Paper detail

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

We use Yosida approximation to find an Itô 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 Lévy process. The functions to which we apply such Itô formula are in $C^{1,2}([0,T]\times H)$, as in the case considered for SDEs in [9]. Using this Itô formula we prove exponential stability and exponential ultimate boundedness properties in mean square sense for mild solutions. We also compare such Itô formula to an Itô formula for mild solutions introduced by Ichikawa in [8], and an Itô formula written in terms of the semigroup of the drift operator [11] which we extend before to the non Gaussian case.