Potential theory in probability

Something about harmonic functions or whatever

February 12, 2020 — August 24, 2023

dynamical systems
functional analysis
Lévy processes
stochastic processes
Figure 1


A theory about something to do with expectations of harmonic functions. Not the same thing as the potentials that arise as an alternative term to factor in factor graphs; that is different. I think? Unsure, because I have not read the literature yet.

I am unfamiliar with potential theory as a thing in itself, but it arises in Markov processes, exp Lévy processes and Markov bridge processes.

The origins of potential theory can be traced to the physical problem of reconstructing a repartition of electric charges inside a planar or a spatial domain, given the measurement of the electrical field created on the boundary of this domain.

Privault (2008):

In mathematical analytic terms this amounts to representing the values of a function \(h\) inside a domain given the data of the values of \(h\) on the boundary of the domain. In the simplest case of a domain empty of electric charges, the problem can be formulated as that of finding a harmonic function \(h\) on \(E\) (roughly speaking, a function with vanishing Laplacian […]) given its values prescribed by a function \(f\) on the boundary \(\partial E\), i.e. as the Dirichlet problem: \[ \begin{cases}\Delta h(y)=0, & y \in E, \\ h(y)=f(y), & y \in \partial E\end{cases} \] Close connections between the notion of potential and the Markov property have been observed at early stages of the development of the theory, see e.g. Joseph L. Doob (2001) and references therein. Thus a number of potential theoretic problems have a probabilistic interpretation or can be solved by probabilistic methods.

1 References

Adams, and Hedberg. 1999. Function Spaces and Potential Theory.
Bogdan, Byczkowski, Kulczycki, et al. 2009. Potential Analysis of Stable Processes and Its Extensions.
Brelot, Gowrisankaran, Murthy, et al. 1967. Lectures on Potential Theory.
Doob, J. L. 1959. Discrete Potential Theory and Boundaries.” Journal of Mathematics and Mechanics.
Doob, Joseph L. 2001. Classical Potential Theory and Its Probabilistic Counterpart.
Doyle, and Snell. 1984. Random Walks and Electric Networks.
Kallenberg. 2002. Foundations of Modern Probability. Probability and Its Applications.
Kuehn. 2008. Introduction to Potential Theory via Applications.” arXiv:0804.4689 [Math].
Privault. 2008. Potential Theory in Classical Probability.” In Quantum Potential Theory.