# Probability Seminar Essen

## Probability Seminar Essen

Covers a wide range of topics in Probability and its applications.

### Current Schedule for the Winter Semester 2017/18

Previous Semesters

 October 17 Sara Mazzonetto (Universität Duisburg-Essen) October 24 Tuan Anh Nguyen (Universität Duisburg-Essen) November 7 Clemens Printz (Universität Duisburg-Essen) November 14 Simon Eberle (Universität Duisburg-Essen) November 21 Michael Weinig (Universität Duisburg-Essen) November 28 17:00 h Sebastian Mentemeier (TU Dortmund) December 05 17:00 h Christoph Schumacher (TU Dortmund) Dezember 12 Larisa Yaroslavtseva (Universität Passau) January 9 Sung-Soo Byun (Seoul National University/                                      Universität Bielefeld) January 16 Sebastian Hummel (Universität Bielefeld) January  23 Thomas Kruse (Universität Duisburg-Essen) January 29 and 30 GRK 2131 Workshop in Bochum (RUB) February 6 Christian Bayer (WIAS)

### Winter Semester 2017/18

 Oct 17 Sara Mazzonetto (Universität Duisburg-Essen) 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. We use this result to present an exact simulation of these diffusions, and comment some (still) open problems. Eventually we consider the exact simulation of Brownian diffusions whose drift admits finitely many jumps. Oct 24 Tuan Anh Nguyen (Universität Duisburg-Essen) The random conductance model under degenerate conditions 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 Jean-Dominique 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 continuous-time 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 Dirichlet-to-Neumann 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 Duisburg-Essen) Stochastic averaging for multiscale Markov processes applied to a Wright-Fisher model with fluctuating 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 $(X_t)_{t\in[0,\infty)}$. With this tool we can generalize the well-known diffusion limit of a Wright-Fisher model with randomly fluctuating selection. Whereas the classical result assumes the selection coefficients to be independent for different generations, we allow the environment to persist with a positive probability. The diffusion limit turns out to depend on this probability. This talk is based on joint work with Martin Hutzenthaler and Peter Pfaffelhuber. Nov 14 Simon Eberle (Universität Duisburg-Essen) Gradient flow formulation and longtime behaviour of a constrained Fokker-Planck equation We consider a Fokker-Planck equation which is coupled to an externally given time-dependent constraint on its first moment. This constraint introduces a Lagrange-multiplier 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 time-constrained 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 time-step $h>0$. 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 Duisburg-Essen) Nested Monte-Carlo Simulation Estimates of Expected Value of Partial Perfect Information The expected value of partial perfect information (EVPPI) expresses the value gaining by acquiring further information on certain unknowns in a decision-making process. In this talk, we will show and compare various approaches to estimate the EVPPI like nested Monte-Carlo simulations, Multi-Level Monte-Carlo simulations, and regression-based algorithms. Further, we introduce an unbiased estimator based on a randomized version multi-level Monte-Carlo algorithm. The EVPPI estimate is mostly used in medical decision making. We will show some numerical result in evaluating the benefit of further information when deciding between two different treatment options. Nov 28 Sebastian Mentemeier (TU Dortmund) Solutions to complex smoothing equations 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 so-called smoothing equations. These are equations of the form $$\label{SFPE} X \stackrel{\text{law}}{=} \sum_{j \ge 1} T_j X_j + C,$$ where $X_1, X_2, \dots$ are i.i.d.~copies of the (unknown)  random variable $X$ and independent of the (given) random variables $(C, T_1, T_2, \dots)$. I will start my talk by presenting several examples where this phenomenon occurs. Then I will present a general theory for solving equations of type (1) in the case where $X$ as well as $C, T_1, T_2, \dots$ are complex-valued random variables. These results are then applied to the examples mentioned in the introduction. This talk is based on joint work with Matthias Meiners, University Innsbruck. Dec 5 Christoph Schumacher (TU Dortmund) The asymptotic behavior of the ground state energy of the Anderson model on large regular trees The Anderson model was invented 1985 by Anderson and describes the quantum mechanical motion of a particle in a random potential in $Z^d$. It is related to random walks in a random environment.  