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Invariant measures for mKdV and KdV in infinite volume

Justin Forlano, Rowan Killip, Monica Visan

Abstract

We construct dynamics for the defocusing real-valued (Miura) mKdV equation on the real line with initial data distributed according to Gibbs measure. We also prove that Gibbs measure is invariant under these dynamics. On the way, we provide a new proof of the invariance of the Gibbs measure under mKdV on the torus. Building on these results, we construct new measure-preserving dynamics for the KdV equation on the whole real line. Samples from this family of measures exhibit the same local regularity as white-noise, but completely different statistics!

Invariant measures for mKdV and KdV in infinite volume

Abstract

We construct dynamics for the defocusing real-valued (Miura) mKdV equation on the real line with initial data distributed according to Gibbs measure. We also prove that Gibbs measure is invariant under these dynamics. On the way, we provide a new proof of the invariance of the Gibbs measure under mKdV on the torus. Building on these results, we construct new measure-preserving dynamics for the KdV equation on the whole real line. Samples from this family of measures exhibit the same local regularity as white-noise, but completely different statistics!
Paper Structure (16 sections, 46 theorems, 346 equations)

This paper contains 16 sections, 46 theorems, 346 equations.

Table of Contents

  1. Introduction
  2. Overview of the paper
  3. Gibbs measures
  4. The KMS condition
  5. Coupling Gibbs measures
  6. Mixing properties
  7. The infinite-volume Gibbs state as a diffusion
  8. Lax operators, Green's functions, and approximating flows
  9. Diagonal Green's functions
  10. The periodic case
  11. Invariance of the periodic Gibbs measure
  12. $H_\kappa$ dynamics and invariance in infinite volume
  13. Dynamics and invariance for mKdV in infinite volume
  14. New invariant measures for KdV in infinite volume
  15. The Miura mapping of Gibbs states
  16. ...and 1 more sections

Key Result

Theorem 1.1

For all values of temperature and chemical potential, solutions of mkdv define a measure-preserving group of transformations of the Gibbs state.

Theorems & Definitions (91)

  • Theorem 1.1
  • Theorem 1.2
  • Proposition 2.1
  • proof
  • Proposition 2.2: Gibbs measure
  • proof
  • Lemma 2.3: Conditional laws
  • proof
  • Lemma 2.4
  • proof
  • ...and 81 more