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

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 t...

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Bibliographic Details
Main Authors Selk, Zachary, Honnappa, Harsha
Format Journal Article
LanguageEnglish
Published 15.06.2021
Subjects
Computer Science - Numerical Analysis
Mathematics - Classical Analysis and ODEs
Mathematics - Numerical Analysis
Mathematics - Probability
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Summary: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.
DOI:10.48550/arxiv.2106.08525