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Working paper

Feynman type formulas for Feller semigroups in Riemannian manifolds

Mazzucchi S., Moretti V., Remizov I., Smolyanov O.
Feynman formulas are representations of solutions to initial value problems, for some parabolic and Schrödinger equations, by the limits of integrals over finite Cartesian powers of some spaces. Two versions of these formulas which were suggested by Feynman himself are associated with names of Trotter and Chernoff respectively. These formulas can be interpreted as approximations for path integrals over spaces of functions of a real variable; the corresponding representations of the solutions to the said equations are usually known as Feynman-Kac formulas. This work presents some new Feynman type formulas, related to the Chernoff theorem, on Riemannian manifolds. The used manifolds are of boundend geometry which include all compact manifolds and also a wide range of non-compact manifolds. Sufficient conditions are established for a class of second order elliptic operators to generate a Feller semigroup on a generally non-compact manifold of bounded geometry. A construction of Chernoff approximations is presented for those Feller semigroups in terms of shift operators. This provides approximations for solutions to initial-value problems for parabolic equations with variable coefficients on the manifold. It also yields the weak convergence of a sequence of random walks on the manifold to the diffusion process associated with the elliptic generator. For parallelizable manifolds this result is applied to the Brownian motion