Asymptotic relation for the density of a multidimensional random evolution with rare poisson switchings - Kolesnik, A. D. (Ukrains’kyi Matematychnyi Zhurnal, 2008)
In the Euclidean space $\mathbb{R}^m,\quad m ≥ 2,$ the symmetric random evolution$X(t) = (X_1(t),...,X_m(t))$ controlled by a homogeneous Poisson process with parameter $λ > 0$ is considered.An asymptotic formula for the transition density $p(x,t),\quad t > 0,$ of the process $X(t)$ for $λ → 0$ is obtained. The behavior of $p(x,t)$ near the boundary of the diffusion area in spaces of various dimensions is described.