Nonlinear martingale theory for processes with values in metric spaces of nonpositive curvature

• 成果类型: Article
• 署名作者: Sturm, KT
• 署名单位: University of Bonn
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• 发表日期: 2002
• 页码: 1195-1222
• 关键词: maps; probability

摘要

We develop a nonlinear martingale theory for time discrete processes (Y-n)(nis an element ofN0). These processes are defined on any filtered probability space (Omega, F, F-n, P)(nis an element ofN0) and have values in a metric space (N, d) of nonpositive curvature (in the sense of A. D. Alexandrov). The defining martingale property for such processes is E(Yn+1\F-n) = Y-n, P-a.s., where the conditional expectation on the left-hand side is defined as the minimizer of the functional Z --> Ed(2)(Z, Yn+1) within the space of F-n-measurable maps Z: Omega --> N. We give equivalent characterization of N-valued martingales (using merely the usual linear conditional expectations) and derive fundamental properties of these martingales, for example, a martingale convergence 4:1 theorem. Finally, we exploit the relation with harmonic maps. It turns out that a map f: M --> N is harmonic w.r.t. a given Markov kernel p on M if and only if it maps Markov chains (X-n)(nis an element ofN) (with transition kernel p) on M onto martingales (f(X-n))(nis an element ofN) with values in N. The nonlinear heat flow f: N-0 X M --> N of a given initial map f(0, (.)): M --> N at time n is obtained as the filtered expectation, f(n, x) := E-x[f(X-n)\\\(F-k)(kgreater than or equal to0)] of the random map f(X-n). Similarly, the unique solution to the Dirichlet problem for a given map g: M --> N and a subset D subset of M is obtained as f(x) := E-x[g(X-tau(D))\\\(F-k)(kgreater than or equal to0)]. In both cases, a crucial role is played by the notion of filtered expectation Ex[(.)\\\(F-k)(kgreater than or equal to0)] which will be discussed in detail. Moreover, we prove Jensen's inequality for expectations and filtered expectations and we prove (weak and strong) laws of large numbers for sequences of i.i.d. random variables with values in N. Our theory is an extension of the classical linear martingale theory and of the nonlinear theory of martingales with values in manifolds as developed, for example, in Emery (1989) and Kendall (1990). The Pal is to extend the previous framework towards processes with values in metric spaces. This will lead to a stochastic approach to the theory of (generalized) harmonic maps with values in such singular spaces as developed by Jost (1994) and Korevaar and Schoen (1993).