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

Abstract

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 consisting of piecewise geodesic paths adapted to partitions $P$ of $[0,1]$. The finite dimensional manifolds of piecewise geodesics carry both an $H^{1}$ and a $L^{2}$ type Riemannian structures $G^i_P$. It is proved that as the mesh of the partition tends to $0$, $$ 1/Z_P^i e^{- 1/2 E(σ)} Vol_{G^i_P}(σ) \to ρ_i(σ)ν(σ) $$ where $E(σ)$ is the energy of the piecewise geodesic path $σ$, and for $i=0$ and $1$, $Z_P^i$ is a ``normalization'' constant, $Vol_{G^i_P}$ is the Riemannian volume form relative $G^i_P$, and $ν$ is Wiener measure on paths on $M$. Here $ρ_1 = 1$ and $$ ρ_0 (σ) = \exp( -1/6 \int_0^1 Scal(σ(s))ds ) $$ where $Scal$ is the scalar curvature of $M$. These results are also shown to imply the well know integration by parts formula for the Wiener measure.