Transience of a pair of local martingales

Abstract:

We consider the process of windings of complex Brownian motion about two points $a$ and $b$ in the complex plane, $\{ ({\theta ^a}(t),{\theta ^b}(t)):t \geq 0\}$. We show that this process is transient in the sense that ${\lim _t} \to \infty |({\theta ^a}(t),{\theta ^b}(t))| = \infty$. This extends a result found in both Lyons and McKean (1984) and McKean and Sullivan (1984). We will mostly use facts and ideas found in the former paper.