Some theorems concerning 2-dimensional Brownian motion

This paper consists of three separate parts(1) which are related mainly in that they treat different stochastic processes which arise in the study of plane brownian motion. §1 is concerned with the process R(t)=|Z(t)|, denoting the distance of the 2-dimensional separable Bachelier-Wiener process Z(t) =X(t)+iY(t) from the origin. We shall derive a law of the so-called strong type concerning the frequency of small values of R(t). This theorem disproves a conjecture of Paul Levy. In the next section we study the process θ(t) =arg Z(t). Results are obtained concerning the transition probabilities and absorption probabilities of θ(t). The limiting distribution of (2−1 log t) − 1 θ(t) is found to be the Cauchy distribution. This problem has also been considered by P. Levy, who showed that the distribution of θ(t) must have infinite variance. The two-sided absorption time is shown to be a random variable which has a finite nth moment if and only if the wedge which constitutes the absorbing barrier has an interior angle β<π/2n. In §3 we point out how plane brownian motion can be used to represent the Cauchy process. A theorem on brownian motion due to P. Levy is then used to gain information about the Cauchy process C(t). If −1 < C(0) = x <1 the probability that C(t)≧1 before C(t)≦−1 is found to be 1/2+π−1 sin−1 x.