• Corpus ID: 235446802

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

@article{Selk2021AFT,
  title={A Feynman-Kac Type Theorem for ODEs: Solutions of Second Order ODEs as Modes of Diffusions},
  author={Zachary Selk and Harsha Honnappa},
  journal={ArXiv},
  year={2021},
  volume={abs/2106.08525}
}
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… 
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References

SHOWING 1-10 OF 31 REFERENCES
Feynman-Kac formula for the heat equation driven by fractional noise with Hurst parameter H < 1/2
In this paper, a Feynman–Kac formula is established for stochastic partial differential equation driven by Gaussian noise which is, with respect to time, a fractional Brownian motion with Hurst
Feynman–kac Formulas for Black–Scholes–Type Operators
There are many references showing that a classical solution to the Black–Scholes equation is a stochastic solution. However, it is the converse of this theorem that is most relevant in applications,
Feynman–Kac formula for heat equation driven by fractional white noise
We establish a version of the Feynman-Kac formula for the multidimensional stochastic heat equation with a multiplicative fractional Brownian sheet. We use the techniques of Malliavin calculus to
The stochastic heat equation: Feynman-Kac formula and intermittence
We study, in one space dimension, the heat equation with a random potential that is a white noise in space and time. This equation is a linearized model for the evolution of a scalar field in a
Branching and interacting particle systems. Approximations of Feynman-Kac formulae with applications to non-linear filtering
This paper focuses on interacting particle systems methods for solving numerically a class of Feynman-Kac formulae arising in the study of certain parabolic differential equations, physics, biology,
A backward particle interpretation of Feynman-Kac formulae
We design a particle interpretation of Feynman-Kac measures on path spaces based on a backward Markovian representation combined with a traditional mean field particle interpretation of the flow of
Numerical Solution of Riccati Equations by the Adomian and Asymptotic Decomposition Methods over Extended Domains
We combine the Adomian decomposition method (ADM) and Adomian’s asymptotic decomposition method (AADM) for solving Riccati equations. We investigate the approximate global solution by matching the
A nonasymptotic theorem for unnormalized Feynman-Kac particle models
We present a nonasymptotic theorem for interacting particle approximations of unnormalized Feynman-Kac models. We provide an original stochastic analysis-based on Feynman-Kac semigroup techniques
Feynman-Kac Functional and the Schrödinger Equation
The Feynman-Kac formula and its connections with classical analysis were inititated in [3]. Recently there has been a revival of interest in the associated probabilistic methods, particularly in
An Introduction to Stochastic PDEs
These notes are based on a series of lectures given first at the University of Warwick in spring 2008 and then at the Courant Institute in spring 2009. It is an attempt to give a reasonably
...
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