ISBN 978-3-8439-2579-2

978-3-8439-2579-2, Reihe Mathematik

Robert Voßhall
Sticky Reflected Diffusion Processes -in view of Stochastic Interface Models and on General Domains-

158 Seiten, Dissertation Technische Universität Kaiserslautern (2016), Hardcover, A5

Zusammenfassung / Abstract

Motivated by the theory on stochastic models for interfaces in two phase systems and especially by the wetting model with delta-pinning and repulsion, the stochastic dynamics of a sticky reflected distorted Brownian motion under mild assumptions on the interaction was constructed via Dirichlet forms in [FGV14]. The state space of the process is given by the n-fold product of the positive half-line. The sticky reflection is characterized by a delayed reflection of each component in zero compared to the immediate reflection in the case of Neumann boundary conditions.

In the first part of the present thesis, a further analysis of the dynamics is provided. We prove an ergodicity theorem and investigate the connection of the Dirichlet form construction to random time changes. The construction of a strong Feller process for a specific class of densities allows to strengthen the previous results. Moreover, we take first steps towards proving convergence of an appropriate scaling to an infinite dimensional sticky reflected Ornstein-Uhlenbeck process. In particular, we construct the limit Dirichlet form and process.

In the second part, the construction is generalized from the positive half-line to general bounded domains. In dimension greater than or equal to two, an optional diffusion along the boundary of the domain is included additionally to the sticky boundary behavior. First, we construct and analyze the dynamics under mild assumptions on the density. Using regularity results for elliptic partial differential equations with Wentzell boundary conditions it is possible to provide under additional assumptions a strong Feller process. Afterwards, we construct a corresponding interacting particle system and consider applications with singular interactions.