Victor Pavlovich Maslov

Jump-type processes and their applications in quantum mechanics

A survey is presented of mathematical methods in the theory of the Feynman path integral. Principal attention is devoted to new results making it possible to represent the solution of the Cauchy problem for the Schrödinger equation and a quasilinear equation of Hartree type in the form of the mathematical expectation of functionals on jump-type Markov processes and to use Monte Carlo methods for solving these equations. A brief survey of results on complex Markov chains is presented.

Second term of the logarithmic asymptotics of path integrals

In the survey results are presented related to the construction of asymptotic expansions of Greens function of the Cauchy problem for the heat equation. The basic attention is devoted to the first two terms of the logarithmic asymptotics which are obtained “locally” by probabilistic methods and “globally” by the method of convolution of the sequence of asymptotic solutions over small time.