Asymptotic behaviour of certain martingales with values in a Riemannian manifold
The author studies the asymptotic behaviour [see also \textit{J.-J. Prat}, C. R. Acad. Sci., Paris, Sér. A 280, 1539-1542 (1975; Zbl 0309.60052)] of the bounded dilation martingales [see also \textit{W. S. Kendall}, Lect. Notes Math. 851, 479-491 (1981; Zbl 0467.60072)], or of the quasi-conformal martingales [see also \textit{S. I. Goldberg} and \textit{C. Mueller}, Bull. Am. Math. Soc., New Ser. 7, 259-263 (1982; Zbl 0501.58016)] with values in a Riemannian manifold. The main idea is that after translation and conditioning with a stopping time the upper processes are also martingales with the same features. The author also uses a comparison method [see also \textit{P. Malliavin}, Proc. Natl. Acad. Sci. USA 71, 381-383 (1974; Zbl 0449.60040)]. Theorems of almost sure convergence of martingales with values in a Riemannian manifold and a generalization of Picard's little theorem on stochastically complete Riemannian manifolds are given.
stochastically complete Riemannian manifold
scientific article; zbMATH DE number 1202433
scientific article