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**Winding of the plane Brownian motion.**
*(English)*
Zbl 0541.60075

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 \(\infty)\).

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 \(\infty)\).

Reviewer: D.Kannan

### MSC:

60J65 | Brownian motion |

58J65 | Diffusion processes and stochastic analysis on manifolds |

60J45 | Probabilistic potential theory |

### Citations:

Zbl 0191.466
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XMLCite

\textit{T. J. Lyons} and \textit{H. P. McKean}, Adv. Math. 51, 212--225 (1984; Zbl 0541.60075)

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### References:

[1] | Bowen, R., Hausdorff dimension of quasi-circles, Inst. Hautes Etudes Sci. Publ. Math., 50, 11-26 (1979) · Zbl 0439.30032 |

[2] | Breadon, A., Inequalities for certain Fuchsian groups, Acta Math., 127, 221-258 (1971) · Zbl 0235.30022 |

[3] | E. T. Copson; E. T. Copson · Zbl 0012.16902 |

[4] | Davis, B., Brownian motion and analytic functions, Ann. Probab., 7, 913-932 (1979) · Zbl 0421.60072 |

[5] | Itô, K.; McKean, H. P., Diffusion Processes (1965), Springer-Verlag: Springer-Verlag Berlin · Zbl 0127.09503 |

[6] | Lévy, P., Stochastic Integrals (1948), Gauthier-Villars: Gauthier-Villars Paris |

[7] | McKean, H. P., Stochastic Integrals (1969), Academic Press: Academic Press New York · Zbl 0191.46603 |

[8] | McKean, H. P.; Sullivan, D., Brownian notion and harmonic functions on the class surface of the thrice-punctured sphere, Advan. in Math., 51, 203-211 (1984) · Zbl 0541.60076 |

[9] | Patterson, S., The limit set of a Fuchsian group, Acta Math., 136, 241-273 (1976) · Zbl 0336.30005 |

[10] | Sullivan, D., The density at infinity of a discrete group of hyperbolic motions Inst, Hautes Etudes Sci. Publ. Math., 50, 419-450 (1979) |

[11] | Sullivan, D., On the ergodic theory at infinity of an arbitrary discrete group of hyperbolic motions, Inst. Hautes Etudes Sci. Publ. Math. (1980) |

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