The classical Kubo–Martin–Schwinger (KMS) condition is a fundamental property of statistical mechanics characterizing the equilibrium of infinite classical mechanical systems. It was introduced in the seventies by G. Gallavotti and E. Verboven as an alternative to the Dobrushin–Lanford–Ruelle (DLR) equation. In this article, we consider this concept in the framework of nonlinear Hamiltonian PDEs and discuss its relevance. In particular, we prove that Gibbs measures are the unique KMS equilibrium states for such systems. Our proof is based on Malliavin calculus and Gross–Sobolev spaces. The main feature of our work is the applicability of our results to the general context of white noise, abstract Wiener spaces and Gaussian probability spaces, as well as to fundamental examples of PDEs like the nonlinear Schrödinger, Hartree, and wave (Klein–Gordon) equations.
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Zied Ammari, Vedran Sohinger, Gibbs measures as unique KMS equilibrium states of nonlinear Hamiltonian PDEs. Rev. Mat. Iberoam. 39 (2023), no. 1, pp. 29–90DOI 10.4171/RMI/1366