On the Density of the Winding Number of Planar Brownian Motion

Stella Brassesco () and Silvana C. García Pire ()
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Stella Brassesco: Instituto Venezolano de Investigaciones Científicas
Silvana C. García Pire: Universidad Nacional Experimental Simón Rodriguez

Journal of Theoretical Probability, 2014, vol. 27, issue 3, 899-914

Abstract: Abstract We obtain a formula for the density $$f(\theta , t)$$ of the winding number of a planar Brownian motion $$Z_t$$ around the origin. From this formula, we deduce an expansion for $$f(\log (\sqrt{t})\,\theta ,\,t)$$ in inverse powers of $$\log \sqrt{t}$$ and $$(1+\theta ^2)^{1/2}$$ which in particular yields the corrections of any order to Spitzer’s asymptotic law (in Spitzer, Trans. Am. Math. Soc. 87:187–197, 1958). We also obtain an expansion for $$f(\theta ,t)$$ in inverse powers of $$\log \sqrt{t}$$ , which yields precise asymptotics as $$t \rightarrow \infty $$ for a local limit theorem for the windings.

Keywords: Planar Brownian motion; Winding number; Transition density; Spitzer’s law; Local limit theorem; Asymptotic expansions; 60J65; 60J60; 58J65 (search for similar items in EconPapers)
Date: 2014
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DOI: 10.1007/s10959-012-0462-z

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