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Flow equation approach to singular stochastic PDEs

Paweł Duch

TL;DR

This work develops a Wilsonian renormalization group framework for subcritical singular SPDEs driven by additive noise with fractional diffusion, proving the existence and universality of a macroscopic scaling limit. By constructing an effective force via a Polchinski-type flow and controlling cumulants through a hierarchical renormalization of relevant coefficients, the authors establish that the macroscopic law depends only on a finite set of renormalization parameters and the limiting initial data, independent of microscopic details. The approach avoids diagrammatic techniques and yields a robust, scale-by-scale renormalization that applies to polynomial nonlinearities in the weakly nonlinear regime, with explicit treatment of regularized noises and initial-value problems. The results illuminate how universal macroscopic behavior emerges from a broad class of microscopic SPDEs, including non-Gaussian noise, within the subcritical regime, and they provide a comprehensive framework for analyzing singular SPDEs via RG flows. The methodology has potential implications for understanding universality classes in stochastic dynamics and for constructing rigorous scaling limits in nonlocal SPDE settings.

Abstract

We prove universality of a macroscopic behavior of solutions of a large class of semi-linear parabolic SPDEs on $\mathbb{R}_+\times\mathbb{T}$ with fractional Laplacian $(-Δ)^{σ/2}$, additive noise and polynomial non-linearity, where $\mathbb{T}$ is the $d$-dimensional torus. We consider the weakly non-linear regime and not necessarily Gaussian noises which are stationary, centered, sufficiently regular and satisfy some integrability and mixing conditions. We prove that the macroscopic scaling limit exists and has a universal law characterized by parameters of the relevant perturbations of the linear equation. We develop a new solution theory for singular SPDEs of the above-mentioned form using the Wilsonian renormalization group theory and the Polchinski flow equation. In particular, in the case of $d=4$ and the cubic non-linearity our analysis covers the whole sub-critical regime $σ>2$. Our technique avoids completely all the algebraic and combinatorial problems arising in different approaches.

Flow equation approach to singular stochastic PDEs

TL;DR

This work develops a Wilsonian renormalization group framework for subcritical singular SPDEs driven by additive noise with fractional diffusion, proving the existence and universality of a macroscopic scaling limit. By constructing an effective force via a Polchinski-type flow and controlling cumulants through a hierarchical renormalization of relevant coefficients, the authors establish that the macroscopic law depends only on a finite set of renormalization parameters and the limiting initial data, independent of microscopic details. The approach avoids diagrammatic techniques and yields a robust, scale-by-scale renormalization that applies to polynomial nonlinearities in the weakly nonlinear regime, with explicit treatment of regularized noises and initial-value problems. The results illuminate how universal macroscopic behavior emerges from a broad class of microscopic SPDEs, including non-Gaussian noise, within the subcritical regime, and they provide a comprehensive framework for analyzing singular SPDEs via RG flows. The methodology has potential implications for understanding universality classes in stochastic dynamics and for constructing rigorous scaling limits in nonlocal SPDE settings.

Abstract

We prove universality of a macroscopic behavior of solutions of a large class of semi-linear parabolic SPDEs on with fractional Laplacian , additive noise and polynomial non-linearity, where is the -dimensional torus. We consider the weakly non-linear regime and not necessarily Gaussian noises which are stationary, centered, sufficiently regular and satisfy some integrability and mixing conditions. We prove that the macroscopic scaling limit exists and has a universal law characterized by parameters of the relevant perturbations of the linear equation. We develop a new solution theory for singular SPDEs of the above-mentioned form using the Wilsonian renormalization group theory and the Polchinski flow equation. In particular, in the case of and the cubic non-linearity our analysis covers the whole sub-critical regime . Our technique avoids completely all the algebraic and combinatorial problems arising in different approaches.

Paper Structure

This paper contains 36 sections, 68 theorems, 528 equations.

Table of Contents

  1. Introduction
  2. Technique of the proof
  3. Flow equation approach
  4. Renormalization problem
  5. Initial value problem
  6. Statement of the result
  7. Assumptions
  8. Main theorem
  9. Regularizing kernels
  10. Function spaces and useful maps
  11. Taylor polynomial and remainder
  12. Effective force coefficients at stationarity
  13. Regularized noise and force
  14. Effective force and flow equation
  15. Support properties
  16. ...and 21 more sections

Key Result

Theorem 1.13

Let $\varPhi_\nu$ be the solution of the macroscopic equation eq:intro_spde_macro with the initial data $\phi_\nu$. Under the assumptions described above the macroscopic scaling limit $\varPhi_0=\lim_{\nu\searrow0}\varPhi_\nu$ exists and its law depends only on the renormalization parameters $\mathf

Theorems & Definitions (343)

  • Definition 1.1
  • Definition 1.2
  • Remark 1.3
  • Definition 1.4
  • Remark 1.5
  • Remark 1.6
  • Remark 1.7
  • Definition 1.8
  • Remark 1.9
  • Remark 1.10
  • ...and 333 more