Singular integrals of subordinators with applications to structural properties of SPDEs

TL;DR

This paper investigates stochastic integrals driven by subordinators, establishing fundamental properties and applying these to analyze the behavior and solutions of SPDEs with jump noise, including inequalities, probabilities, and approximation methods.

Contribution

It introduces new results on stochastic integrals with subordinators and applies them to derive structural properties of SPDEs driven by jump noise.

Findings

Maximal inequality for stochastic convolution $Z_t$

Small ball probability estimates for $Z_t$

Existence of invariant measures and zero accessibility for SPDEs

Abstract

We study stochastic integrals driven by a general subordinator and establish a zero-one law for the finiteness of the resulting integral as well as moment estimates. As an application, we use these results to obtain structural properties of SPDEs driven by multiplicative pure jump noise, which include (1) a maximal inequality for a multiplicative stochastic convolution $Z_t$, (2) a small ball probability of $Z_t$, (3) the existence of invariant measures and accessibility to zero of SPDEs, and (4) a Galerkin approximation of solutions to SPDEs.