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

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