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Published online by Cambridge University Press: 07 November 2024
The intimate connection between Brownian motion of classical potential theory is described in Chapter 11. The first topic is again the representation of solutions to the Dirichlet problem in terms of the exit distribution of Brownian paths from a region. In particular, it is shown that, with probability 1, Brownian paths exit through regular points. This is followed by a discussion of the Poisson problem and its relationship, depending on dimension, to the transience or recurrence of Brownian paths. Among other things, a proof is given of F. Riesz’s representation theorem for superharmonic functions, and this result is used to introduce the concept of capacity. K. L. Chung’s formula for the capacitory potential in term of the last exit distribution of Brownian paths is derived and used to prove Wiener’s test for regularity in terms of capacity. Finally, the chapter concludes with two interesting connections, one made by F. Spitzer and the other by G. Hunt, between Brownian paths and capacity.
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