Exact asymptotics of the density of the transition probability for discontinuous Markov processes

In this paper we consider some Kolmogorov- Feller equations with a small pa rameter h. We present a method for constructing the exact (exponential) asy mptotics of the fundamental solution of these equations for finite time int ervals uniformly with respect to h. This means that we construct an asympto tics of the density of the transition probability for discontinuous Markov processes. We justify the asymptotic solutions constructed. We also present an algorithm for constructing all terms of the asymptotics of the logarith mic limit (logarithmic asymptotics) of the fundamental solution as t --> +0 uniformly with respect to h. We write formulas of the asymptotics of the l ogarithmic limit for some special cases as t --> +0. The method presented i n this paper also allows us to construct exact asymptotics of solutions of initial-boundary value problems that are of probability meaning.