Differentiability of multiplicative processes related to branching random walks
The purpose of this paper is to study the almost sure differentiability of the terminal value of a certain martingale valued in the set of continuous functions from an open subinterval of \(\mathbb{R}\) on \(\mathbb{R}^+\). The latter martingale is obtained through concatenation and summation of a certain random sequence, and was initiated by \textit{B. B. Mandelbrot} [J. Fluid Mech. 62, 331-358 (1974; Zbl 0289.76031) and C. R. Acad. Sci., Paris, Sér. A 278, 289-292 (1974; Zbl 0276.60096)] and \textit{J. F. C. Kingman} [Ann. Probab. 3, 790-801 (1975; Zbl 0325.60079)], for different purposes. The question of the differentiability addressed here was recently studied by \textit{J. D. Biggins} [in: Applied probability. IMS Lect. Notes, Monogr. Ser. 18, 159-173 (1991; Zbl 0770.60077)]. In this article, the conditions ensuring the differentiability are (partly) slightly weaker than in the article of Biggins, and the method is also slightly different, relying on the study of a certain functional equation.
scientific article; zbMATH DE number 1525671