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Operator semigroups and infinite dimensional analysis applied to problems from mathematical physics
(2020)

In this dissertation we treat several problems from mathematical physics via methods from functional analysis and probability theory and in particular operator semigroups. The thesis consists thematically of two parts.
In the first part we consider so-called generalized stochastic Hamiltonian systems. These are generalizations of Langevin dynamics which describe interacting particles moving in a surrounding medium. From a mathematical point of view these systems are stochastic differential equations with a degenerated diffusion coefficient. We construct weak solutions of these equations via the corresponding martingale problem. Therefore, we prove essential m-dissipativity of the degenerated and non-sectorial It\^{o} differential operator. Further, we apply results from the analytic and probabilistic potential theory to obtain an associated Markov process. Afterwards we show our main result, the convergence in law of the positions of the particles in the overdamped regime, the so-called overdamped limit, to a distorted Brownian motion. To this end, we show convergence of the associated operator semigroups in the framework of Kuwae-Shioya. Further, we established a tightness result for the approximations which proves together with the convergence of the semigroups weak convergence of the laws.
In the second part we deal with problems from infinite dimensional Analysis. Three different issues are considered. The first one is an improvement of a characterization theorem of the so-called regular test functions and distribution of White noise analysis. As an application we analyze a stochastic transport equation in terms of regularity of its solution in the space of regular distributions. The last two problems are from the field of relativistic quantum field theory. In the first one the $ (\Phi)_3^4 $-model of quantum field theory is under consideration. We show that the Schwinger functions of this model have a representation as the moments of a positive Hida distribution from White noise analysis. In the last chapter we construct a non-trivial relativistic quantum field in arbitrary space-time dimension. The field is given via Schwinger functions. For these which we establish all axioms of Osterwalder and Schrader. This yields via the reconstruction theorem of Osterwalder and Schrader a unique relativistic quantum field. The Schwinger functions are given as the moments of a non-Gaussian measure on the space of tempered distributions. We obtain the measure as a superposition of Gaussian measures. In particular, this measure is itself non-Gaussian, which implies that the field under consideration is not a generalized free field.