Ergodic and quasideterministic properties of finite-dimensional stochastic systems

Abstract

The ergodic and stability properties of certain stochastic models are studied. Each model is described by a finite-dimensional stochastic processxλ(t) satisfyingdxλ=ℱλ(xλ,t)dt+ λdz(t), where ℱλ represents a “secular force” andz(t) is a stochastic process with given statistical properties. Such a model may represent a reduced description of an infinite-particle system. Thenxλ(t) may be either a set of macrovariables fluctuating about thermal equilibrium or the macrostate of a system maintained through pumping in a nonequilibrium state. Two Markovian models for whichz(t) is Wiener and ℱλ(y, t) = G(λ,y(t)) for someG nonlinear iny(t) are shown to possess a unique stationary probability density which is approached by any other density ast → ∞. For one of these models, which is of Hamiltonian type, the stationary state is given by the Maxwell-Boltzmann distribution. A particular form of non-Markovian model is also proved to have the above mixing property with respect to the Maxwell-Boltzmann distribution. Finally, the behavior of the sample paths ofxλ(t) for small values of the parameter A is investigated. In the case whenz(t) is Wiener and ℱλ(y, t) = G(y(t), it is shown thatxλ(t) will remain close to the deterministic trajectoryx0(t) (corresponding to λ = 0) for allt ⩾= 0 if and only ifx0(t) is highly stable with respect to small perturbations of the initial conditions.