On the spectrum of the generator of an infinite system of interacting diffusions

R. A. Minlos, Iouri M. Soukhov

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21 Scopus citations


We study the spectrum of the operator Lf(Q) = - ∑x∈ℤd (∂2f/∂q2x) (Q) - β ∑x∈ℤd (∂H/∂qx) (Q) (∂f/∂qx) (Q), Q = {qx}, generating an infinite-dimensional diffusion process Ξ(t), in space L2 (ℝℤd, dv(Q)). Here v is a "natural" Ξ(t)-invariant measure on ℝℤd 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 H1 ⊂ L2(ℝℤd, dv(Q)) such that L | H1 has a distinctive character related to a "quasi-particle" picture. In particular, L | H1 has a Lebesgue spectrum separated from the rest of the spectrum of L and concentrated near a point K1 > 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 L2-convergence to equilibrium for the process Ξ(t) for small values of β.

Original languageEnglish (US)
Pages (from-to)463-489
Number of pages27
JournalCommunications In Mathematical Physics
Issue number2
StatePublished - Jan 1 1999

All Science Journal Classification (ASJC) codes

  • Statistical and Nonlinear Physics
  • Mathematical Physics


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