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On the connected components of the complement of a two-dimensional Brownian path. (English) Zbl 0748.60073

Random walks, Brownian motion, and interacting particle systems, Festschr. in Honor of Frank Spitzer, Prog. Probab. 28, 323-338 (1991).
[For the entire collection see Zbl 0733.00027.]
Certain a.s. estimate for the number \(N(u,v)\) of connected components of the complement of Brownian motion \(\xi (t)\) in the complex plane as \(t\in [u,v)\) is obtained. The result sharpens T. S. Mountford’s statement [Stochastics Stochastics Rep. 28, No. 3, 177-188 (1989; Zbl 0686.60087)] and answers B. B. Mandelbrot’s question on the limit behaviour of \(N_{[u,\infty)}\) as \(u\to 0\) [The fractal geometry of nature (1982; Zbl 0504.28001), Chapter 4].

MSC:

60J65 Brownian motion
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