SPDEs with α-Stable Lévy Noise: A Random Field Approach

Abstract

This article is dedicated to the study of an SPDE of the form Lu(t, x) = σ(u(t, x))Ż(t, x) t > 0, x ∈ O with zero initial conditions and Dirichlet boundary conditions, where σ is a Lipschitz function, L is a second-order pseudo-differential operator, O is a bounded domain in R d , andŻ is an α-stable Lévy noise with α ∈ (0, 2), α = 1 and possibly non-symmetric tails. To give a meaning to the concept of solution, we develop a theory of stochastic integration with respect to Z, by generalizing the method of [11] to higher dimensions and non-symmetric tails. The idea is to first solve the equation with "truncated" noiseŻ K (obtained by removing from Z the jumps which exceed a fixed value K), yielding a solution u K , and then show that the solutions u L , L > K coincide on the event t ≤ τ K , for some stopping times τ K ↑ ∞ a.s. A similar idea was used in [22] in the setting of Hilbert-space valued processes. A major step is to show that the stochastic integral with respect to Z K satisfies a p-th moment inequality, for p ∈ (α, 1) if α < 1, and p ∈ (α, 2) if α > 1. This inequality plays the same role as the Burkholder-Davis-Gundy inequality in the theory of integration with respect to continuous martingales.MSC 2000 subject classification: Primary 60H15; secondary 60H05, 60G60

Random field solutions to linear SPDEs driven by symmetric pure jump Lévy space-time white noises

“…The criterion for the existence of the mild solution to the linear stochastic heat equation (6.1) is known (see [1]). However, we can also obtain this from (H2').…”

Space-time fractional stochastic partial differential equations with Lévy noise

“…The area of SPDEs is interesting to mathematicians because it contains a lot of hard open problems. However, not much have been done for equations driven by discontinuous noise even though this situation has started to change recently, see, for example, [5] and references therein.…”

Stable cylindrical Lévy processes and the stochastic Cauchy problem

“…Another approach to generalise the model of the driving noise is based on the Gaussian space-time white noise leading to a Lévy space-time white noise; see Albeverio et al [1] or Applebaum and Wu [3]. In this framework, equation (1.1) (and even with a multiplicative noise) driven by an α-stable Lévy white noise is considered in the work [4] by Balan. We show that the α-stable Lévy white noise in [4] corresponds to a canonical α-stable cylindrical Lévy process in the same way as it is known in the Gaussian setting (see [7,Th.3.2.4]).…”

“…In this framework, equation (1.1) (and even with a multiplicative noise) driven by an α-stable Lévy white noise is considered in the work [4] by Balan. We show that the α-stable Lévy white noise in [4] corresponds to a canonical α-stable cylindrical Lévy process in the same way as it is known in the Gaussian setting (see [7,Th.3.2.4]). However, and very much in contrast to the Gaussian setting, it turns out that the corresponding cylindrical Lévy process is defined on a Banach space different from the underlying Hilbert space U .…”