Elliptic analogs of multiple zeta values
Abstract
Multiple zeta values are generalizations of the special values of Riemann's zeta function at positive integers. They satisfy a large number of algebraic relations some of which were already known to Euler. More recently, the interpretation of multiple zeta values as periods of mixed Tate motives has led to important new results. However, this interpretation seems insufficient to explain the occurrence of several phenomena related to modular forms.
The aim of this talk is to describe an analog of multiple zeta values for complex elliptic curves introduced by Enriquez. We will see that these define holomorphic functions on the upper half-plane which degenerate to multiple zeta values at cusps. If time permits, we will explain how some of the rather mysterious modular phenomena pertaining to multiple zeta values can be interpreted directly via the algebraic structure of their elliptic analogs.
Numerical approximation of BSDEs with polynomial growth driver
Abstract
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.
Support characterisation for path-dependent SDEs
Abstract
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.
Reinforcement and random media
Abstract
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.
Angles of Random Polytopes
Abstract
We will consider some problems on calculating the average angles of random polytopes. Some of them are open.
Cut off phenomenon for the weakly asymmetric simple exclusion process
Abstract
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).
The branching-ruin number, the once-reinforced random walk, and other results
Abstract
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.
Small time asymptotics for Brownian motion with singular drift
Abstract
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.
'Semilinear PDE and hydrodynamic limits of particle systems on fractals'
Abstract
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.