Stable cylindrical Lévy processes and the stochastic Cauchy problem

Markus Riedle

doi:10.1214/18-ecp134

Abstract

In this work, we consider the stochastic Cauchy problem driven by the canonical $\alpha $-stable cylindrical Lévy 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.

Highlights

One of the most fundamental stochastic partial differential equations is a linear evolution equation perturbed by an additive noise of the form dX(t) = AX(t) dt + dL(t) for t ∈ [0, T ], (1.1)where A is the generator of a strongly continuous semigroup (T (t))t≥0 on a Hilbert space U
We consider the stochastic Cauchy problem driven by the canonical α-stable cylindrical Lévy process
We introduce the canonical α-stable cylindrical Lévy process as a natural generalisation of the cylindrical Brownian motion

Summary

Introduction

Equation (1.1) in Banach spaces with an α-stable noise, or even slightly more general with a subordinated cylindrical Brownian motion, has already been considered by Brzezniak and Zabczyk in [5] Their approach is based on embedding the underlying Hilbert space U in a larger space such that the cylindrical noise becomes a genuine Lévy process. 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 This leads to the new phenomena that the necessary and sufficient condition for the existence of a solution of the heat equation in the space-time white noise approach differs from our condition (1.4) in the cylindrical approach

The canonical α-stable cylindrical Lévy process

Two representations

The stochastic Cauchy problem

Irregularities of the trajectories

A Appendix