The authors study the small time asymptotics of the density of the transition probability of a Markov process , having both continuous and jump components. The forward Kolmogorov equation for the density of this process is written in the form
They construct the function which is an asymptotic approximation of the density and prove the convergence in the sense
Here , , and is a smooth given function. To this end they apply the Hamilton approach developed by V. Maslov and a special representation of the Dirac delta-function.
In Section 2 the authors construct a formal asymptotics to the symbol of the exact solution to the equation (1) rewritten in the Hamilton form
with initial data being a parameter and . The asymptotics is constructed in the form
where is proved to be a solution to the Hamilton-Jacobi equation
with initial data and satisfy corresponding transport equations. The expression for the leading term of the exponential asymptotics (as ) for is derived as well. In Section 3 the authors study the behavior of the function as and illustrate the results by several examples. The last section deals with the justification of the constructed asymptotics.