Ito formula for mild solutions of SPDEs with Gaussian and non-Gaussian noise and applications to stability properties

TL;DR

This paper develops an Itô formula for mild solutions of SPDEs driven by Gaussian and non-Gaussian noise, enabling analysis of stability properties and extending existing formulas to more general noise types.

Contribution

It introduces a new Itô formula for SPDEs with Gaussian and non-Gaussian noise using Yosida approximation, extending previous formulas to broader noise settings.

Findings

Proves exponential stability in mean square sense.

Establishes exponential ultimate boundedness.

Compares new Itô formula with existing ones.

Abstract

We use Yosida approximation to find an It\^o formula for mild solutions $\left\{X^x(t), t\geq 0\right\}$ of SPDEs with Gaussian and non-Gaussian coloured noise, the non Gaussian noise being defined through compensated Poisson random measure associated to a L\'evy process. The functions to which we apply such It\^o formula are in $C^{1,2}([0,T]\times H)$, as in the case considered for SDEs in [9]. Using this It\^o formula we prove exponential stability and exponential ultimate boundedness properties in mean square sense for mild solutions. We also compare such It\^o formula to an It\^o formula for mild solutions introduced by Ichikawa in [8], and an It\^o formula written in terms of the semigroup of the drift operator [11] which we extend before to the non Gaussian case.