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

Abstract

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 Schrodinger 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 better adapted for derivation of Feynman formulas. The main idea of the proof of the central result of this paper is reducing the derivation of Feynman formu1 For the heat equation, these multiple integrals coincide with integrals being finite-dimensional approximations to integrals with respect to the Wiener measure. For the Schrodinger equation, such integrals coincide with those used in the definition (which goes back to Feynman himself) of sequential Feynman path integrals. Therefore, the limits of multiple integrals in the Feynman formulas are integrals with respect to the Wiener measure in the former case and (sequential) Feynman path integrals in the latter case, and in both cases, the Feynman‐Kac-type formulas are consequences of the Feynman formulas being discussed. las for equations on a manifold to the derivation of similar formulas for equations on a vector space. For equations on finite-dimensional manifolds (containing the usual finite-dimensional Laplacians), this approach was suggested in [4] and developed in [6]. The Riemannian manifold under consideration was embedded in a suitable Euclidean space (this can always be done by the Nash theorem), and the technique of surface measures developed in [4, 7] was applied. An essential point in the proof was the application of the Chernoff formula (generalizing the Trotter formula), which is related to obtaining representations of solutions to evolution equations on manifolds (and to representations of solutions to equations on vector spaces in terms of path integrals in the phase space [5]) in the same way as the Trotter formula is related to representations of the solution to the simplest Schrodinger equation with potential in terms of path integrals in the configuration space. The remark made in the footnote means that the results obtained in this paper contain the construction