UC San Diego

Let M be a Riemannian manifold, o 2 M be a xed base point, Wo (M) be the space of continuous paths from [0; 1] to M starting at o 2 M; and let x denote Wiener measure on Wo (M) conditioned to end at x 2 M: The goal of this thesis is to give a rigorous interpretation of the informal path integral expression for x;

dx () \ = "x ( (1)) 1 Z e􀀀1 2E()D , 2 Wo (M) :

In this expression E () is the "energy" of the path ; x is the { function based at x; D is interpreted as an innite dimensional volume "measure" and Z is a certain \normalization" constant. We will interpret the above path integral expression as a limit of measures, 1 P ;x; indexed by partitions, P of [0; 1]. The measures 1 P ;x are constructed by restricting the above path integral expression to the nite dimensional manifolds, HP;x (M) ; of piecewise geodesics in Wo (M) which are allowed to have jumps in their derivatives at the partition points and end at x. The informal volume measure, D; is then taken to be a certain Riemannian volume measure on HP;x (M) : When M is a symmetric space of non{compact type, we show how to naturally interpret the pinning condition, i.e. the { function term, in such a way that 1 P;x; are in fact well dened nite measures on HP;x (M) : The main theorem of the this thesis then asserts that 1 P ;x ! x (in a weak sense) as the mesh size of P tends to zero. Along the way we develop a number of integration{by{parts arguments for the approximate measures, 1 P;x; which are analogous to those known for the measures, vx.