Some Theorems Concerning 2-Dimensional Brownian Motion

Abstract

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 Lévy. 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. Lévy, 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. Lévy 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.