Renormalized integrals and a path integral representation of the heat kernel

We propose a concept of renormalized integrals which generalizes integrals on measure spaces. This is a suitable framework for defining path integrals without having to construct an actual measure on path space. We show how this works for the heat kernel of a Laplace type operator on a compact Riemannian manifold. The hope is that this approach can circumvent problems with the standard approach using Wiener measure when one wants to deal with the Schrödinger equation instead of the heat equation.