Stochastic evolution equations driven by cylindrical stable noise

TL;DR

This paper establishes the existence and uniqueness of solutions for stochastic evolution equations driven by cylindrical stable noise, expanding the understanding of such equations in infinite-dimensional spaces.

Contribution

It introduces a novel approach to prove existence and uniqueness of solutions driven by cylindrical stable Lévy processes in Hilbert spaces.

Findings

Proved existence and uniqueness of mild solutions.

Constructed solutions via weak limits of Picard iterations.

Extended Yamada--Watanabe theorem to this setting.

Abstract

We prove existence and uniqueness of a mild solution of a stochastic evolution equation driven by a standard $\alpha$-stable cylindrical L\'evy process defined on a Hilbert space for $\alpha \in (1,2)$. The coefficients are assumed to map between certain domains of fractional powers of the generator present in the equation. The solution is constructed as a weak limit of the Picard iteration using tightness arguments. Existence of strong solution is obtained by a general version of the Yamada--Watanabe theorem.