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Holley--Stroock uniqueness method for the $\varphi^4_2$ dynamics

Roland Bauerschmidt, Benoit Dagallier, Hendrik Weber

TL;DR

This work extends the Holley–Stroock method to the continuum Phi^4_2 dynamics and proves uniqueness of the infinite-volume invariant measure for μ>μ_c(λ) by leveraging a uniform log-Sobolev inequality on finite-volume tori, a propagation-speed estimate comparing finite- and infinite-volume dynamics, and a finite-time relative-entropy bound. The approach combines Da Prato–Debussche decompositions, Wick renormalisation, and sharp Besov–Hölder regularity to control nonlinear terms and renormalisation effects in the continuum limit. The results establish exponential convergence to a unique invariant measure and provide a robust framework for connecting lattice-based log-Sobolev inequalities with continuum SPDE dynamics, while also highlighting the role of critical temperature via susceptibility. The methodology offers a template for extending to higher dimensions and related SPDEs, and clarifies how boundary-free Holley–Stroock arguments can be deployed in the continuum with controlled finite-volume approximations and entropy bounds.

Abstract

The approach initiated by Holley--Stroock establishes the uniqueness of invariant measures of Glauber dynamics of lattice spin systems from a uniform log-Sobolev inequality. We use this approach to prove uniqueness of the invariant measure of the $\varphi^4_2$ SPDE up to the critical temperature (characterised by finite susceptibility). The approach requires three ingredients: a uniform log-Sobolev inequality (which is already known), a propagation speed estimate, and a crude estimate on the relative entropy of the law of the finite volume dynamics at time $1$ with respect to the finite volume invariant measure. The last two ingredients are understood very generally on the lattice, but the proofs do not extend to SPDEs, and are here established in the instance of the $\varphi^4_2$ dynamics.

Holley--Stroock uniqueness method for the $\varphi^4_2$ dynamics

TL;DR

This work extends the Holley–Stroock method to the continuum Phi^4_2 dynamics and proves uniqueness of the infinite-volume invariant measure for μ>μ_c(λ) by leveraging a uniform log-Sobolev inequality on finite-volume tori, a propagation-speed estimate comparing finite- and infinite-volume dynamics, and a finite-time relative-entropy bound. The approach combines Da Prato–Debussche decompositions, Wick renormalisation, and sharp Besov–Hölder regularity to control nonlinear terms and renormalisation effects in the continuum limit. The results establish exponential convergence to a unique invariant measure and provide a robust framework for connecting lattice-based log-Sobolev inequalities with continuum SPDE dynamics, while also highlighting the role of critical temperature via susceptibility. The methodology offers a template for extending to higher dimensions and related SPDEs, and clarifies how boundary-free Holley–Stroock arguments can be deployed in the continuum with controlled finite-volume approximations and entropy bounds.

Abstract

The approach initiated by Holley--Stroock establishes the uniqueness of invariant measures of Glauber dynamics of lattice spin systems from a uniform log-Sobolev inequality. We use this approach to prove uniqueness of the invariant measure of the SPDE up to the critical temperature (characterised by finite susceptibility). The approach requires three ingredients: a uniform log-Sobolev inequality (which is already known), a propagation speed estimate, and a crude estimate on the relative entropy of the law of the finite volume dynamics at time with respect to the finite volume invariant measure. The last two ingredients are understood very generally on the lattice, but the proofs do not extend to SPDEs, and are here established in the instance of the dynamics.

Paper Structure

This paper contains 40 sections, 47 theorems, 477 equations.

Table of Contents

  1. Introduction and main results
  2. Preliminaries on the SPDE
  3. Local Besov--Hölder norms
  4. Scaling and test function
  5. Norms
  6. Periodisation and weights
  7. Space-periodic dynamics
  8. Gaussian estimates
  9. Da Prato--Debussche decomposition
  10. Local a priori bounds and consequences
  11. Markov process of the $\varphi^4_2$ dynamics
  12. Log-Sobolev inequality
  13. Main argument
  14. Propagation speed estimate
  15. Proof of Proposition \ref{['prop_finite_PS']}
  16. ...and 25 more sections

Key Result

theorem 1

For any $\lambda>0$ and $\mu>\mu_c(\lambda)$, the infinite volume $\varphi^4$ SPDE e:SPDE on $\R^2$ has a unique invariant measure $\nu$ supported on $C^{-\alpha}(\rho)$, where $\alpha,\sigma>0$ and $\rho(x)=(1+|x|^2)^{-\sigma/2}$. Moreover, for any deterministic initial condition $\varphi_0 \in C^{ where $\gamma>0$ is a uniform bound on the log-Sobolev constant of $\nu_L$ and $o(1)$ is independen

Theorems & Definitions (87)

  • theorem 1
  • proposition 1
  • proposition 2
  • theorem 2
  • theorem 3
  • corollary 1
  • proof
  • corollary 2
  • proof
  • corollary 3
  • ...and 77 more