Feynman formula for a class of second-order parabolic equations in a bounded domain

A representation of a solution to the boundary value Cauchy-Dirichlet problem for a class of second-order parabolic equations with coordinate-dependent coefficient is obtained with the help of a limit of finite-dimensional integrals of elementary functions depending on the coefficients of the equation and the initial conditions. Such representation for the solution to the Cauchy problem for evolutionary equations is termed as the Feynman formula. The finite-dimensional integrals in the 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 explored. The equations that describe the diffusion of particles with a mass depending on the particle's coordinate was considered. Instead to transitional probabilities of diffusion processes generated by such equations, their approximations expressed by elementary functions were used.