Algebraic and arithmetic area for \(m\) planar Brownian paths. (English) Zbl 1456.60207
Summary: The leading and next to leading terms of the average arithmetic area \(\langle S\)(m)\( \rangle\) enclosed by \(m\to \infty\) independent closed Brownian planar paths, with a given length t and starting from and ending at the same point, are calculated. The leading term is found to be \(\langle S\)(m)\( \rangle \sim (\pi\) t/2)lnm and the 0-winding sector arithmetic area inside the \(m\) paths is subleading in the asymptotic regime. A closed form expression for the algebraic area distribution is also obtained and discussed.
MSC:
| 60J65 | Brownian motion |
References:
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