Prof. Dr. Andreas Eberle

Biography

1998 PhD, University of Bielefeld 1998 - 1999 Postdoc, Paul Sabatier University (Toulouse III), France 1999 - 2000 DFG research grant, University of California, San Diego, CA, USA 2000 - 2001 Teaching Assistant, University of Bielefeld 2001 - 2003 Lecturer, University of Oxford and Worcester College, UK 2009 - 2016 Head of Examination Board for Bachelor and Master Studies, Bonn Since 2003 Professor (C3), University of Bonn My research is based on the combination of methods from probability theory and other branches of mathematics, including differential equations and functional analysis, numerical analysis, geometry, and mathematical physics. A current focus is on coupling methods for stochastic processes on continuous state spaces. Here, a common goal is to quantify stability properties and convergence to equilibrium, for example for stochastic differential equations, systems with mean-field interactions, processes with high and infinite dimensional state spaces, numerical approximations, and both Markov Chain Monte Carlo and sequential Monte Carlo methods. An important tool is an approach developed in recent years that is based on contraction properties for combinations of reflection couplings and other couplings in specifically adjusted Kantorovich distances. Both the underlying metric and the coupling are adapted carefully to the corresponding problem, thus providing quantitative non-asymptotic bounds that are often relatively precise. The approach has first been applied successfully to non-degenerate diffusion processes. More recently, it has been extended to mean-field systems and nonlinear equations with weak interactions, and variants have been applied to numerical approximations and a class of MCMC methods.  Markov Chain Monte Carlo methods are the source of a variety of non-trivial mathematical problems. One example of current interest is the observation that often non-reversible processes seem to approach equilibrium faster than the more standard reversible ones. The question how to implement non-reversible processes in MCMC in the most effective way is still widely open. This is complemented by a much more incomplete mathematical understanding of the long time behavior of non-reversible Markov processes compared to reversible ones. Coupling methods are not based on reversibility. Therefore, they might help to clarify these important questions. First steps in this direction are made in current work in progress which shows that a similar coupling approach as described above yields qualitatively new bounds for convergence to equilibrium of (kinetic) Langevin equations. A goal of my future research is to extend these results to related Monte Carlo methods, and also to other stochastic systems with degenerate noise. Another important question, arising for example in the study of sequential Monte Carlo methods, is how to quantify the deviation of a mean-field approximation from a corresponding nonlinear SDE. Coupling methods might help to gain new insight. More generally, coupling approaches are natural for deriving long-time stable bounds for the difference between two different stochastic dynamics. First steps in this direction are done in current work in progress on sticky couplings.

Research Interest

Research Area G My central research topics are related to couplings of Markov processes on continuous state spaces in both discrete and continuous time, long-time stability of time-dependent stochastic systems and convergence to equilibrium of time homogeneous stochastic processes, spectral gaps and functional inequalities, and stochastic equations in high dimensions and on function spaces. Many problems are motivated by applications to Markov Chain Monte Carlo methods, sequential Monte Carlo methods and systems with mean-field interactions, nonlinear stochastic differential equations, and stochastic systems with degenerate noise.