Classical Potential Theory and Brownian Motion
نویسنده
چکیده
Pioneering work of Doob, Kac, and Kakutani showed that there was a beautiful and deep connection between certain problems in the study of Brownian motion and those of classical potential theory. This work stimulated much research on the theory of Markov processes. In spite of all this work, however, there doesn't appear anywhere in the literature any reasonably complete treatment of the connection between potential theory and Brownian motion. In this paper and its companion "Logarithmic Potentials and Planar Brownian Motion" which follows in this volume, we present this connection in a way that is both elementary and essentially selfcontained. Our treatment here is not complete and will be expanded upon in a forthcoming monograph. This paper, being basically expository in nature, contains essentially nothing new. Its novelty (if any) consists in the treatment given to the topics discussed. In one place, however, we do seem to have a result that is new. This is in finding all bounded solutions of the modified Dirichlet problem for any arbitrary open set 0, and in giving a necessary and sufficient condition for there to be a unique such solution. In this paper, we consider a Brownian motion process X, in n _ 2 dimensional Euclidean space R'. Let B be a Borel set and set
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