TY  - INPR
A1  - Bär, Christian
T1  - Renormalized integrals and a path integral formula for the heat kernel on a manifold
N2  - We introduce renormalized integrals which generalize conventional measure theoretic integrals. One approximates the integration domain by measure spaces and defines the integral as the limit of integrals over the approximating spaces. This concept is implicitly present in many mathematical contexts such as Cauchy's principal value, the determinant of operators on a Hilbert space and the Fourier transform of an L^p function. We use renormalized integrals to define a path integral on manifolds by approximation via geodesic polygons. The main part of the paper is dedicated to the proof of a path integral formula for the heat kernel of any self-adjoint generalized Laplace operator acting on sections of a vector bundle over a compact Riemannian manifold.
T3  - Preprints des Instituts für Mathematik der Universität Potsdam - 1(2012)21 
KW  - Renormalized integral
KW  - path integral
KW  - Feynman-Kac formula
KW  - generalized Laplace operator
KW  - Riemannian manifold
Y1  - 2012
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-60052
ER  - 
TY  - INPR
A1  - Bär, Christian
A1  - Pfäffle, Frank
T1  - Wiener measures on Riemannian manifolds and the Feynman-Kac formula
N2  - This is an introduction to Wiener measure and the Feynman-Kac formula on general Riemannian manifolds for Riemannian geometers with little or no background in stochastics. We explain the construction of Wiener measure based on the heat kernel in full detail and we prove the Feynman-Kac formula for Schrödinger operators with bounded potentials. We also consider normal Riemannian coverings and show that projecting and lifting of paths are inverse operations which respect the Wiener measure.
T3  - Preprints des Instituts für Mathematik der Universität Potsdam - 1(2012)17 
KW  - Wiener measure
KW  - conditional Wiener measure
KW  - Brownian motion
KW  - Brownian bridge
KW  - Riemannian manifold
Y1  - 2012
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-59998
ER  -