This thesis deals with the study of linear stochastic partial differential equations driven by cylindrical Levy processes. Cylindrical Levy processes were recently introduced as a natural generalisation of cylindrical Wiener processes. A good part of the thesis is to obtain necessary and sufficient conditions for the existence of a weak and mild solution of the abstract stochastic Cauchy problem driven by a cylindrical Levy process in a separable Hilbert space. The methods employed are to use the techniques of strongly continuous semi-group theory and the recently developed stochastic integration theory for integrating deterministic functions with respect to cylindrical Levy processes. These techniques are first employed to prove a stochastic version of the Fubini theorem to stochastic integrals with respect to cylindrical Levy processes, which in turn is used to prove the existence of the weak solution. Some further theoretical properties of the solution such as the Markov property and stochastic continuity are derived. The necessary and sufficient conditions for the existence of the invariant measure for the stochastic Cauchy problem are obtained when the semi-group is stable. For specific examples including the stochastic heat equation driven by cylindrical Levy process, necessary and sufficient conditions as generalisation of the log moment condition for genuine Levy processes are obtained, which are satisfied by many examples of cylindrical Levy processes.
The Stochastic Cauchy Problem driven by a cylindrical Levy Process
Umesh, U. (Author). 2018
Student thesis: Doctoral Thesis › Doctor of Philosophy