Feynman formulas for evolution equations with Levy Laplacians on infinite-dimensional manifolds

The Feynman formulas to solve Cauchy problems for the Schrodinger equation and heat equation with Levy Laplacian on the infinite-dimensional manifold of mappings from a closed real interval to a Riemannian manifold are obtained. The aim of the problem is to reduce the derivation of Feynman formulas for equations on a manifold to the derivation of similar formulas for equations on a vector space. The Laplace-Volterra operator and the Laplace-Levy operator is defined on the spaces of functions on a Riemannian manifold G that is either the entire Eucledian space or a compact space. The Feynman formula assumes that the Riemannian G is the submanifold of a Euclidean space with norm 1. The formula is used to solve the Cauchy problems for the heat equation and the Schrodinge equation with Levy Laplacian on the space of functions. The approximations of semigroups generated by Levy Laplacian are obtained by considering Laplace-Levy equation as the space of two different functions.