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 equation 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.

中文翻译：

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ODE 的 Feynman-Kac 类型定理：作为扩散模式的二阶 ODE 的解

在本文中，我们证明了一类广泛的二阶常微分方程的 Feynman-Kac 型结果。经典的 Feynman-Kac 定理说，一类广泛的二阶抛物线方程的解是特定扩散的平均值。在我们的情况下，我们表明二阶常微分方程组的解是扩散模式，通过 Onsager-Machlup 形式主义定义。我们的结果的一个潜在用途是使用蒙特卡罗方法来估计常微分方程的解。我们以我们的结果示例结束，说明它在数值求解线性二阶常微分方程中的效用。