Nevanlinna theory through the Brownian motion
In this paper Nevanlinna theory using stochastic calculus is considered. This study is related to the works of \textit{B. Davis} [Trans. Am. Math. Soc. 213, 353--362 (1975; Zbl 0292.60126)], \textit{T. K. Carne} [Proc. Lond. Math. Soc. (3) 52, 349--368 (1986; Zbl 0562.60079)] and \textit{A. Atsuji} [J. Funct. Anal. 132, No. 2, 473--510 (1995; Zbl 0872.32019); Am. Math. Soc. 215, 109--123 (2005); translation from Sūgaku 54, No. 3, 235--248 (2002; Zbl 1082.60069); J. Math. Soc. Japan 60, No. 2, 471--493 (2008; Zbl 1145.32008); J. Math. Soc. Japan 69, No. 2, 477--501 (2017; Zbl 1369.32011)], etc. In particular, the proof of the first and second main theorems of Nevanlinna for meromorphic functions, through probabilistic methods, is given. The proof is founded on the statement of Levy that an analytic map transforms one Brownian motion into a new process which is also a Brownian motion, but modified by changing the time scale along each path. Also, a proof of Cartan's second main theorem is given (through the Ahlfors' method) for holomorphic curves into projective spaces using the stochastic calculus. Namely, let \(H_1,\dots,H_q\) be hyperplanes in \(\mathbb{P}^n(\mathbb{C})\) in general position. Let \(f:\mathbb{C}\rightarrow\mathbb{P}^n(\mathbb{C})\) be a linearly non-degenerated holomorphic curve. Then, for any positive \(\varepsilon\) and \(\delta\) \[ \sum\limits_{j=1}^qm_f(r,H_j)+N_W(r,0)\leqslant(n +1+\varepsilon)T_f(r)+\delta\log r\|_E, \] where \(N_W(r,0)\) is the counting function for the zeros of the Wronskian determinant \(W=W(f_0,\dots,f_n)\).
scientific article; zbMATH DE number 7137526
scientific article