The statistical equilibrium of an isotropic stochastic flow with negative lyapounov exponents is trivial

Abstract

It is well known that every stochastic flow on R_d, whose one-point motion has an invariant measure m, gives rise to a measure-valued process (v_t, t≥0) with v₀=m, which converges almost surely to a random measure ℝ_d, called the statistical equilibrium. We prove here that if the flow is spatially homogeneous and isotropic, and if either the covariance is smooth and the top Lyapounov exponent is strictly negative, or if the flow is “of coalescing type” (these phenomena can only occur when d≤3), then v_∞=0 a.s.