Construction and justification of the asymptotics of fundamental solutions of parabolic equations

The report will talk about some generalization of the WKB method for construction of fundamental solutions of parabolic equations. In contrast to the hyperbolic case, the fundamental solution of the parabolic equation has a real phase function. This leads to the well-known problem about the initial data that make it possible to represent the Dirac δ-function as an exponential with a real phase function. Such a representation can be obtained by expanding the phase space. As a result, an IOF with a non-oscillating exponent arises. This representation is a modification of the construction of the canonical operator proposed by V.P. Maslov to solve similar tasks.
As an example, we consider the construction and justification of the asymptotic of the fundamental solution of the degenerate in the main part parabolic equation.