Stochastic dynamics and the Polchinski equation: an introduction

TL;DR

This paper introduces a renormalisation group perspective on stochastic dynamics, connecting various recent developments like stochastic localisation, variational formulas, and measure transportation, to deepen understanding of log-Sobolev inequalities.

Contribution

It provides an overview linking stochastic dynamics, renormalisation group methods, and recent advances in measure transportation and variational approaches.

Findings

Connects stochastic localisation with renormalisation group techniques.

Explains relationships between F"ollmer process, variational formulas, and measure transportation.

Highlights classical analogues for Hamilton--Jacobi equations in mean-field limits.

Abstract

This introduction surveys a renormalisation group perspective on log-Sobolev inequalities and related properties of stochastic dynamics. We also explain the relationship of this approach to related recent and less recent developments such as Eldan's stochastic localisation and the F\"ollmer process, the Bou\'e--Dupuis variational formula and the Barashkov--Gubinelli approach, the transportation of measure perspective, and the classical analogues of these ideas for Hamilton--Jacobi equations which arise in mean-field limits.