Linear Evolution Equations with Cylindrical L\'evy Noise: Gradient Estimates and Exponential Ergodicity

TL;DR

This paper studies linear evolution equations driven by cylindrical Le9vy noise, providing gradient estimates and coupling properties that lead to exponential ergodicity, extending recent results for cylindrical symmetric b1-stable processes.

Contribution

It introduces explicit coupling and gradient estimates for such equations, advancing understanding of their ergodic behavior and extending prior work on cylindrical symmetric b1-stable processes.

Findings

Established explicit coupling properties.

Proved gradient estimates for transition semigroups.

Demonstrated exponential ergodicity of solutions.

Abstract

Explicit coupling property and gradient estimates are investigated for the linear evolution equations on Hilbert spaces driven by an additive cylindrical L\'evy process. The results are efficiently applied to establish the exponential ergodicity for the associated transition semigroups. In particular, our results extend recent developments on related topic for cylindrical symmetric $\alpha$-stable processes.