Dear colleagues, I would like to invite you to the following online seminar organized by the Probability group of the University of Pisa. The two talks will be accessible under the link Click here to join<https://teams.microsoft.com/l/meetup-join/19:af3d635091e049579e555a84219ab378@thread.tacv2/1622580937570?context={"Tid":"c7456b31-a220-47f5-be52-473828670aa1","Oid":"dfd1e5f6-331d-43e0-a180-4bb6ce727fb7"}> Best regards, Giacomo Tuesday, June 8, 15:00 Speaker: Josué Corujo (Université Paris Dauphine) Title: Spectrum and ergodicity of a neutral multi-allelic Moran model Abstract: We will present some recent results on the study of a neutral multi-allelic Moran model, which is a finite continuous-time Markov process. For this process, it is assumed that the individuals interact according to two processes: a mutation process where they mutate independently of each other according to an irreducible rate matrix, and a Moran type reproduction process, where two individuals are uniformly chosen, one dies and the other is duplicated. During this talk we will discuss some recent results for the spectrum of the generator of the neutral multi-allelic Moran process, providing explicit expressions for its eigenvalues in terms of the eigenvalues of the rate matrix that drives the mutation process. Our approach does not require that the mutation process be reversible, or even diagonalizable. Additionally, we will discuss some applications of these results to the study of the speed of convergence to stationarity of the Moran process for a process with general mutation scheme. We specially focus on the case where the mutation scheme satisfies the so called "parent independent" condition, where (and only where) the neutral Moran model becomes reversible. In this later case we can go further and prove the existence of a cutoff phenomenon for the convergence to stationarity. This presentation is based on a recently submitted work, for which a preprint is available at https://arxiv.org/abs/2010.08809. Tuesday, June 8, 16:00 Speaker: Willem Van Zuijlen (WIAS) Title: Total mass asymptotics of the parabolic Anderson model Abstract: We consider the parabolic Anderson model with a white noise potential in two dimensions. This model is also called the stochastic heat equation with a multiplicative noise. We study the large time asymptotics of the total mass of the solution. Due to the irregularity of the white noise, in two dimensions the equation is a priori not well-posed. Using paracontrolled calculus or regularity structures one can make sense of the equation by a renormalisation, which can be thought of as ''subtracting infinity of the potential''. To obtain the asymptotics of the total mass we use the spectral decomposition, an alternative Feynman-Kac type representation and heat-kernel estimates which come from joint works with Khalil Chouk, Wolfgang König and Nicolas Perkowski. ************************ Giacomo Di Gesù Dipartimento di Matematica Università di Pisa Largo Bruno Pontecorvo 5 56127 - Pisa, Italy giacomo.digesu@unipi.it<mailto:giacomo.digesu@unipi.it> https://sites.google.com/site/giacomodigesu/