

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 meanfield 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 nonasymptotic bounds that are often relatively precise. The approach has first been applied successfully to nondegenerate diffusion processes. More recently, it has been extended to meanfield 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 nontrivial mathematical problems. One example of current interest is the observation that often nonreversible processes seem to approach equilibrium faster than the more standard reversible ones. The question how to implement nonreversible 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 nonreversible 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 meanfield approximation from a corresponding nonlinear SDE. Coupling methods might help to gain new insight. More generally, coupling approaches are natural for deriving longtime 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.


ProcopeProject: Quantitative convergence rates for diffusions by coupling methods


Research Area G In recent years, my research has focused on designing coupling approaches for quantifying convergence to equilibrium and stability properties for various classes of Markov processes on continuous state spaces. A corresponding general approach based on reflection couplings and contraction properties in Kantorovich distances w.r.t. underlying concave distance functions has been introduced in [1] and [2], where it has been shown to provide relatively sharp bounds for a variety of examples. Recently, the approach has been extended significantly in several directions. In [3] it has been combined with Lyapunov function arguments to provide quantitative versions of Harris' theorem that apply both to linear and nonlinear diffusions. A new sticky coupling process introduced in [4] allows to bound distances between the laws of diffusions with different drifts over long time intervals. Finally, in [5] the different approaches have been combined to construct a coupling for second order Langevin equations that yields both qualitatively and quantitatively new bounds on the distance to equilibrium. All the results should also be relevant for Markov Chain Mote Carlo methods. First steps in this direction are carried out in [6]. 


[ 1] Andreas Eberle
Reflection coupling and Wasserstein contractivity without convexity C. R. Math. Acad. Sci. Paris , 349: (1920): 11011104 2011[ 2] Andreas Eberle
Reflection couplings and contraction rates for diffusions Probab. Theory Related Fields , 166: (34): 851886 2016[ 3] A. Eberle, A. Guillin, R. Zimmer
Quantitative Harris type theorems for diffusions and McKeanVlasov processes ArXiv eprints 2016[4] A. Eberle, R. Zimmer
Sticky couplings of multidimensional diffusions with different drifts ArXiv eprints 2016 [5] A. Eberle, A. Guillin, R. Zimmer
Couplings and quantitative contraction rates for Langevin dynamics ArXiv eprints 2017 [6] Andreas Eberle
Error bounds for MetropolisHastings algorithms applied to perturbations of Gaussian measures in high dimensions Ann. Appl. Probab. , 24: (1): 337377 2014 [ 7] Andreas Eberle, Carlo Marinelli
Quantitative approximations of evolving probability measures and sequential Markov chain Monte Carlo methods Probab. Theory Related Fields , 155: (34): 665701 2013[ 8] Andreas Eberle, Carlo Marinelli
L^{p} estimates for FeynmanKac propagators with timedependent reference measures J. Math. Anal. Appl. , 365: (1): 120134 2010[ 9] Andreas Eberle
Local spectral gaps on loop spaces J. Math. Pures Appl. (9) , 82: (3): 313365 2003[ 10] Andreas Eberle
Absence of spectral gaps on a class of loop spaces J. Math. Pures Appl. (9) , 81: (10): 915955 2002[ 11] Andreas Eberle
Uniqueness and nonuniqueness of semigroups generated by singular diffusion operators of Lecture Notes in Mathematics : viii+262 Publisher: SpringerVerlag, Berlin 1999 ISBN: 3540666281





• Annals of Applied Probability (Associate Editor, since 2014)


2000  Stochastic analysis and applications, Lisbon, Portugal  2002  Stochastic analysis, Beijing, China  2004  Stochastic partial differential equations and applications, Levico, Italy  2011  Filtering, MCMC, ABC, Lille, France  2016  Computational statistics and molecular simulation, Paris, France 


Nikolaus Schweizer (2012): “Nonasymptotic Error Bounds for Sequential MCMC Methods”,
now Assistant Professor, Tilburg University, Netherlands
Daniel Gruhlke (2014): “Convergence of multilevel MCMC methods on path spaces”
Raphael Zimmer (2017): “Couplings and contractions with explicit rates for diffusions”
Mateusz Majka (2017): “Stability of stochastic differential equations with jumps by the coupling method”


 Master theses: 14, currently 8
 Diplom theses: 30
 PhD theses: 4


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