Foundations of Modern Probability pp 470-489 | Cite as

# Connections with PDEs and Potential Theory

Chapter

## Abstract

Backward equation and Feynman-Kac formula; uniqueness for SDEs from existence for PDEs; harmonic functions and DirichleVs problem; Green functions as occupation densities; sweeping and equilibrium problems; dependence on conductor and domain; time reversal; capacities and random sets

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## References

- The fundamental solution to the heat equation in terms of the Gaussian kernel was obtained by Laplace (1809). A century later Bachelier (1900, 1901) noted the relationship between Brownian motion and the heat equation. The PDE connections were further explored by many authors, including Kolmogorov (1931a), Feller (1936), Kac (1951), and Doob (1955). A first version of Theorem 24.1 was obtained by Kac (1949), who was in turn inspired by Feynman’s (1948) work on the Schrödinger equation. Theorem 24.2 is due to Stroock and Vàradhan (1969).Google Scholar
- Green (1828), in his discussion of the Dirichlet problem, introduced the functions named after him. The Dirichlet, sweeping, and equilibrium problems were all studied by Gauss (1840) in a pioneering paper on electrostatics. The rigorous developments in potential theory began with Poincaré (1890–99), who solved the Dirichlet problem for domains with a smooth boundary. The equilibrium measure was characterized by Gauss as the unique measure minimizing a certain energy functional, but the existence of the minimum was not rigorously established until Frostman (1935).Google Scholar
- The first probabilistic connections were made by Phillips and Wiener (1923) and Courant et al. (1928), who solved the Dirichlet problem in the plane by a method of discrete approximation, involving a version of Theorem 24.5 for a simple symmetric random walk. Kolmogorov and Leontovich (1933) evaluated a special hitting distribution for two-dimensional Brownian motion and noted that it satisfies the heat equation. Kakutani (1944b, 1945) showed how the harmonic measure and sweeping kernel can be expressed in terms of a Brownian motion. The probabilistic methods were extended and perfected by Doob (1954, 1955), who noted the profound connections with martingale theory. A general potential theory was later developed by Hunt (1957–58) for broad classes of Markov processes.Google Scholar
- The interpretation of Green functions as occupation densities was known to Kac (1951), and a probabilistic approach to Green functions was developed by Hunt (1956). The connection between equilibrium measures and quitting times, implicit already in Spitzer (1964) and Itô and McKean (1965), was exploited by Chung (1973) to yield the explicit representation of Theorem 24.14.Google Scholar
- Time reversal of diffusion processes was first considered by Schrödinger (1931). Kolmogorov (1936b, 1937) computed the transition kernels of the reversed process and gave necessary and sufficient conditions for symmetry. The basic role of time reversal and duality in potential theory was recognized by Doob (1954) and Hunt (1958). Proposition 24.15 and the related construction in Theorem 24.21 go back to Hunt, but Theorem 24.19 may be new. The measure
*v*in Theorem 24.21 is related to the “Kuznetsov measures,” discussed extensively in Getoor (1990). The connection between random sets and alternating capacities was established by Choquet (1953–54), and a corresponding representation of infinitely divisible random sets was obtained by Matheron (1975).Google Scholar - Elementary introductions to probabilistic potential theory appear in Bass (1995) and Chung (1995), and to other PDE connections in Karatzas and Shreve (1991). A detailed exposition of classical probabilistic potential theory is given by Port and Stone (1978). Doob (1984) provides a wealth of further information on both the analytic and probabilistic aspects. Introductions to Hunt’s work and the subsequent developments are given by Chung (1982) and Dellacherie and Meyer (1975–87). More advanced treatments appear in Blumenthal and Getoor (1968) and Sharpe (1988).Google Scholar

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