Summary
The two stochastic flows studied are (i) the canonical stochastic flow on the orthonormal frame bundle of hyperbolic space (which gives stochastic parallel translation along Brownian paths in hyperbolic space) and (ii) a stochastic flow on the sphereS n-1 arising from its embedding as the unit sphere in ℝn. Both flows are controlled by the same stochastic differential equation in a finite-dimensional Lie group. In each case the Lyapunov exponents are computed and a complete description is given of the local and global stability of the flow.
Article PDF
Similar content being viewed by others
References
Arnold, V.I., Avez, A.: Ergodic problems of classical mechanics. New York: Benjamin 1968
Baxendale, P.H.: Wiener processes on manifolds of maps. Proc. Royal Soc. Edinburgh87 A, 127–152 (1980)
Baxendale, P.H.: Recurrence of a diffusion process to a shrinking target. J. London Math. Soc.32, 166–176 (1985)
Bismut, J-M.: Mécanique aléatoire. Lecture Notes in Math.866. Berlin Heidelberg New York: Springer 1981
Carverhill, A.P.: Flows of stochastic dynamical systems: ergodic theory. Stochastics14, 273–318 (1985)
Elworthy, K.D.: Stochastic differential equations on manifolds. Cambridge University Press (1982)
Ichihara, K., Kunita, H.: A classification of the second order degenerate elliptic operators and its probability characterization. Z. Wahrscheinlichkeitstheor. Verw. Geb.30, 235–254 (1974)
Ikeda, N., Watanabe, S.: Stochastic differential equations and diffusion processes. Amsterdam-Tokyo: North Holland/Kodansha 1981
Itô, K., McKean, H.P.: Diffusion processes and their sample paths. Berlin-Heidelberg-New York. Springer 1965
Klingenberg, W.: Riemannian geometry. Berlin: de Gruyter 1982
Kobayashi, S., Nomizu, K.: Foundations of differential geometry, Vol. I. New York: Interscience 1963
Kunita, H.: On backward stochastic differential equations. Stochastics6, 293–313 (1982)
Le Jan, Y.: Equilibrium state for a turbulent flow of diffusion. Proceedings “Stochastic processes and infinite dimensional analysis”. Bielefeld 1983. Pitman Research Notes in Mathematics 124 (1985)
Malliavin, M-P. and P.: Factorisations et lois limites de la diffusion horizontale au-dessus d'un espace Riemannien symetrique. Théorie du potential et analyse harmonique. Lecture Notes in Mathematics404, 164–217 Berlin-Heibelberg-New York: Springer-Verlag 1974
Oseledec, V.I.: A multiplicative ergodic theorem. Ljapunov characteristic numbers for dynamical systems. Trans. Moscow Math. Soc.19, 197–231 (1968)
Pinsky, M.A.: Stochastic Riemannian geometry, Probabilistic analysis and related topics. Vol. 1, pp. 199–236. New York: Academic Press 1978
Prat, J-J,.: Etude asymptotique et convergence angulaire du mouvement Brownien sur une variété a courbure negative. C.R. Acad. Sci. Paris Sér. A280, A1539-A1542 (1975)
Ruelle, D.: Ergodic theory of differentiable dynamical systems. Inst. Hautes Etudes Sci. Publ. Math.50, 275–305 (1979)
Wong, E., Zakai, M.: Riemann-Stieltijes approximations of stochastic intergrals. Z. Wahrscheinlichkeitstheor. Verw. Geb.12, 87–97 (1969)
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Baxendale, P.H. Asymptotic behaviour of stochastic flows of diffeomorphisms: Two case studies. Probab. Th. Rel. Fields 73, 51–85 (1986). https://doi.org/10.1007/BF01845993
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01845993