Weak convergence of the distributions of Markovian random evolutions in two and three dimensions

We consider Markovian random evolutions performed by a particle moving in $R^2$ and $R^3$ with some finite constant speed $v$ randomly changing its directions at Poisson-paced time instants of intensity $\lambda>0$ uniformly on the $S_2$ and $S_3$-spheres, respectively. We prove that under the Kac condition
$$ v\to\infty,\qquad \lambda\to\infty,\qquad\frac{v^2}{\lambda}\to c,\qquad c>0 $$
the transition laws of the motions weakly converge in an appropriate Banach space to the transition law of the two- and three-dimensional Wiener process, respectively, with explicitly given generators.