On the asymptotics of the transition probability density of processes with small diffusion

Let $x_s^{\varepsilon}$ be a diffusion process with the infinitesimal operator given by (3), and let $p^{\varepsilon}(t,x,y)$ be the transition probability density of $x_s^{\varepsilon}$. The aim of the article is to prove that the asymptotics of $p^{\varepsilon}(t,x,y)$ has the form of (4) if $t$ and the distance between $x$ and $y$ are sufficiently small. We calculate the principal term of the asymptotics and deduce recurrent formulas for the others.