On the Feynman-Kac Formula

In this article, given $y:[0,\eta )\to H$ a continuous map into a Hilbert space $H$ we study the equation $$\widehat{y}(t)=e^{{\int }_0^{t}c(s,\widehat{y})}y(t)$$ where $c(s,\cdot )$ is a given `potential’ on $C([0,\eta ),H)$. Applying the transformation $y\to \widehat{y}$ to the solutions of the SPDE and PDE underlying a diffusion, we study the Feynman-Kac formula.