A limit theorem for symmetric Markovian random evolution in ${\mathbb R}^m$

We consider the symmetric Markovian random evolution ${\mathbf X}(t)$ performed by a particle that moves with constant finite speed c in the Euclidean space ${\mathbb R}^m, m\geq 2(t).$ Its motion is subject to the control of a homogeneous Poisson process of rate $\lambda>0.$ We show that, under the Kac condition $c\to\infty, \lambda\to\infty, (c^2/\lambda)-\rho, \rho>0,$ the transition density of ${\mathbf X}(t)$ converges to the transition density of the homogeneous Wiener process with zero drift and the diffusion coefficient $\sigma^2=2\rho/m$.