulsive Poisson or alpha stable). This allows for a conceptual decoupling between the correlation properties of the process, which are imposed by the white...

Based on the theory of independently scattered random measures, we introduce a natural generalisation of Gaussian space-time white noise to a Levy-type setting, which we call Levy-valued random measures. We determine the subclass of cylindrical Levy processes which correspond to Levy-valued random measures, and describe the elements of this subclass uniquely by their characteristic function. We embed the Levy-valued random measure, or the corresponding cylindrical Levy proces...

The main result of the paper is the existence of a solution of the nonlinear Schr\"odinger equation with a \levy noise with infinite activity. To be more precise, let $A=\Delta$ be the Laplace operator with $D(A)=\{ u\in L ^2 (\mathbb{R} ^d): \Delta u \in L ^2 (\mathbb{R} ^d)\}$. Let $Z\hookrightarrow L ^2(\mathbb{R} ^d)$ be a function space and $\eta$ be a Poisson random measure on $Z$, let $g:\mathbb{R}\to\mathbb{C}$ and $h:\mathbb{R}\to\mathbb{C}$ be some given functions, ...

We present an $L_{p}$-theory ($p\geq 2$) for time-fractional stochastic partial differential equations driven by L\'evy processes of the type $$ \partial^{\alpha}_{t}u=\sum_{i,j=1}^d a^{ij}u_{x^{i}x^{j}} +f+\sum_{k=1}^{\infty}\partial^{\beta}_{t}\int_{0}^{t} (\sum_{i=1}^d\mu^{ik} u_{x^i} +g^k) dZ^k_{s} $$ given with nonzero intial data. Here $\partial^{\alpha}_t$ and $\partial^{\beta}_t$ are the Caputo fractional derivatives, $\alpha\in (0,2), \beta\in (0,\alpha+1/p)$, and $\...

In this article, we examine a stochastic partial differential equation (SPDE) driven by a symmetric $\alpha$-stable (S$\alpha$S) L\'evy noise, that is multiplied by a linear function $\sigma(u)=u$ of the solution. The solution is interpreted in the mild sense. For this models, in the case of the Gaussian noise, the solution has an explicit Wiener chaos expansion, and is studied using tools from Malliavin calculus. These tools cannot be used for an infinite-variance L\'evy noi...

We consider the stochastic heat equation of the following form \frac{\partial}{\partial t}u_t(x) = (\sL u_t)(x) +b(u_t(x)) + \sigma(u_t(x))\dot{F}_t(x)\quad \text{for}t>0, x\in \R^d, where $\sL$ is the generator of a L\'evy process and $\dot{F}$ is a spatially-colored, temporally white, gaussian noise. We will be concerned mainly with the long-term behavior of the mild solution to this stochastic PDE. For the most part, we work under the assumptions that the initial data $u_0...

This article is dedicated to the study of an SPDE of the form $$Lu(t,x)=\sigma(u(t,x))\dot{Z}(t,x) \quad t>0, x \in \cO$$ with zero initial conditions and Dirichlet boundary conditions, where $\sigma$ is a Lipschitz function, $L$ is a second-order pseudo-differential operator, $\cO$ is a bounded domain in $\bR^d$, and $\dot{Z}$ is an $\alpha$-stable L\'evy noise with $\alpha \in (0,2)$, $\alpha\not=1$ and possibly non-symmetric tails. To give a meaning to the concept of solut...

We trace the evolution of the theory of stochastic partial differential equations from the foundation to its development, until the recent solution of long-standing problems on well-posedness of the KPZ equation and the stochastic quantization in dimension three.

An explicit formula for the chaotic representation of the powers of increments, (X_{t+t_0}-X_{t_0})^n, of a Levy process is presented. There are two different chaos expansions of a square integrable functional of a Levy process: one with respect to the compensated Poisson random measure and the other with respect to the orthogonal compensated powers of the jumps of the Levy process. Computationally explicit formulae for both of these chaos expansions of (X_{t+t_0}-X_{t_0})^n ...

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 solution. These conditions are verified by the stochastic heat and wave equations. We introduce the basic elements of Malliavin calculus with respect to the compensated Poisson random measure associated with the L\'evy white noise. If $\sigma$ is ...