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Renormalisation of Singular SPDEs with Correlated Coefficients

Nicolas Clozeau, Harprit Singh

Abstract

We show local well-posedness of the g-PAM and the $φ^{K+1}_2$-equation for $K\geq 1$ on the two-dimensional torus when the coefficient field is random and correlated to the driving noise. In the setting considered here, even when the model in the sense of Hairer (2014) is stationary, naive use of renormalisation constants in general leads to variance blow-up. Instead, we prove convergence of renormalised models choosing random renormalisation functions analogous to the deterministic variable coefficient setting. The main technical contribution are stochastic estimates on the model in this correlated setting which are obtained by a combination of heat kernel asymptotics, Gaussian integration by parts formulae and Hairer-Quastel type bounds.

Renormalisation of Singular SPDEs with Correlated Coefficients

Abstract

We show local well-posedness of the g-PAM and the -equation for on the two-dimensional torus when the coefficient field is random and correlated to the driving noise. In the setting considered here, even when the model in the sense of Hairer (2014) is stationary, naive use of renormalisation constants in general leads to variance blow-up. Instead, we prove convergence of renormalised models choosing random renormalisation functions analogous to the deterministic variable coefficient setting. The main technical contribution are stochastic estimates on the model in this correlated setting which are obtained by a combination of heat kernel asymptotics, Gaussian integration by parts formulae and Hairer-Quastel type bounds.

Paper Structure

This paper contains 18 sections, 19 theorems, 71 equations.

Table of Contents

  1. Introduction
  2. Solution theory based on Rough Analysis
  3. Main results
  4. Notations
  5. Solution theory
  6. Regularity structures and models
  7. Solutions to the g-PAM equation
  8. Canonical and renormalised models
  9. Proof of Theorem \ref{['thm:g-pam']}
  10. Solutions to the $\boldsymbol{\phi^{K+1}_2}$-equation
  11. Stochastic estimates for the g-PAM equation
  12. Stochastic estimates on $\boldsymbol{\hat{\Pi}_\star ^\delta (\Xi \cdot I\Xi)}$
  13. Stochastic estimates on $\boldsymbol{\hat{\Pi}_\star ^\delta (\partial_i I \Xi \cdot \partial_j I\Xi)}$
  14. Stochastic estimates for the $\boldsymbol{\phi^{K+1}_2}$-equation
  15. Proof of Proposition \ref{['prop:varblowup']}: Variance blow-up
  16. ...and 3 more sections

Key Result

Proposition 1.3

In the setting of Assumption AssumCoef for the g-PAM equation, assume furthermore that $\sigma \neq 0$ and $\det (A)$ is not constant. Let $u_\delta$ be the solution to SolutionStochasticHeatEq with the white noise $\xi$ replaced by $\xi_\delta$ defined in eq:regularised noise. Then, for any determ

Theorems & Definitions (30)

  • Remark 1.2
  • Proposition 1.3: variance blow-up
  • Remark 1.4
  • Remark 1.5
  • Remark 1.6
  • Theorem 1.7
  • Remark 1.8
  • Theorem 1.9
  • Remark 1.10
  • Remark 1.11
  • ...and 20 more