The Iwasawa decomposition and the limiting behaviour of Brownian motion on a symmetric space of non-compact type. (English) Zbl 0658.58041

Taylor, J. C.

Geometry of random motion, Proc. AMS-IMS-SIAM Jt. Summer Res. Conf., Ithaca/NY 1987, Contemp. Math. 73, 303-332 (1988).

[For the entire collection see Zbl 0649.00018.]
Let G be a semisimple Lie group with maximal compact subgroup K. The Iwasawa decomposition yields a representation of Brownian motion on the symmetric space G/K as n(t) exp(H(t)). n(\(\cdot)\) and H(\(\cdot)\) are diffusions on the nilpotent subgroup of G and the Lie algebra of the Abelian part, respectively. The present article is an exposition of the following result of M. P. Malliavin and P. Malliavin [Théorie du Potentiel, Analyse Harmon. Lect. Notes Math. 404, 164-217 (1974; Zbl 0291.53025)]: n(t) converges a.s. and \(H(t)=D\cdot t+\) a term bounded by \(\sqrt{2t \log \log t}\). (D being a fixed element of the Lie algebra.) The proofs are mainly based on techniques of stochastic calculus.

Reviewer: K.-U.Schaumlöffel

MSC:

58J65	Diffusion processes and stochastic analysis on manifolds
43A85	Harmonic analysis on homogeneous spaces
22E30	Analysis on real and complex Lie groups
60B15	Probability measures on groups or semigroups, Fourier transforms, factorization
60H10	Stochastic ordinary differential equations (aspects of stochastic analysis)

Keywords:

Iwasawa decomposition; representation of Brownian motion; stochastic calculus