On the Spectrum of the Generator of an Infinite System of Interacting Diffusions

Abstract:

We study the spectrum of the operator

generating an infinite-dimensional diffusion process Ξ (t), in space . Here ν is a “natural”Ξ (t)-invariant measure on which is a Gibbs distribution corresponding to a (formal) Hamiltonian H of an anharmonic crystal, with a value of the inverse temperature β > 0. For β small enough, we establish the existence of an L-invariant subspace such that has a distinctive character related to a “quasi-particle” picture. In particular, has a Lebesgue spectrum separated from the rest of the spectrum of L and concentrated near a point κ₁>0 giving the smallest non-zero eigenvalue of a limiting problem associated with β= 0.

An immediate corollary of our result is an exponentially fast L ₂-convergence to equilibrium for the process Ξ(t) for small values of β.