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

Abstract

We study the notions of mild solution and generalized solution to a linear stochastic partial differential equation driven by a pure jump symmetric Lévy white noise. We identify conditions for existence for these two kinds of solutions, and we identify conditions under which they are essentially equivalent. We establish a necessary condition for the existence of a random field solution to a linear SPDE, and we apply this result to the linear stochastic heat, wave and Poisson equations driven by a symmetric α-stable noise.

Extremes of the stochastic heat equation with additive Lévy noise

“…To our best knowledge, this result is new. While many works have studied sufficient conditions for existence [1,4,5,26], necessary and sufficient conditions have only been derived for multiplicative noise [2] or for specific types of noises such as α-stable noise [11]. Introduce the measure η as…”

Extremes of the stochastic heat equation with additive Lévy noise

“…To the best of our knowledge, this result is new. While many works have studied sufficient conditions for existence [1,5,6,28], necessary and sufficient conditions have only been derived for multiplicative noise [2] or for specific types of noises such as α-stable noise [12]. Introduce the measure η as…”

Modelling Lévy space‐time white noises

“…Both generalisations, cylindrical Lévy processes and Lévy space-time white noises, serve as a model for random perturbations of complex dynamical systems. These applications can be found for cylindrical Lévy processes, for example, in the monograph in Peszat and Zabczyk [34] or in Kumar and Riedle [30], and for Lévy space-time white noise in Applebaum and Wu [3], Chong [11], Chong and Kevei [12] and Dalang and Humeau [15], among many others. Another approach to model such perturbed dynamical systems, for example, parabolic stochastic partical differential equations, is provided by the recently introduced ambit fields, presented in the monograph [6] by Barndorff-Nielsen, Benth and Veraart, and their relations to SPDE investigated in [7] by the same authors.…”