Scaling Limits of Solutions of SPDE Driven by L\'evy White Noises

TL;DR

This paper investigates the asymptotic behavior of solutions to SPDEs driven by Lévy white noise, revealing conditions under which rescaled solutions converge to self-similar processes at different scales.

Contribution

It provides new theoretical results on the scaling limits of SPDE solutions driven by Lévy noise, linking homogeneity and Lévy indices to self-similarity.

Findings

Rescaled solutions converge to self-similar processes under certain conditions.

The self-similarity order H depends on operator homogeneity and Lévy indices.

Results apply to various classes of random processes and fields.

Abstract

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 out of the process s. More precisely, we give sufficient conditions on L and w so that the rescaled versions of s converges in law to a self-similar process of order H at coarse scales and at fine scales. The parameter H depends on the homogeneity order of the operator L and the Blumenthal-Getoor indices associated to the L\'evy white noise w. Finally, we apply our general results to several notorious classes of random processes and random fields.