Ergodic properties of stochastic processes

Stochastic processes have been a very successful tool for modelling systems that either inherently include uncertainties in their description or evolve under the influence of an external force which is only known through its statistical properties. A natural problem is to try to understand the long time behaviour of such systems. In particular, one is interested in finding criteria that ensure that a given system relaxes over time to a stationary state independent of its initial condition. This and related questions are by now well understood for Markov processes (i.e. processes that have no memory of their past given their present state) with a finite number of degrees of freedom. They are however currently under active investigation for infinite-dimensional systems and the corresponding theory for non-Markov processes is still in its infancy.The proposed research will both build on existing theories and develop new methods to investigate the asymptotic behaviour for a wide class of stochastic systems. Emphasis will be put on non-Markovian systems with both intrinsic and extrinsic memory, infinite-dimensional systems, hypoelliptic systems, and the analysis of related differential operators.This will lead to the preparation of a monograph on the ergodicity of stochastic processes in finite and infinite dimensions. It will complement existing literature in the field by putting an emphasis on recent techniques that are either still under development or have been developed over the last five years.