Brownian Motion and its Applications to Mathematical Analysis
École d'Été de Probabilités de Saint-Flour XLIII – 2013
Article
We prove Archimedes’ principle for a macroscopic ball in ideal gas consisting of point particles with non-zero mass. The main result is an asymptotic theorem, as the number of point particles goes to infinity ...
Article
We give a new upper bound \(K_+\) K + ...
Article
Let \(\Omega \subset {\mathbb {R}}^n\) Ω ⊂ ...
Article
We consider random labelings of finite graphs conditioned on a small fixed number of peaks. We introduce a continuum framework where a combinatorial graph is associated with a metric graph and edges are identi...
Article
We present a game inspired by research on the possible number of billiard ball collisions in the whole Euclidean space. One player tries to place n static “balls” with zero radius (i.e., points) in a way that wil...
Article
We prove by example that the number of elastic collisions of n balls of equal mass and equal size ind-dimensional space can be greater than n3/27 for \({n \geq 3}\)n≥3 and \({d \geq 2}\)d≥2. The previously known ...
Article
We consider light ray reflections in a d-dimensional semi-infinite tube, for \(d\ge 3\) ...
Article
We show that there exists an ergodic conductance environment such that the weak (annealed) invariance principle holds for the corresponding continuous time random walk but the quenched invariance principle doe...
Article
The meteor process is a model for mass redistribution on a graph. The case of finite graphs was analyzed in Billey et al. (On meteors, earthworms and WIMPs. Ann Appl Probab, 2014). This paper is devoted to the me...
Chapter and Conference Paper
Reflection of a path is a perturbation that is sufficiently powerful to substantially change many properties of a stochastic process and yet sufficiently structured to be amenable to rigorous analysis. There s...
Article
We consider processes which have the distribution of standard Brownian motion (in the forward direction of time) starting from random points on the trajectory which accumulate at
Article
The purpose of this paper is to study the convergence in distribution of two subsequences of the signed cubic variation of the fractional Brownian motion with Hurst parameter
Book
École d'Été de Probabilités de Saint-Flour XLIII – 2013
Chapter
This chapter is devoted to new probabilistic proofs of results previously proved using analytic techniques.
Chapter
This chapter contains some simple facts and more advanced results on Nuemann eigenfunctions related to the hot spots conjecture.
Chapter
The hot spots problem is closely related to the problem of finding the location of the nodal line of the first non-constant eigenfunction. The chapter contains a few results on the latter problem.
Chapter
The chapter provides a short general review of Brownian motion and its place in probability theory. We also review some basic facts and formulas.
Chapter
This chapter is devoted to a general overview of the “hot spots” conjecture. To this day, the conjecture has been proved only for a very limited family of domains. Hence, it has a great potential as a source o...
Chapter
This chapter presents the definition and an application of the “scaling coupling.” This clever coupling links trajectories of two reflected Brownian motions at different times.
Chapter
This chapter contains the probabilistic proof of the claim that the Neumann heat kernel in a ball, evaluated on the diagonal, is a monotone function of the distance from the center.