Miur - Prin 2015
Spring day in probability
Morning lecturer: Frank den Hollander (Leiden University)
Title: Large deviations for the Wiener sausage
Abstract: The Wiener sausage is the 1-environment of Brownian motion. It is an important mathematical object because it is one of the simplest non-Markovian functionals of Brownian motion. The Wiener sausage has been studied intensively since the 1970's. It plays a key role in the study of various stochastic phenomena, including heat conduction, trapping in random media, spectral properties of random Schrödinger operators, and Bose-Einstein condensation. In these lectures we look at two specific quantities: the volume and the capacity. After an introduction to the Wiener sausage, we show that both the volume and the capacity satisfy a downward large deviation principle. We identify the rate and the rate function, and analyse the properties of the rate function. We also explain how the large deviation principles are proved with the help of the skeleton approach. Joint work with Michiel van den Berg (Bristol) and Erwin Bolthausen (Zurich).
Afternoon lecturer: Giovanni Peccati (University of Luxembourg)
Title: Stein's method and stochastic geometry
Abstract: The so-called 'Stein's method' for probabilistic approximations is a collection of powerful analytical techniques, allowing one to explicitly assess the distance between the distributions of two random objects, by using caracterizing differential operators. Originally developed by Ch. Stein at the end of the sixties for dealing with one-dimensional normal approximations under weak dependence assumptions, Stein's method has rapidly become a crucial tool in many areas of modern stochastic analysis, ranging from random matrix theory and random graphs, to mathematical physics, geometry, combinatorics and statistics. In the first part of my talk, I will provide a self-contained introduction to Stein's method for normal approximations, by focussing on some connection with generalised integration by parts formulae, both in a continuous and discrete setting. In the second part of my talk, I will present some recent applications of Stein's method in stochastic geometry, with specific emphasis on the geometry of random fields, and on random geometric graphs.