Stable cylindrical L\'evy processes and the stochastic Cauchy problem

TL;DR

This paper investigates the stochastic Cauchy problem driven by cylindrical Lévy processes, providing conditions for solutions and analyzing their temporal irregularity, thus extending understanding beyond Gaussian noise models.

Contribution

It introduces necessary and sufficient conditions for solutions to the stochastic Cauchy problem with cylindrical Lévy noise and studies their temporal irregularity.

Findings

Derived conditions for existence of solutions

Established temporal irregularity of solutions

Generalized Gaussian noise to Lévy processes

Abstract

In this work, we consider the stochastic Cauchy problem driven by the canonical $\alpha$-stable cylindrical L\'evy process. This noise naturally generalises the cylindrical Brownian motion or space-time Gaussian white noise. We derive a sufficient and necessary condition for the existence of the weak and mild solution of the stochastic Cauchy problem and establish the temporal irregularity of the solution.