Diffusion Processes with Reflection
In recent years diffusion processes with reflection have been subject of active research in the field of probability theory and stochastic analysis, where such reflected processes arise in quite various manners. The present work deals with two rather different types of reflected diffusion processes. In the first part we prove pathwise differentiabilty results for Skorohod SDEs with respect to the initial condition, in particular we consider processes on convex polyhedrons with oblique reflection at the boundary as well as processes on bounded smooth domains with normal reflection. In the second part a particle approximation of the Wasserstein diffusion is established, where the approximating process can be intepreted as a system of interacting Bessel processes with small Bessel dimension. More precisely, we introduce a reversible particle system, whose associated empirical measure process converges weakly to the Wasserstein diffusion in the high-density limit. Moreover, we prove regularity properties of the approximating system, in particular Feller properties, using tools from harmonic analysis on weighted Sobolev spaces.