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A flow approach to the generalized KPZ equation

Ajay Chandra, Léonard Ferdinand

TL;DR

The paper advances local well-posedness for the generalized KPZ equation with subcritical noise by extending Duch’s flow RG framework to non-polynomial nonlinearities. It introduces coordinates based on elementary differentials to handle non-polynomial right-hand sides, and constructs a truncated Polchinski-type flow combined with a remainder equation to control small-space-time scales. A two-tier analysis is performed: probabilistic control of generalized force cumulants and a deterministic fixed-point scheme for the remainder, leading to convergence of regularized solutions as the regularization vanishes. The methodology connects a continuous RG flow with a detailed coordinate system and renormalization to yield a robust local solvability theory for gKPZ and its parabolic Anderson model variant, with time-uniform renormalization constants.

Abstract

We show that the flow approach of Duch [Duc21] can be adapted to prove local well-posedness for the generalized Kardar-Parisi-Zhang equation. The key step is to extend the flow approach so that it can accommodate semi-linear equations involving smooth, non-polynomial, functions of the solution - this is accomplished by introducing coordinates for the flow built out of elementary differentials.

A flow approach to the generalized KPZ equation

TL;DR

The paper advances local well-posedness for the generalized KPZ equation with subcritical noise by extending Duch’s flow RG framework to non-polynomial nonlinearities. It introduces coordinates based on elementary differentials to handle non-polynomial right-hand sides, and constructs a truncated Polchinski-type flow combined with a remainder equation to control small-space-time scales. A two-tier analysis is performed: probabilistic control of generalized force cumulants and a deterministic fixed-point scheme for the remainder, leading to convergence of regularized solutions as the regularization vanishes. The methodology connects a continuous RG flow with a detailed coordinate system and renormalization to yield a robust local solvability theory for gKPZ and its parabolic Anderson model variant, with time-uniform renormalization constants.

Abstract

We show that the flow approach of Duch [Duc21] can be adapted to prove local well-posedness for the generalized Kardar-Parisi-Zhang equation. The key step is to extend the flow approach so that it can accommodate semi-linear equations involving smooth, non-polynomial, functions of the solution - this is accomplished by introducing coordinates for the flow built out of elementary differentials.
Paper Structure (20 sections, 35 theorems, 201 equations)

This paper contains 20 sections, 35 theorems, 201 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. The flow approach
  4. Notations
  5. Setting up the flow approach
  6. Introducing the effective scale and remainder flow
  7. Coordinates and the approximate flow
  8. Multi-indices and elementary differentials
  9. The flow equation for the stationary force coefficients
  10. Norms for stochastic objects
  11. Deterministic analysis
  12. Construction of the remainder
  13. Convergence of the solution
  14. Probabilistic analysis
  15. Generalized force coefficients
  16. ...and 5 more sections

Key Result

Theorem 1.8

Fix $\alpha\in ( 0\vee(1/2-n/4),1 ]$ and $\kappa\in(0,\kappa_0]$. Suppose that for $1 \leqslant i \leqslant j \leqslant n$, we have $b, d_i, g_{ij}, h \in C^{1+\Gamma+3N_1^{3\Gamma+1}}({{\hbox{\bfR}}})$, and that $(\varpi_\varepsilon)_{\varepsilon\in[0,1]}$ is as per Assumption assump:IC. Then, for

Theorems & Definitions (89)

  • Remark 1.2
  • Remark 1.3
  • Remark 1.4
  • Theorem 1.8
  • Theorem 1.9
  • Definition 1.10
  • Lemma 1.11
  • Definition 2.1
  • Remark 2.2
  • Remark 2.3
  • ...and 79 more