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

  • Sergio Albeverio , Leszek Gawarecki , Vidyadhar Mandrekar , Barbara Rüdiger EMAIL logo and Barun Sarkar
Published/Copyright: May 13, 2017

Abstract

We use the Yosida approximation to find an Itô formula for mild solutions {Xx(t),t0} 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 C1,2([0,T]×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.


Communicated by Mykola Portenko


Acknowledgements

We are very grateful to Prof. P. Sundar (Louisiana State University) for useful discussions related to this work.

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Received: 2016-12-30
Accepted: 2017-2-22
Published Online: 2017-5-13
Published in Print: 2017-6-1

© 2017 Walter de Gruyter GmbH, Berlin/Boston

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