Let $d = 2$, and consider the domain $D = \mathbb{H}$, the upper half-plane. Can someone give me a reference to a proof that the Poisson kernel is the Cauchy distribution?
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$\begingroup$ here is a reference (page 60 and following): diss.fu-berlin.de/diss/servlets/MCRFileNodeServlet/… $\endgroup$– Carlo BeenakkerCommented Oct 23, 2015 at 17:52
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One place to find this and much more is "Brownian motion and martingales in analysis" by Rick Durrett. The result is originally due to Frank Spitzer, and can be found near the end of his article "Some theorems concerning 2-dimensional Brownian motion", Trans. Amer. Math. Soc., vol. 87, (1958) pp. 187–197.
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1$\begingroup$ the Dirichlet problem on the upper half-plane was not solved until 1958 ??? $\endgroup$ Commented Oct 24, 2015 at 7:48
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1$\begingroup$ I misspoke. The stochastic connection between the two-dimensional Brownian motion and the one-dimensional Cauchy process (which involves the Poisson kernel in a dynamic way) was established by Spitzer. Of course, Poisson's formula (in the disk) is much earlier--both he and Green were dead by 1841. $\endgroup$ Commented Oct 24, 2015 at 16:04