Probability Seminar Essen
Covers a wide range of topics in Probability and its applications.
Covers a wide range of topics in Probability and its applications.
October 17  Sara Mazzonetto (Universität DuisburgEssen) 
October 24  Tuan Anh Nguyen (Universität DuisburgEssen) 
November 7  Clemens Printz (Universität DuisburgEssen) 
November 14  Simon Eberle (Universität DuisburgEssen) 
November 21  Michael Weinig (Universität DuisburgEssen) 
November 28 17:00 h  Sebastian Mentemeier (TU Dortmund) 
December 05  
Dezember 12  Larisa Yaroslavtseva (Universität Passau) 
January 9  
January 16  Sebastian Hummel (Universität Bielefeld) 
January 23  Thomas Kruse (Universität DuisburgEssen) 
January 29 and 30  GRK 2131 Workshop in Bochum (RUB) 
February 6  Christian Bayer (WIAS) 
Oct 17  Sara Mazzonetto (Universität DuisburgEssen)
About some skewed Brownian diffusions: explicit representation of their transition densities and exact simulation In this talk we first discuss an explicit representation of the transition density of Brownian dynamics undergoing their motion through semipermeable and semireflecting barriers, called skewed Brownian motions.

Oct 24 
Tuan Anh Nguyen (Universität DuisburgEssen) The aim of the talk is to briefly introduce some ideas in my PhD thesis. Random motions in random media is an interesting topic that has been studied intensively since several decades. Although these models are relatively simple mathematical objects, they have a wide variety of interesting properties from the theoretical point of view. In the talk I would like to draw your attention to an important branch within this topic, namely reversible random walks moving among nearest neighbour random conductances on $\mathbb{Z}^d$  the random conductance model. Reversibility provides the model a variety of interesting connections with other fields in mathematics, for instance, percolation theory and especially stochastic homogenization. Many questions coming from this model have been answered by techniques from partial differential equations and harmonic analysis. As seen in the name of the talk, I would like to consider this model under ''degenerate conditions''. Here, ''degenerate'' has essentially two meanings. First, the conductances are not assumed to be bounded from above and below and stochastically independent. Second, we also consider the case of zero conductances, where the random walk can only move on a subgraph of $\mathbb{Z}^d$. Since there are percolation clusters, where the existence of the infinite cluster does not rely on stochastic independence, it is reasonable to accept the lack of stochastic independence. In the first part of the talk I introduce quenched invariance principles (joint work with JeanDominique Deuschel and Martin Slowik). We assume that the positive conductances have some certain moment bounds, however, not bounded from above and below, and give rise to a unique infinite cluster and prove a quenched invariance principle for the continuoustime random walk among random conductances under relatively mild conditions on the structure of the infinite cluster. An essential ingredient of our proof is a new anchored relative isoperimetric inequality. In the second part I would like to talk about Liouville principles. As in the first part, I also assume some moment bounds and prove a first order Liouville property for this model. Using the corrector method introduced by Papanicolaou and Varadhan, the first and the second part are closely related to each other at the technical level. I also introduce a discrete analogue of the DirichlettoNeumann estimate, which compares the tangential and normal derivatives of a harmonic function on the boundary of a domain. Although it is a purely deterministic classical result, it is used in the second part and perhaps useful for numerical analysis. 
Nov 7 
Clemens Printz (Universität DuisburgEssen)
We present a new result on stochastic averaging for sequences of bivariate Markov processes $((X_t^n,Z_t^n)_{t\in[0,\infty)})_{n\in\mathbb{N}}$ whose components evolve on different time scales. Under suitable conditions, convergence of certain functionals of the fast variables $Z^n$ guarantees convergence of the (not necessarily Markovian on its own) slow variables $X^n$ to a limiting Markov process $(X_t)_{t\in[0,\infty)}$. 
Nov 14  Simon Eberle (Universität DuisburgEssen)
Gradient flow formulation and longtime behaviour of a constrained FokkerPlanck equation We consider a FokkerPlanck equation which is coupled to an externally given timedependent constraint on its first moment. This constraint introduces a Lagrangemultiplier which renders the equation nonlocal and nonlinear. In this talk we exploit an interpretation of this equation as a Wasserstein gradient flow of a free energy ${\mathcal{F}}$ on a timeconstrained manifold. First, we prove existence of solutions by passing to the limit in an explicit Euler scheme obtained by minimizing $h {\mathcal{F}}(\varrho)+W_2^2(\varrho^0,\varrho)$ among all $\varrho$ satisfying the constraint for some $\varrho^0$ and timestep $h>0$. Second, we provide quantitative estimates for the rate of convergence to equilibrium when the constraint converges to a constant. The proof is based on the investigation of a suitable relative entropy with respect to minimizers of the free energy chosen according to the constraint. The rate of convergence can be explicitly expressed in terms of constants in suitable logarithmic Sobolev inequalities. 
Nov 21 
Michael Weinig (Universität DuisburgEssen) The expected value of partial perfect information (EVPPI) expresses the value gaining by acquiring further information on certain unknowns in a decisionmaking process. In this talk, we will show and compare various approaches to estimate the EVPPI like nested MonteCarlo simulations, MultiLevel MonteCarlo simulations, and regressionbased algorithms. 
Nov 28 
Sebastian Mentemeier (TU Dortmund) In many models of Applied Probability and Statistical Physics quantities of interest satisfy distributional limit theorems (i.e. convergence in law) which are nonstandard in the sense that neither normal nor $\alpha$stable laws appear as limiting distributions. Sometimes, there is even no convergence in law, but periodical fluctuations centered at some limit law occur. Common to all these models is that the limiting distributions satisfy socalled smoothing equations. These are equations of the form 
Dec 12 
Larisa Yaroslavtseva (Universität Passau) We consider the problem of strong approximation of the solution of a stochastic differential equation (SDE) at the final time based on finitely many evaluations of the driving Brownian motion $W$. While the majority of results for this problem deals with equations that have globally Lipschitz continuous coefficients, such assumptions are typically not met for real world applications. In recent years a number of positive results for this problem has been established under substantially weaker assumptions on the coefficients such as global monotonicity conditions: new types of algorithms have been constructed that are easy to implement and still achieve a polynomial rate of convergence under these weaker assumptions. 
Feb 6 
Christian Bayer (WIAS) 