An Occupation Time Theorem for the Angular Component of Plane Brownian Motion

Abstract

Let $Z(t) = (X(t), Y(t)), t \geqq 0$, be the standard plane Brownian motion process. Let $(R(t), \theta_1(t))$ be the polar coordinates of $Z(t)$. Suppose that $R(0) = r > 0$ with probability 1, that is $Z(\cdot)$ starts away from the origin. We shall define the process $\theta(t), t \geqq 0$, as the total algebraic angle traveled on the continuous path $Z(s), 0 \leqq s \leqq t$; we take $\theta(0) = \theta_1(0)$; we have $\theta(\cdot) \equiv \theta_1(\cdot) \mod 2\pi$. Let $f = f(\theta), 0 \leqq \theta \leqq 2\pi$, be a bounded measurable function such that $f(0) = f(2\pi)$. For $r_1 > r$ let $\tau$ be the first passage time of $R(\cdot)$ to the point $r_1$. This note is devoted to the computation of the functional \begin{equation*}\tag{1.1}L(f) = E_{r,\theta}\{ \int^\tau_0 f(\theta_1(t)) dt\},\end{equation*} where $E_{r,\theta}\{\cdot \cdot\}$ is the expectation operator under the condition $R(0) = r, \theta_1(0) = \theta$. This is interpreted as the expected occupation time of a measurable subset of $\lbrack 0, 2\pi\rbrack$ if $f$ is the indicator function of the subset. We find an explicit formula for $L(f)$ as a linear functional on the Hilbert space $L_2\lbrack 0, 2\pi\rbrack$. A preliminary result of interest is presented in Section 2: the random variable $\lbrack \theta(\tau) - \theta(0)\rbrack/|\log (r_1/r)|$ has a Cauchy distribution for any positive numbers $r, r_1$ with $r \neq r_1$. This recalls the independent result of Spitzer that $\lbrack \theta (t) - \theta(0)\rbrack/\frac{1}{2} \log t$ has a limiting Cauchy distribution for $t \rightarrow \infty$ [4]. I thank the referee for his constructive remarks and for the alternate proof of Theorem 2.1 given in Section 6.