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

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.

arXiv commentComments welcome!

References in corpus (6)

• MAP Estimators and Their Consistency in Bayesian Nonparametric Inverse Problems
• An Introduction to Stochastic PDEs
• Feynman-Kac formula for heat equation driven by fractional white noise
• Feynman--Kac formula for the heat equation driven by fractional noise with Hurst parameter $H<1/2$
• Explicit solution of relative entropy weighted control
• Maximum a Posteriori State Path Estimation: Discretization Limits and their Interpretation