2023-24

In this talk, we will present some recent progress on using optimal transport theory and PDE methods to analyze the empirical large-time behavior of stochastic processes. Specifically, we studied the asymptotic properties of the occupation measure for Brownian motion on flat tori, by deriving an upper bound that we conjecture to be sharp, in terms of Brownian interlacements on Euclidean space. Our techniques show the potential for generalized application to other types of diffusion processes on weighted manifolds, and a similar asymptotic analysis may be possible for a variety of stochastic models arising in physics, biology, and beyond. Based on a joint work with M. Mariani (arXiv:2307.10325).

We prove that the fluctuations of the eigenvalues converge to the Gaussian Free Field (GFF) on the unit disk. These fluctuations appear on a non-natural scale, due to strong correlations between the eigenvalues. We also study the space--time correlations of those eigenvalues.

Then, motivated by the long-time behaviour of the ODE \dot{u}=Xu, we give a precise estimate on the eigenvalue with the largest real part and on the spectral radius of X, and prove the universality of their Gumbel fluctuations.

The Averaging process (a.k.a. repeated averages) is a mass redistribution model over the vertex set of a graph. Given a graph G, the process starts with a non-negative mass associated to each vertex. The edges of G are equipped with Poissonian clocks: when an edge rings, the masses at the two extremes of the edge are equally redistributed on these two vertices. Clearly, as time grows to infinity, the state of the system will converge (in some sense) to a flat configuration in which all the vertices have the same mass. This very simple process has been introduced to the probabilistic community by Aldous and Lanoue in 2012. However, up to few years ago, there was no graph for which sharp quantitative results on the time needed to reach equilibrium were available. Indeed, the analysis of this process requires different tools compared to the classical Markov chain framework, and even in the case of seemingly straightforward geometries—such as the complete graph or the 1-d torus—it can be handled only by means of non trivial probabilistic and functional analytic techniques. During the talk, I’ll try to give a broad overview of the problem and of its difficulties, and I'll present the few examples that have been completely settled. 

Based on joint work with P. Caputo (Roma Tre) and F. Sau (Università di Trieste)

Abstract:  We formulate the central limit theorem for the winding number and free energy for a directed polymer on 1 + 1 dimensional cylinder. We give formulas for the asymptotic variance in each case and discuss the relation between them. This is a joint work with A. Dunlap (Courant Institute) and Y. Gu (Univ. of Maryland). 

[1] Y. Gu, T. Komorowski„ KPZ on torus: Gaussian fluctuation, to appear in Ann. Inst. H. Poincare, Prob. and Stat., https://arxiv.org/abs/2104.13540

[2] Y. Gu, T. Komorowski„ Fluctuations of the winding number of a directed polymer on a cylinder, to appear in SIAM Journ. Math. Anal., https://arxiv.org/abs/2207.14091

[3] A. Dunlap, Y. Gu, T. Komorowski„ Fluctuations of the KPZ equation on a large torus, to appear in Comm. on Pure and Applied Math., https://arxiv.org/abs/2111.03650