"Finite dimensional approximations to Wiener measure and path integral
formulas on manifolds," by Lars Andersson and B. Driver
(UCSD Preprint, July 1998 )
The manuscript is available as a DVI
file (261K) or as a (PDF) file (625K)
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 $\mathrm{H}_{\mathcal{P}}(M)$
consisting of piecewise geodesic paths adapted to partitions $\mathcal{P}$
of $[0,1].$ The finite dimensional manifolds $\mathrm{H}_{\mathcal{P}}(M)$
carry both an $H^{1}$ and a $L^{2}$ type Riemannian structures, $G_{\mathcal{%
P}}^{1}$ and $G_{\mathcal{P}}^{0}$ respectively. It is proved that
\begin{equation*}
\frac{1}{Z_{\mathcal{P}}^{i}}e^{-\frac{1}{2}E(\sigma )}d\mathrm{Vol}_{G_{%
\mathcal{P}}^{i}}(\sigma )\rightarrow \rho _{i}(\sigma )d\nu (\sigma
)\text{
as mesh}(\mathcal{P)}\rightarrow 0,
\end{equation*}
where $E(\sigma )$ is the energy of the piecewise geodesic path $\sigma
\in
\mathrm{H}_{\mathcal{P}}(M),$ and for $i=0$ and $1,$ $Z_{\mathcal{P}}^{i}$
is a ``normalization'' constant, $\mathrm{Vol}_{G_{\mathcal{P}}^{i}}
$ is
the Riemannian volume form relative $G_{\mathcal{P}}^{i},$ and $\nu
$ is
Wiener measure on paths on $M.$ Here $\rho _{1}(\sigma )\equiv 1$ and
$\rho
_{0}(\sigma )=\exp \left( -\frac{1}{6}\int_{0}^{1}\mathrm{Scal} (\sigma
(s))ds\right) $ where $\mathrm{Scal}$ is the scalar curvature of $M.$
These
results are also shown to imply the well know integration by parts
formula
for the Wiener measure.
July, 1998 |
