A Feynman-Kac Type Theorem for ODEs: Solutions of Second Order ODEs as Modes of Diffusions

TL;DR

This paper extends the Feynman-Kac theorem to second order ODEs, showing solutions as modes of diffusions, which enables Monte Carlo methods for numerical solutions.

Contribution

It introduces a Feynman-Kac type theorem for second order ODEs, linking solutions to diffusion modes via the Onsager-Machlup formalism.

Findings

Solutions of second order ODEs can be represented as modes of diffusions.

Monte Carlo methods can be used to estimate ODE solutions.

Numerical examples demonstrate the practical utility of the approach.

Abstract

In this article, we prove a Feynman-Kac type result for a broad class of second order ordinary differential equations. The classical Feynman-Kac theorem says that the solution to a broad class of second order parabolic equations is the mean of a particular diffusion. In our situation, we show that the solution to a system of second order ordinary differential equations is the mode of a diffusion, defined through the Onsager-Machlup formalism. One potential utility of our result is to use Monte Carlo type methods to estimate the solutions of ordinary differential equations. We conclude with examples of our result illustrating its utility in numerically solving linear second order ODEs.