23/04/2025, 17:00 — 18:00 — Online
Benoit Dagallier, CEREMADE, Universite Paris Dauphine-PSL, France
Uniqueness of the invariant measure of the phi42 dynamics in infinite volume
I will discuss the phi42 dynamics, a singular stochastic partial differential formally corresponding to the Langevin dynamics for the phi42 field theory model. The goal is to characterise infinite volume invariant measures for these dynamics. The associated phi42 theory is known to undergo phase transitions as parameters in the model are varied, with the field transitioning from short-distance correlations (in the sense that the susceptibility of the measure defined on a finite box is bounded uniformly on the box size) to long-distance correlations. One expects this behaviour to be reflected in the dynamics, with a unique infinite volume invariant measure in absence of phase transition and possibly more than one in the strongly correlated regime. We prove that this picture is indeed correct in the sense that there is a unique invariant measure for the dynamics whenever the susceptibility is finite. This is done by adapting to the field-theory setting the Holley-Stroock-Zegarlinski approach to uniqueness for statistical mechanics models. This approach is based on a volume-independent bound on the log-Sobolev constant of the associated dynamics, together with crude, model-independent bounds. In the phi42 case the log-Sobolev has recently been established, but substantial work is required to adapt the other parts of the argument. The talk is based on joint work in progress with R. Bauerschmidt and H. Weber.
