Finite dimensional approximations to Wiener measure and path integral formulas on manifolds

Certain natural geometric approximation schemes are developed for Wiener measure on a compact Riemannian manifold. These approximations closely mimic the informal path integral formulas used in the physics literature for representing the heat semi-group on Riemannian manifolds. The path space is approximated by finite dimensional manifolds H-P(M) consisting of piecewise geodesic paths adapted to partitions P of [0, 1]. The finite dimensional manifolds H-P(M) carry both an H-1 and a L-2 type Riemannian structures, G(P)(1) and G(P)(0), respectively. It is proved that (1/Z(P)(i)) e(-(1/2)E(sigma)) d Vol(GP)(i)(sigma) --> rho(i)(sigma) d nu(sigma) as mesh(P) --> 0, where E(sigma) is the energy of the piecewise geodesic path sigma is an element of H-P(M), and for i = 0 and 1, Z(P)(i) is a "normalization" constant, Vol(GP)(i) is the Riemannian volume form relative to GPi, and nu is Wiener measure on paths on M. Here rho(1)(sigma) = 1 and rho(0)(sigma) = exp(-1/6 integral(0)(1) Scal(sigma(s)) ds) where Scal is the scalar curvature of M. These results are also shown to imply the well known integration by parts formula for the Wiener measure. (C) 1999 Academic Press.