Feynman formula for a diffusion of particles with a variable mass in a domain

Abstract

In the present note we consider a class of second order parabolic equations with position dependent coefficients; such equations describe a diffusion of (quasi) particles with a variable mass. We represent a solution of Cauchy-Dirichlet problem for such class of equations in a bounded domain in the form of a limit of finite dimensional integrals of elementary functions. Such kind of a representation is usually called Feynman formula and can be used for calculations. Finite dimensional integrals in our Feynman formula give approximations for a functional integral over a probability measure on a set of trajectories in the domain where the solution of the considered problem is investigated; this measure is generated by a diffusion process with variable diffusion coefficient and absorption on the boundary, hence, to get Feynman formula also means to get a representation of the solution of the considered problem with the help of a functional integral (such kind of a representation is usually called Feynman-Kac formula).