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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| Main Authors | , |
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| Format | Journal Article |
| Language | English |
| Published |
15.06.2021
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| Online Access | Get full text |
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| 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. |
|---|---|
| AbstractList | 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. |
| Author | Selk, Zachary Honnappa, Harsha |
| Author_xml | – sequence: 1 givenname: Zachary surname: Selk fullname: Selk, Zachary – sequence: 2 givenname: Harsha surname: Honnappa fullname: Honnappa, Harsha |
| BackLink | https://doi.org/10.48550/arXiv.2106.08525$$DView paper in arXiv |
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| Snippet | 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... |
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| SubjectTerms | Computer Science - Numerical Analysis Mathematics - Classical Analysis and ODEs Mathematics - Numerical Analysis Mathematics - Probability |
| Title | A Feynman-Kac Type Theorem for ODEs: Solutions of Second Order ODEs as Modes of Diffusions |
| URI | https://arxiv.org/abs/2106.08525 |
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