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Description:
This thesis deals with different topics in probability theory. We are interested in infinitely divisible distributions and their densities, the class of quasi-infinitely divisible distributions and Lévy driven stochastic partial differential equations. In Chapter 2 we deal with infinitely divisible distributions and their densities. We obtain bounds of the integral modulus of continuity in terms of the characteristic triplet. We then apply our results to stochastic integrals. In Chapter 3 we study the class of probability measures \mu(dx)=\mu_{ld}(dx)+\mu_{ac}(dx), where \mu_{ld} is a discrete lattice distribution and \mu_{ac} is absolutely continuous. We prove that if \hat{\mu}_{ld}(z)\neq 0 for all z\in R then \mu is quasi-infinitely divisible if and only if \hat{\mu}(z)\neq 0 for all z\in R. As an application of this result we study certain variance mixtures and prove that they are quasi-infinitely divisible. In Chapter 4 we give sufficient conditions for the existence of a generalized solution s in the space of distributions of the stochastic partial differential equation p(D)s=q(D)\dot{L}, where p and q are polynomials in C^d and \dot{L} is a so called Lévy white noise. Furthermore, we give sufficient conditions for the existence of a mild solution and provide a sufficient condition when the mild solution can be identified with a generalized solution. Chapter 5 deals with linear and semilinear Lévy driven stochastic partial differential equations. In the linear case we work with different distributional spaces and show existence and uniqueness results under different assumptions. As a next step we analyze a semilinear partial differential equation driven by Lévy white noise in weighted Besov spaces. In Chapter 6 we prove central limit theorems for the sample mean and autocovariance of a moving average random field. We use a sampling scheme on a grid, which can be deterministic or random.