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Lagrangian Feynman formulae for second order parabolic equations in bounded and unbounded domains
In this note a class of second-order parabolic equations with variable coefficients, depending on coordinate, is considered in bounded and unbounded domains. Solutions of the Cauchy–Dirichlet and the Cauchy problems are represented in the form of a limit of finite-dimensional integrals of elementary functions (such representations are called Feynman formulas). Finite-dimensional integrals in the Feynman formulas give approximations for functional integrals in the corresponding Feynman–Kac formulas, representing solutions of these problems. Hence, these Feynman formulas give an effective tool to calculate functional integrals with respect to probability measures generated by diffusion processes with a variable diffusion coefficient and absorption on the boundary.