How to correct the statistical properties of a stochastic evolution? An answer based on the Gibbs principle.

2023/12/01 金 15:30 - 17:00
Louis-Pierre Chaintron
ENS Paris

Stochastic diffusions are widely used to model physical phenomena that evolve in time like swarming birds, opinion dynamics or neuron activation… However, the proposed model is always an approximation that cannot exactly reproduce all the features of the system (mean, variance, higher order moment...). From the law of large numbers, such a statistic can be seen as an average of random realizations that one wants to bias to select the desired rare event. The Gibbs conditioning principle proposes a theoretical procedure to answer this problem, whose implementation to concrete dynamics remains challenging. In this talk, I will present a generalized version of Gibbs conditioning principle for stochastic diffusions. I will show how to describe the corrected process in a general way, and I will detail some concrete examples. An interesting PDE structure emerges from the correction procedure, that allows for connections with stochastic control. This latter point of view enables extensions to interacting systems of mean-field type. The long-time evolution of the corrected process yields surprising behaviours questioning the mechanism of selection of the equilibrium states. This ongoing work is a collaboration with Giovanni Conforti (CMAP, Ecole Polytechnique) and Julien Reygner (Cermics, ENPC).