L3

The zeta-function of a manifold varies with the parameters and may be evaluated in terms of the periods. For a one parameter family of CY manifolds, the periods satisfy a single 4th order differential equation. Thus there is a straight and, it turns out, readily computable path that leads from a differential operator to a zeta-function. Especially interesting are the specialisations to singular manifolds, for which the zeta-function manifests modular behaviour. We are also able to find, from the zeta function, attractor points. These correspond to special values of the parameter for which there exists a 10D spacetime for which the 6D corresponds to a CY manifold and the 4D spacetime corresponds to an extremal supersymmetric black hole. These attractor CY manifolds are believed to have special number theoretic properties. This is joint work with Xenia de la Ossa, Mohamed Elmi and Duco van Straten.

Backward Stochastic Differential Equations (BSDEs) provide a systematic way to obtain Feynman-Kac formulas for linear as well as nonlinear partial differential equations (PDEs) of parabolic and elliptic type, and the numerical approximation of their solutions thus provide Monte-Carlo methods for PDEs. BSDEs are also used to describe the solution of path-dependent stochastic control problems, and they further arise in many areas of mathematical finance. 

In this talk, I will discuss the numerical approximation of BSDEs when the nonlinear driver is not Lipschitz, but instead has polynomial growth and satisfies a monotonicity condition. The time-discretization is a crucial step, as it determines whether the full numerical scheme is stable or not. Unlike for Lipschitz driver, while the implicit Bouchard-Touzi-Zhang scheme is stable, the explicit one is not and explodes in general. I will then present a number of remedies that allow to recover a stable scheme, while benefiting from the reduced computational cost of an explicit scheme. I will also discuss the issue of numerical stability and the qualitative correctness which is enjoyed by both the implicit scheme and the modified explicit schemes. Finally, I will discuss the approximation of the expectations involved in the full numerical scheme, and their analysis when using a quasi-Monte Carlo method.

By viewing a stochastic process as a random variable taking values in a path space, the support of its law describes the set of all attainable paths. In this talk, we show that the support of the law of a solution to a path-dependent stochastic differential equation is given by the image of the Cameron-Martin space under the flow of mild solutions to path-dependent ordinary differential equations, constructed by means of the vertical derivative of the diffusion coefficient. This result is based on joint work with Rama Cont and extends the Stroock-Varadhan support theorem for diffusion processes to the path-dependent case.

Abstract: The edge reinforced random walk is a self-interacting process, in which the random walker prefer visited edges with a bias proportional to the number of times the edges were visited. We will gently introduce this model and talk about some of its histories and recent progresses.

We will consider some problems on calculating  the average  angles of random polytopes. Some of them are open.

Consider the asymmetric simple exclusion process with k particles on a linear lattice of N sites. I will present results on the asymptotic of the time needed for the system to reach its equilibrium distribution starting from the worst initial configuration (also called mixing time). Two main regimes appear according to the strength of the asymmetry (in terms of k and N), and in both regimes, the system displays a cutoff phenomenon: the distance to equilibrium falls abruptly from 1 to 0. This is a joint work with Hubert Lacoin (IMPA).

In a joint-work with Andrea Collevecchio and Vladas Sidoravicius,  we study  phase transitions in the recurrence/transience of a class of self-interacting random walks on trees, which includes the once-reinforced random walk. For this purpose, we define the branching-ruin number of a tree, which is  a natural way to measure trees with polynomial growth and therefore provides a polynomial version of the branching number defined by Furstenberg (1970) and studied by R. Lyons (1990). We prove that the branching-ruin number of a tree is equal to the critical parameter for the recurrence/transience of the once-reinforced random walk on this tree. We will also mention two other results where the branching-ruin number arises as critical parameter: first, in the context of random walks on heavy-tailed random conductances on trees and, second, in the case of Volkov's M-digging random walk.

We consider Brownian motion with Kato class measure-valued drift.   A small time large deviation principle and a Varadhan type asymptotics for the Brownian motion with singular drift are established. We also study the existence and uniqueness of the associated Dirichlet boundary value problems.

We first give a short introduction to analysis and stochastic processes on fractal state spaces and the typical difficulties involved.

We then discuss gradient operators and semilinear PDE. They are used to formulate the main result which establishes the hydrodynamic limit of the weakly asymmetric exclusion process on the Sierpinski gasket in the form of a law of large numbers for the particle density. We will explain some details and, if time permits, also sketch a corresponding large deviations principle for the symmetric case.

In this talk I present work in progress with Wolfgang König and Nicolas Perkowski on the parabolic Anderson model (PAM) with white noise potential in 2d. We show the behavior of the total mass as the time tends to infinity. By using partial Girsanov transform and singular heat kernel estimates we can obtain the mass-asymptotics by using the eigenvalue asymptotics that have been showed in another work in progress with Khalil Chouk.