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 $G_P^i$. It is proved that as the mesh of the partition tends to $0$, $$1/Z_P^i{e}^{-1/2E(\sigma )}Vo{l}_{G_P^i}(\sigma )\to {\rho }_i(\sigma )\nu (\sigma )$$ where $E(\sigma )$ is the energy of the piecewise geodesic path $\sigma$, and for $i=0$ and $1$, $Z_P^i$ is a ``normalization'' constant, $Vo{l}_{G_P^i}$ is the Riemannian volume form relative $G_P^i$, and $\nu$ is Wiener measure on paths on $M$. Here ${\rho }_1=1$ and $${\rho }_0(\sigma )=\exp (-1/6{\int }_0^{1}Scal(\sigma (s))ds)$$ 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.

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