Winding of the plane Brownian motion (Q794350)

The plane Brownian motion \((BM)X=a+ib\), \((i=\sqrt{-1})\), is recurrent and therefore visits every disc infinitely often (i.o.) as \(t\uparrow \infty\). If we puncture the complex plane \({\mathbb{C}}\) once (say at 0) and \(M_ 1\) is the universal cover of \({\mathbb{C}}\backslash 0\), then the BM X winds arbitrarily many times about 0 in the clockwise and anticlockwise directions but returns i.o. to the vicinity of X(0) unwound. But the situation is different if we puncture \({\mathbb{C}}\) twice (or, equivalently, if we consider a BM on a thrice punctured sphere). If \(M_ 2\) is the universal cover of \({\mathbb{C}}\backslash \{0,1\}\) and \(X\not\in \{0,1\}\), then X gets tangled up in its winding about 0 and 1. The more subtle homological counterpart of this fact (w.r.t \(M_ 2)\) is the content of this article which also corrects the error claiming the opposite in the second author's book ''Stochastic integrals.'' (1969; Zbl 0191.466). Let M be a compact Riemann surface with a ''suitable'' metric g and X be a BM with infinite first exit time from M (by choosing g so), then the Levy's alternative says that either X is recurrent and the expected occupation time \(e(D)=\infty\) for every disc D of M or X is transient and \(e(D)<\infty\) for every disc D. The Levy's alternative for the BM \(X_ 3\) on \(M_ 3\) is that \(X_ 3\) is recurrent or transient according as the Poincaré sum \(\sum_{k\in K_ 2}\exp [-d(i,ki)]\) diverges or not, where d is a hyperbolic distance and \(K_ 2\) is the commutator subgroup of \(G_ 2\). In his book, the second author concluded the divergence of this sum by using an inapplicable result. This article corrects this error and establishes the convergence of this Poincaré sum (so that \(X_ 3\) wanders off to \(\infty)\).