# On the Feynman-Kac Formula

## TL;DR

This paper investigates the Feynman-Kac formula by analyzing a specific integral equation involving a potential function in a Hilbert space, connecting solutions of SPDEs and PDEs to this probabilistic representation.

## Contribution

It introduces a new approach to studying the Feynman-Kac formula through an integral equation in a Hilbert space setting, extending its application to SPDEs and PDEs.

## Key findings

- Establishes a link between solutions of SPDEs and PDEs via the integral equation.
- Provides a framework for applying the Feynman-Kac formula in infinite-dimensional spaces.

## Abstract

In this article, given $y :[0,\eta)\rightarrow H$ a continuous map into a Hilbert space $H$ we study the equation \[\hat y(t) = e^{\int_0^tc(s,\hat y)}y(t)\] where $c(s,\cdot)$ is a given `potential' on $C([0,\eta),H)$. Applying the transformation $y \rightarrow \hat y$ to the solutions of the SPDE and PDE underlying a diffusion, we study the Feynman-Kac formula.