Two-dimensional Brownian motion and harmonic functions

The purpose of this paper is to investigate the properties of two-dimensional Brownian motions ' and to apply the results thus ob- tained to the theory of harmonic functions in the Gaussian plane. Our starting point is the following theorem" Le D be a domain in the Gaussian plane R2, and let E be a closed set on the boundary Bd(D) of D. Then, under certain assumptions on D and E, the probability P(, E, D), that the Brownian motion starting from a point e D will enter into E without entering into the other part Bd(D)-E of the boundary of D before it, is equal to the harmonic measure in the sense of R. Nevanlinna of E with respect to the domain D and the point It is expected that, by means of this method, many of the known results in the theory of harmonic or analytic functions will be inter- preted from the standpoint of the theory of probability. We shall here give only the fundamental results and a few of its applications, leaving the detailed discussions of further applications to another occasion.