Noticias en español
Random burning of the Euclidean lattice.
Resumen: The burning number of a graph is the minimal number of steps that are needed to burn all of its vertices, with the following burning procedure: at each step, one can choose a point to set on fire, and the fire propagates constantly at unit speed along the edges of the graph. In joint work with Alice Contat, following Mitsche, Prałat and Roshanbin, we consider two natural random burning procedures in the discrete Euclidean torus $\mathbb{T}_n^d$, in which the points that we set on fire at each step are random variables. Our main result deals with the case where at each step, the...
Read MorePoisson representation of Brownian bridge.
Resumen: We consider Brownian motion $(B(t))$, for $t\in[0,1]$, and Brownian bridge $BB(t)$, the Brownian motion conditioned to return to $0$ at time~$1$. The following identity is well known,(1)\,\hfill law of $(BB(t))_{t\in[0,1]}= $ law of $(B(t)- tB(1))_{t\in[0,1]}$. \hfill\ A centered and rescaled Poisson point process $B^\varepsilon(t)$ converges to Brownian motion, where $\varepsilon$ is the scaling parameter going to $0$. For each $\varepsilon>0$, we construct a coupling $(B^\varepsilon(t),BB^\varepsilon (t))$ satisfying an almost sure version of (1). Taking $\varepsilon\to0$...
Read MoreDomination criterion for some positive operators and quasi-stationary distributions.
Resumen: After a short introduction to the concept of quasi-stationary distributions, I will present the typical and well known “finite state space” convergence results. In a second time, I will present domination criteria for the quasi-compactness of positive operators and show some applications of these spectral theoretical results for the study of quasi-stationary distributions. The talk will conclude with an illustration on the interplay between these results and recent ones on weighted branching processes, obtained in collaboration with Nicolas Zalduendo.
Read MoreCentral limit theorems for structured branching processes
Resumen: In this talk I will discuss recent progress on central limit theorems for supercritical branching Markov processes in infinite-dimensional settings. The class of processes under consideration allows for spatial dependence and branching mechanisms that need not be local. A key feature of our approach is that it only requires a fourth moment condition together with exponential convergence of the mean semigroup in a weighted total variation norm. This assumption is mild in that it does not rely on symmetry or detailed spectral information. The resulting central limit theorems capture...
Read MoreStochastic processes, transport of mass, and functional inequalities.
Resumen: Functional inequalities have proven to be a ubiquitous tool in mathematics, especially in probability theory. For example, they are closely related to the concentration of measure phenomenon, and they help quantify the rate at which ergodic Markov processes converge to equilibrium. Prominent examples of those inequalities include the families of logarithmic Sobolev, Poincaré, and transport-entropy inequalities. In the first part of the talk, I will provide an introduction to this topic, highlighting the classical examples, results, and applications. In the second part of the talk,...
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