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

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.