FUNCTIONAL INTEGRALS OVER SMOLYANOV SURFACEMEASURES FOR EVOLUTIONARY EQUATIONS ON ARIEMANNIAN MANIFOLD

摘要

In this paper several results obtained for some evolutionary equations on a compact Riemannian manifold are presented. In particular, representations of the solution of the Cauchy-Dirichlet problem for the heat equation in a domain of a manifold are obtained in the form of limits of finite-dimensional integrals. These limits coincide with integrals over Smolyanov surface measures on the set of trajecto-ries in a manifold and over the Wiener measure, generated by Brownian motion in a domain with absorption on the boundary. Integrands are combinations of elementary functions of coefficients of the equation and geometric characteristics of the manifold. Also representations of the solution of the Cauchy problem for the Schroedinger equation on a compact Riemannian manifold are obtained in the form of functional integrals over Smolyanov surface measures. In the proof a substantial role is played by Smolyanov-Weizsaecker-Wittich asymptotic estimates for Gaussian integrals over a manifold, by the Chernoff theorem and by the method of transition from the Schroedinger to the heat equation going back to Doss.