The Anderson model on regular tree was introduced 1973 by Abou-Chacra, Thouless and Anderson. We give a detailed description of the ground state energy on large finite symmetric subtrees. This is joint work with Francisco Hoecker-Escuti (TU Hamburg-Harburg). Dec 12 Larisa Yaroslavtseva (Universität Passau) On sub-polynomial lower error bounds for strong approximation of SDEs 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. In our talk we present negative results for this problem. First   we show that there exist  SDEs  with bounded smooth coefficients such that their solutions can not be approximated by means of any kind of adaptive method with a polynomial rate of convergence. Even worse, we show that for any sequence $(a_n)_{n \in \mathbb N}\subset (0, \infty)$, which may converge to zero arbitrarily slowly, there exists an SDE  with bounded smooth coefficients such that no approximation method based on $n$ adaptively chosen evaluations of $W$ on average can achieve a smaller absolute mean error than the given number $a_n$. While the diffusion coefficients of these pathological SDEs are globally Lipschitz continuous, the first order partial derivatives of the drift coefficients are, essentially, of exponential growth. In the second part of the talk we show that sub-polynomial rates of convergence may happen even when the first order partial derivatives of the coefficients have at most polynomial growth, which is one of the typical assumptions in the literature on numerical approximation of SDEs with globally monotone coefficients. The talk is based on joint work with Arnulf Jentzen (ETH Zürich) and Thomas Müller-Gronbach (University of Passau). Jan 9 Sung-Soo Byun (Seoul National University/Universität Bielefeld) CFT and SLE in a doubly connected domain In this talk, I will present certain implementations of conformal field theory (CFT) in a doubly connected domain. The statistical fields in these implementations are generated by central charge modifications of the Gaussian free field with excursion reflected/Dirichlet boundary conditions. I will explain Ward’s equation in terms of a stress energy tensor, Lie derivative operators and the modular parameter. Combining Ward’s equation with the level 2 degeneracy equation for the boundary condition changing operator, I will outline the relation between CFT and Schramm-Loewner Evolution (SLE) theory. As applications, I will present a version of restriction property and Friedrich-Werner’s formula for annulus SLE and explain how to apply the method of screening to find explicit solutions to the partial differential equations for the annulus SLE partition functions introduced by Lawler and Zhan. This is based on joint work with Nam-Gyu Kang and Hee-Joon Tak. Jan 16 Sebastian Hummel (Universität Bielefeld) Lines of descent in a deterministic model with mutation, selection and pairwise interaction We consider a classical deterministic model for the evolution of a haploid population with two allelic types which is subject to mutation, selection, and a special form frequency-dependent selection. The deterministic model arises (also) as the large population limit of the Moran model, in which neither parameters nor time are rescaled. Despite the deterministic nature of this limiting process, the ancestry of single individuals in the population is still stochastic. In the case with mutation and selection, we describe it via a killed ancestral selection graph and connect it to the deterministic process via duality; this leads to a stochastic representation of the deterministic solution. In particular, the stationary state obtains a nice probabilistic interpretation. We generalise the construction to the case with frequency-dependent selection. Jan 23 Thomas Kruse (Universität Duisburg-Essen) Multilevel Picard approximations for high-dimensional nonlinear parabolic partial differential equations In this talk we present a family of new approximation methods for high-dimensional PDEs and BSDEs. A key idea of our methods is to combine multilevel approximations with Picard fixed-point approximations. Thereby we obtain a class of multilevel Picard approximations. Our error analysis proves that for semi-linear heat equations, the computational complexity of one of the proposed methods is bounded by $O(d\,\eps^{-(4+\delta)})$ for any $\delta > 0$, where $d$ is the dimensionality of the problem and $\eps\in(0,\infty)$ is the prescribed accuracy. We illustrate the efficiency of one of the proposed approximation methods by means of numerical simulations presenting approximation accuracy against runtime for several nonlinear PDEs from physics (such as the Allen-Cahn equation) and financial engineering (such as derivative pricing incorporating default risks) in the case of $d=100$ space dimensions. The talk is based on joint work with Weinan E., Martin Hutzenthaler and Arnulf Jentzen. Jan 29/30 GRK 2131 Workshop in Bochum (RUB) Transformations and phase transitions Feb 6 Christian Bayer (WIAS) Solving linear parabolic rough partial differential equations