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Integration by parts and invariant measure for KPZ

Yu Gu, Jeremy Quastel

TL;DR

This work provides a direct proof that spatial white noise is invariant under the stochastic Burgers/KPZ dynamics by developing two Gaussian integration-by-parts formulas and applying Stein's method. The authors leverage a polymer-representation-based IBP identity and a hidden cancellation mechanism to show that the white-noise law is preserved under evolution, avoiding purely discrete approximations. Central to the argument is a delicate mollification strategy that yields a convergent limit with a vanishing antisymmetric Itô term, captured by a key cancellation identity. The approach offers a conceptually transparent route with potential applications to polymer coalescence and broader nonlinear SPDEs.

Abstract

Using Stein's method and a Gaussian integration by parts, we provide a direct proof of the known fact that drifted Brownian motions are invariant measures (modulo height) for the KPZ equation.

Integration by parts and invariant measure for KPZ

TL;DR

This work provides a direct proof that spatial white noise is invariant under the stochastic Burgers/KPZ dynamics by developing two Gaussian integration-by-parts formulas and applying Stein's method. The authors leverage a polymer-representation-based IBP identity and a hidden cancellation mechanism to show that the white-noise law is preserved under evolution, avoiding purely discrete approximations. Central to the argument is a delicate mollification strategy that yields a convergent limit with a vanishing antisymmetric Itô term, captured by a key cancellation identity. The approach offers a conceptually transparent route with potential applications to polymer coalescence and broader nonlinear SPDEs.

Abstract

Using Stein's method and a Gaussian integration by parts, we provide a direct proof of the known fact that drifted Brownian motions are invariant measures (modulo height) for the KPZ equation.
Paper Structure (12 sections, 11 theorems, 113 equations)

This paper contains 12 sections, 11 theorems, 113 equations.

Table of Contents

  1. Background
  2. Formal argument ParisiSpohn
  3. Smoothed out version MR3350451
  4. Discrete version
  5. WASEP MR1462228
  6. Generator
  7. KdV
  8. Integration by parts method
  9. Main results
  10. Integration by parts
  11. Proof of Proposition \ref{['p22']}
  12. Gaussian integration by parts: Proofs of Prop. \ref{['p.yt120']} and Prop. \ref{['Prop13']}

Key Result

Theorem 2.1

The stochastic Burgers flow preserves white noise: If $u_0$ is a white noise, then $u_t$ is a white noise.

Theorems & Definitions (13)

  • Theorem 2.1
  • Proposition 2.2: Integration by parts for KPZ
  • Proposition 2.3
  • Remark 2.4
  • Remark 2.5
  • Proposition 3.1
  • Proposition 3.2
  • Lemma 4.1
  • Lemma 4.2
  • Lemma 4.3
  • ...and 3 more