Feynman Formulas and Functional Integrals for Diffusion with Drift in a Domain on a Manifold

We obtain representations for the solution of the Cauchy–Dirichlet problem for the diffusion equation with drift in a domain on a compact Riemannian manifold as limits of integrals over the Cartesian powers of the domain; the integrands are elementary functions depending on the geometric characteristics of the manifold, the coefficients of the equation, and the initial data. It is natural to call such representations Feynman formulas. Besides, we obtain representations for the solution of the Cauchy–Dirichlet problem for the diffusion equation with drift in a domain on a compact Riemannian manifold as functional integrals with respect to Weizsäcker–Smolyanov surface measures and the restriction of the Wiener measure to the set of trajectories in the domain; such a restriction of the measure corresponds to Brownian motion in a domain with absorbing boundary. In the proof, we use Chernoff's theorem and asymptotic estimates obtained in the papers of Smolyanov, Weizsäcker, and their coauthors.