Infinite Dimensional Dynamics Associated to Quadratic Hamiltonians

#### O. Garet

2000, v.6, №2, 205-237

ABSTRACT

We study here $R^{Z^d}$-valued gradient diffusions associated to quadratic interactions. We establish that each Gaussian Gibbs measure associated to this interaction can be obtained as a limit in time of the solution to the linear diffusion equation for a set of initial deterministic conditions which we describe. Thus the absence of phase transition corresponds to the ergodicity of the system. Moreover, we study the influence of a phase transition on the speed of convergence. Finally, we prove that the invariant measures for these gradient diffusions are exactly the associated Gibbs measures.

Keywords: Gibbsian field,Gaussian field,phase transition,ergodicity,infinite-dimensional diffusion,invariant measure