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

Abstract

Abstract We use the Yosida approximation to find an Itô formula for mild solutions <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo>{</m:mo> <m:mrow> <m:mrow> <m:mrow> <m:msup> <m:mi>X</m:mi> <m:mi>x</m:mi> </m:msup> <m:mo>⁢</m:mo> <m:mrow> <m:mo>(</m:mo> <m:mi>t</m:mi> <m:mo>)</m:mo> </m:mrow> </m:mrow> <m:mo>,</m:mo> <m:mi>t</m:mi> </m:mrow> <m:mo>≥</m:mo> <m:mn>0</m:mn> </m:mrow> <m:mo>}</m:mo> </m:mrow> </m:math> {\{X^{x}(t),t\geq 0\}} of SPDEs with Gaussian and non-Gaussian colored noise, with the non-Gaussian noise being defined through a compensated Poisson random measure associated to a Lévy process. The functions to which we apply such Itô formula are in <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msup> <m:mi>C</m:mi> <m:mrow> <m:mn>1</m:mn> <m:mo>,</m:mo> <m:mn>2</m:mn> </m:mrow> </m:msup> <m:mo>⁢</m:mo> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mrow> <m:mo>[</m:mo> <m:mn>0</m:mn> <m:mo>,</m:mo> <m:mi>T</m:mi> <m:mo>]</m:mo> </m:mrow> <m:mo>×</m:mo> <m:mi>H</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:mrow> </m:math> {C^{1,2}([0,T]\times H)} , as in the case considered for SDEs in [15]. Using this Itô formula, we prove exponential stability and exponential ultimate boundedness properties, in the mean square sense, for mild solutions. We also compare this Itô formula to an Itô formula for mild solutions introduced by Ichikawa in [12], and an Itô formula written in terms of the semigroup of the drift operator [5], which we extend to the non-Gaussian case.