Stochastic Analysis with Lévy Noise in the Dual of a Nuclear Space.

In this thesis we introduce a new theory of stochastic analysis with respect to Lévy processes in the strong dual of a nuclear space.

First we prove some extensions of the regularization theorem of Itô and Nawata to show conditions for the existence of continuous and càdlàg versions to cylindrical and stochastic processes in the dual of a nuclear space. Sufficient conditions for the existence of continuous and càdlàg versions taking values in a Hilbert space continuously included on the dual space are also provided. Then, we apply these results to prove the Lévy-Itô decomposition and the Lévy-Khintchine formula for Lévy processes taking values in
the dual of a complete, barrelled, nuclear space.

Later, we introduce a theory of stochastic integration for operator-valued processes taking values in the strong dual of a quasi-complete, bornological, nuclear space with
respect to some classes of cylindrical martingale-valued measures. The stochastic integrals are constructed by means of an application of the regularization theorems. In
particular, this theory allows us to introduce stochastic integrals with respect to Lévy processes via Lévy-Itô decomposition. Finally, we use our theory of stochastic integration to study stochastic evolution equations driven by cylindrical martingale-valued measure noise in the dual of a nuclear space. We provide conditions for the existence
and uniqueness of weak and mild solutions. Also, we provide applications of our theory to the study of stochastic evolution equations driven by Lévy processes.