### A LIMIT THEOREM FOR SYMMETRIC MARKOVIAN RANDOM EVOLUTION IN $Rˆm$

**ALEXANDER D. KOLESNIK**

*Theory of Stochastic Processes*

*Vol. 14 (30), no. 1, 2008, pp. 69–75*

We consider the symmetric Markovian random evolution $\bold X(t)$ performed by a particle that moves with constant finite speed $c$ in the Euclidean space $\Bbb R^m, \ m\ge 2.$ Its motion is subject to the control of a homogeneous Poisson process of rate $\lambda>0$.

We show that, under the Kac condition

$c\to\infty, \ \lambda\to\infty, \ (c^2/\lambda)\to\rho, \ \rho>0,$

the transition density of $\bold X(t)$ converges to the transition density of the homogeneous Wiener process with zero drift and the diffusion coefficient

$\sigma^2=2\rho/m$.

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