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

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dc.creatorAndersson, Lars
dc.creatorDriver, Bruce K.
dc.date1998-07-19
dc.date.accessioned2026-07-07T05:25:26Z
dc.date.available2026-07-07T05:25:26Z
dc.descriptionCertain 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.
dc.description48 pages, latex2e using amsart and amssymb
dc.identifierhttps://arxiv.org/abs/math/9807098
dc.identifierhttp://arxiv.org/abs/math/9807098
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/77177
dc.subjectDifferential Geometry
dc.subjectProbability
dc.subject60H07, 58D30 (Primary) 58D20 (Secondary)
dc.titleFinite dimensional approximations to Wiener measure and path integral formulas on manifolds
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