Path integral formulations of SDEs

Feynman path integrals, esp for stochastic processes

2021-04-19 — 2021-04-19

Wherein path integral methods are applied to stochastic differential equations, the Onsager–Machlup action is formulated for Fokker–Planck dynamics, and distinctions between Itô and Stratonovich discretizations are examined.

dynamical systems
probability
SDEs
statmech
stochastic processes
time series
Figure 1

Applications of the famous Feynman-style path integral for quantum systems to statistical systems of interest.

Path integrals are given by the sum over all paths satisfying some boundary conditions and can be understood as extensions to an infinite number of integration variables of usual multi-dimensional integrals.

Nothing to say here yet. Keywords: Onsager-Machlup, Fokker-Planck. Does it connect to Feynman-Kac formulae? I think so, looking at Wio (2013).

TODO: mention how they break differently in Itô vs Stratonovich.

1 References

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Borgerding, and Schniter. 2016. “Onsager-Corrected Deep Networks for Sparse Linear Inverse Problems.” arXiv:1612.01183 [Cs, Math].
Buzinov. 2017. “Feynman Formulas for Semigroups Generated by an Iterated Laplace Operator.” Russian Journal of Mathematical Physics.
Chow, and Buice. 2015. “Path Integral Methods for Stochastic Differential Equations.” The Journal of Mathematical Neuroscience (JMN).
Dickman, and Vidigal. 2003. “Path Integrals and Perturbation Theory for Stochastic Processes.” Brazilian Journal of Physics.
Hasegawa, and Van Vu. 2019. “Uncertainty Relations in Stochastic Processes: An Information Inequality Approach.” Physical Review E.
Li, Duan, and Liu. 2021. “Machine Learning Framework for Computing the Most Probable Paths of Stochastic Dynamical Systems.” Physical Review E.
Malory, and Sherlock. 2016. “Residual-Bridge Constructs for Conditioned Diffusions.”
Onsager, and Machlup. 1953. “Fluctuations and Irreversible Processes.” Physical Review.
Sethna. 2006. Statistical Mechanics: Entropy, Order Parameters, and Complexity.
Westbroek, King, Vvedensky, et al. 2018. “User’s Guide to Monte Carlo Methods for Evaluating Path Integrals.” American Journal of Physics.
Whitaker, Golightly, Boys, et al. 2017. “Improved Bridge Constructs for Stochastic Differential Equations.” Statistics and Computing.
Wio. 2013. Path Integrals for Stochastic Processes: An Introduction.
Zinn-Justin. 2009. “Path Integral.” Scholarpedia.