A limit theorem for symmetric Markovian random evolution in R^m

Abstract

We consider the symmetric Markovian random evolution X(t) performed by a particle that moves with constant finite speed c in the Euclidean space R^m, m >= 2. Its motion is subject to the control of a homogeneous Poisson process of rate λ > 0. We show that, under the Kac condition c → ∞, λ →∞, (c^2/λ) → ρ, ρ > 0, the transition density of X(t) converges to the transition density of the homogeneous Wiener process with zero drift and the diffusion coefficient σ^2 = 2ρ/m.