On invariant Gibbs measures for the generalized KdV equations

Abstract

Abstract. We consider the generalized KdV equations on the circle. In particular, we construct global-in-time solutions with initial data distributed according to the Gibbs measure and show that the law of the random solutions, at any time, is again given by the Gibbs measure. In handling a nonlinearity of an arbitrary high degree, we make use of the Hermite polynomials and the white noise functional.

A pedestrian approach to the invariant Gibbs measures for the 2-d defocusing nonlinear Schrödinger equations

“…In [ 6 , 38 ], Bourgain ( ) and Richards ( ) proved invariance of the Gibbs measures for the generalized KdV equation (gKdV) on the circle: In [ 36 ], the authors and Richards studied the problem for . In particular, by following the approach in [ 12 ] and this paper, we proved almost sure global existence and invariance of the Gibbs measures in some mild sense analogous to Theorem 1.4 for (i) all in the defocusing case and (ii) in the focusing case.…”

Invariant Gibbs measures for the 2-d defocusing nonlinear wave equations

“…Moreover, it also allows us to establish invariance of the Gibbs measure P In [30], we proved an analogous result for the defocusing Wick ordered nonlinear Schrödinger equations on M. Theorem 1.7 follows from repeating the argument presented in [30] with systematic modifications and thus we omit details. See also [1,8,11,29]. The main ingredient for Theorem 1.7 is to establish tightness (= compactness) of measures ν N on space-time functions, emanating from the truncated Gibbs measure P…”

Metastability of the Nonlinear Wave Equation: Insights from Transition State Theory

“…Rather than the dynamics of a single trajectory, here we consider the average crossing frequency of a collection of trajectories with initial distribution given by the invariant measure (6). We start by considering the finite dimensional system (18), with initial conditions selected from the invariant measure (23), and then justify the infinite dimensional limit. The same calculation for the microcanonical invariant measure (21) is considered in appendix C. 4.1.…”