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Renormalization of singular elliptic stochastic PDEs using flow equation

Paweł Duch

TL;DR

The paper develops a flow-equation–based renormalization framework for singular elliptic SPDEs with a fractional Laplacian, additive white noise, and a cubic nonlinearity in the subcritical regime. By introducing an effective force $F_{\kappa,\mu}$ and recursively defined force coefficients $F^{i,m}_{\kappa,\mu}$ (and generalized $F^{i,m,a}_{\kappa,\mu}$), it combines deterministic bounds with probabilistic cumulant estimates to control near-critical behavior and construct solutions via $\varPhi_\kappa = G_\kappa * F_{\kappa,1}[0]$, proving convergence as $\kappa \to 0$ in Besov spaces. A key innovation is the systematic use of Wilsonian RG flow equations, regularizing kernels, and a Taylor-analytic framework to manage both UV and long-range contributions, yielding finite sets of relevant coefficients and uniform cumulant bounds. The results establish existence, uniqueness up to renormalization, and almost-sure convergence, providing a robust renormalization scheme for nonlocal singular SPDEs beyond previous regularity-structure approaches.

Abstract

We develop a solution theory for singular elliptic stochastic PDEs with fractional Laplacian, additive white noise and cubic non-linearity. The method covers the whole sub-critical regime. It is based on the Wilsonian renormalization group theory and the Polchinski flow equation.

Renormalization of singular elliptic stochastic PDEs using flow equation

TL;DR

The paper develops a flow-equation–based renormalization framework for singular elliptic SPDEs with a fractional Laplacian, additive white noise, and a cubic nonlinearity in the subcritical regime. By introducing an effective force and recursively defined force coefficients (and generalized ), it combines deterministic bounds with probabilistic cumulant estimates to control near-critical behavior and construct solutions via , proving convergence as in Besov spaces. A key innovation is the systematic use of Wilsonian RG flow equations, regularizing kernels, and a Taylor-analytic framework to manage both UV and long-range contributions, yielding finite sets of relevant coefficients and uniform cumulant bounds. The results establish existence, uniqueness up to renormalization, and almost-sure convergence, providing a robust renormalization scheme for nonlocal singular SPDEs beyond previous regularity-structure approaches.

Abstract

We develop a solution theory for singular elliptic stochastic PDEs with fractional Laplacian, additive white noise and cubic non-linearity. The method covers the whole sub-critical regime. It is based on the Wilsonian renormalization group theory and the Polchinski flow equation.
Paper Structure (16 sections, 25 theorems, 144 equations)

This paper contains 16 sections, 25 theorems, 144 equations.

Table of Contents

  1. Introduction
  2. Effective force
  3. Construction of the effective force coefficients
  4. Regularizing kernels
  5. Function spaces for the coefficients
  6. Deterministic bounds for the irrelevant coefficients
  7. Generalized effective force coefficients
  8. Taylor polynomial and remainder
  9. Deterministic bounds for the relevant coefficients
  10. Deterministic bounds for the long-range part of the coefficients
  11. Deterministic construction of the solution
  12. Cumulants of the effective force coefficients
  13. Probabilistic analysis
  14. Function spaces for cumulants
  15. Flow equation for cumulants
  16. ...and 1 more sections

Key Result

Theorem 1.1

There exists a choice of counterterms and a random variable $\lambda_0$ such that $\mathbb{E}(\lambda_0^{-n})<\infty$ for every $n\in\mathbb{N}_+$ and for every random variable $\lambda\in[-\lambda_0,\lambda_0]$ and $\kappa\in(0,1/2]$ Eq. eq:intro_mild has a periodic solution $\varPhi_\kappa\in C^\i

Theorems & Definitions (115)

  • Theorem 1.1
  • proof
  • Remark 1.2
  • Remark 1.3
  • Remark 1.4
  • Definition 2.1
  • Remark 2.2
  • Remark 2.3
  • Remark 3.1
  • Remark 3.2
  • ...and 105 more