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Stochastic dynamics and the Polchinski equation: an introduction

Roland Bauerschmidt, Thierry Bodineau, Benoit Dagallier

TL;DR

The paper develops a renormalisation-group framework for deriving log-Sobolev inequalities in stochastic dynamics by evolving a renormalised potential $V_t$ along the Polchinski equation. It connects this multiscale Bakry–Émery approach to Eldan's stochastic localisation, Föllmer's variational viewpoint, and transport-based methods, providing a unifying view of entropy decomposition across scales. The framework yields quantitative LSI bounds in diverse models, including Euclidean field theories, lattice and continuum $\varphi^4$ models, sine-Gordon, and Ising systems, with uniform guarantees in volume and, in many cases, independence from regularisation. The approach offers concrete pathwise, variational, and transport interpretations, enabling both entropy contraction proofs and Lipschitz transport bounds that transfer Gaussian refinements to interacting measures. This has broad implications for understanding near-critical dynamics, scaling, and continuum limits in statistical mechanics and quantum field theory, where classical convexity criteria fail and multiscale structure dominates relaxation behavior.

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 Föllmer process, the Boué--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.

Stochastic dynamics and the Polchinski equation: an introduction

TL;DR

The paper develops a renormalisation-group framework for deriving log-Sobolev inequalities in stochastic dynamics by evolving a renormalised potential along the Polchinski equation. It connects this multiscale Bakry–Émery approach to Eldan's stochastic localisation, Föllmer's variational viewpoint, and transport-based methods, providing a unifying view of entropy decomposition across scales. The framework yields quantitative LSI bounds in diverse models, including Euclidean field theories, lattice and continuum models, sine-Gordon, and Ising systems, with uniform guarantees in volume and, in many cases, independence from regularisation. The approach offers concrete pathwise, variational, and transport interpretations, enabling both entropy contraction proofs and Lipschitz transport bounds that transfer Gaussian refinements to interacting measures. This has broad implications for understanding near-critical dynamics, scaling, and continuum limits in statistical mechanics and quantum field theory, where classical convexity criteria fail and multiscale structure dominates relaxation behavior.

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 Föllmer process, the Boué--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.
Paper Structure (49 sections, 39 theorems, 422 equations, 1 figure)

This paper contains 49 sections, 39 theorems, 422 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Background on stochastic dynamics
  3. Motivation: Spin models and their stochastic dynamics
  4. Continuous spins
  5. Discrete spins
  6. Generalities on Glauber--Langevin dynamics
  7. Continuous spins
  8. Discrete spins
  9. Log-Sobolev inequality
  10. Bakry--Émery theorem
  11. Decomposition and properties of the entropy
  12. Difficulties arising from statistical physics perspective
  13. Difficulties arising from continuum perspective
  14. Gaussian integration and the Polchinski equation
  15. Gaussian integration
  16. ...and 34 more sections

Key Result

proposition 1

Consider the (continuous spin) stochastic dynamics eq: Langevin equation with invariant measure $\nu$ and Dirichlet form $D_{\nu}$ defined in eq: Dirichlet continuous. Then for $F_t(\varphi) = \E_{\varphi_0=\varphi} \qa{F(\varphi_t)}$ as in eq: distribution at time t, where the Fisher information is defined in terms of the Dirichlet form eq: Dirichlet continuous:

Figures (1)

  • Figure 1: In the renormalisation group approach, the small scales $\zeta$ are averaged out and one considers the projection of the measure to the variables $\varphi$ encoding the large scales; in the figure above, the Polchinski flow goes from $0$ to $+\infty$. In fact, $\zeta$ and $\varphi$ play symmetric roles: in particular for $t=0$ the original measure is coded by $\varphi$, while instead for $t =+\infty$ the original measure is coded by $\zeta$. Stochastic localisation puts the emphasis on the variable $\zeta$ and therefore flows in the opposite direction (depicted by the thick arrow).

Theorems & Definitions (86)

  • proposition 1: de Bruijn identity
  • proof
  • definition 1
  • theorem 1: Hypercontractivity MR0420249
  • proposition 2: Spectral gap inequality
  • proposition 3: de Bruijn identity
  • theorem 2: Bakry--Émery MR889476
  • proof : Proof of Theorem \ref{['thm:BE']}
  • proof
  • proposition 4: Entropy inequality
  • ...and 76 more