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

A Feynman formula is a representation of the solution to the Cauchy problem for an evolution partial differential (or pseudodifferential) equation in terms of the limit of a sequence of multiple integrals with multiplicities tending to infinity. The integrands are products of the initial condition and Gaussian (or complex Gaussian) exponentials 1 [5]. In this paper, we obtain Feynman formulas for the solutions to the Cauchy problems for the Schrödinger equation and the heat equation with Levy Laplacian on the infinite-dimensional manifold of mappings from a closed real interval to a Riemannian manifold. The definition of the Levi Laplacian acting on functions on such a manifold is obtained by combining the methods of papers [3] and [7]. In the former, Levi Laplacians in the space of functions on an infinitedimensional vector space were considered, and in the latter, Volterra Laplacians in the space of functions on the above infinite-dimensional manifold were examined. This definition of a Levi Laplacian is equivalent to that given in [2], but it is