A study of SPDEs w.r.t. compensated Poisson random measures and related topics

This thesis consists of two parts. In the first part, we define stochastic integrals w.r.t. the compensated Poisson random measures in a martingale type p, 1 ≤ p ≤ 2 Banach space and establish a
certain continuity, in substitution of the Ita isometry property, for the stochastic integrals .. A version
of Ita formula, as a generalization of the case studies in Ikecla and Watanabe [40], is derived. This
Itô formula enables us to treat certain Levy processes without Gaussion components. Moreover,
using ideas in [63] a version of stochastic Fubini theorem for stochastic integrals W.r. t. compensated
Poisson random measures in martingale type spaces is established. In addition, if we assume that
E is a martingale type p Banach space with the q-th, q ≥ p, power of the norm in C2-class, then
we prove a maximal inequality for a cadlag modification u of the stochastic convolution w.r.t. the
compensated Poisson random measures of a contraction Co-semigroups.
The second part of this thesis is concerned with the existence and uniqueness of global mild
solutions for stochastic beam equations w.r.t. the compensated Poisson random measures. In view
of Khas'minskii's test for nonexplosions, the Lyapunov function technique is used via the Yosida
approximation approach. Moreover, the asymptotic stability of the zero solution is proved and the
Markov property of the solution is verified.