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Nevanlinna theory, Fuchsian functions and Brownian motion windings. (English) Zbl 1055.60077

Let \(f\) be a meromorphic function in the unit disk \(D\), denote by \(D_r\) the Euclidian disk centered at \(O\) with radius \(r<1\). The fundamental quantity in Nevanlinna theory is the Nevanlinna characteristic \(T_f(r)= T(f,r)=(1 /\pi) \int_{D_r}{ {| f'(z)| ^2}\over {(| f(z)| ^2+1)^2}} \log(r/| z| )\,dx\,dy.\) The paper deals with the A. Atsuji proposal [J. Math. Sci. Univ. Tokyo 3, 45–56 (1996; Zbl 1054.30518)] to study \(T(f,r)\) with the help of the parametrised family of integrals \(J_{\alpha}(f)=\int_0^{\sigma} | f(z_s)| ^{\alpha}\, ds \), \(\alpha>0\), where \(\sigma\) is the first hitting time of the boundary of \(D\) by a complex Brownian motion \((z_s, s \geq 0)\) started in \(D\). The author demonstrates that Atsuji’s criterion does not apply to the basic case when \(f\) is a modular elliptic function. The presented example of the modular function is strongly connected with windings of the Brownian motion on the thrice punctured sphere. The divergence of similar integrals computed along the geodesic flow is also proved.

MSC:

60J65 Brownian motion
30D35 Value distribution of meromorphic functions of one complex variable, Nevanlinna theory
30F45 Conformal metrics (hyperbolic, Poincaré, distance functions)

Citations:

Zbl 1054.30518
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