Summary
We extend Furstenberg's theorem to the case of an i.i.d. random composition of incompressible diffeomorphisms of a compact manifold M. The original theorem applies to linear maps {X i} i ∈N on ℝm with determinant 1, and says that the highest Lyapunov exponent
is strictly positive unless there is a probability measure on the projective (m-1)-space which is a.s. invariant under the action of X i . Our extension refers to a probability measure on the projective bundle over M.
We show that when our diffeomorphism is the flow of a stochastic differential equation, the criterion for β>0 is ensured by a Lie algebra condition on the induced system on the principal bundle over M.
References
Baxendale, P.H.: Lyapunov exponents and relative entropy for stochastic flows of diffeomorphisms. Preprint. University of Aberdeen, Scotland (1986)
Carverhill, A.P.: Flows of stochastic dynamical systems: ergodic theory. Stochastics 14, 273–317 (1985)
Carverhill, A.P.: A formaula for the Lyapunov numbers of a stochastic flow. Application to a perturbation theorem. Stochastics 14, 209–226 (1985)
Carverhill, A.P., Elworthy, K.D.: Flows of stochastic dynamical systems: the functional analytic approach. Z. Wahrscheinlichkeitstheor. Verw. Geb. 65, 245–267 (1983)
Elworthy, K.D.: Stochastic differential equations on manifolds. LMS Lecture Notes Series 70, New York: Cambridge University Press 1982
Furstenberg, H.: Noncommuting random products. Trans. AMS 108, 377–428 (1963)
Furstenberg, H., Kifer, Y.: Random matrix products and measures on projective spaces. Isr. J. Math. 46, 12–32 (1983)
Ichihara, K., Kunita, H.: A classification of the second order degenerate elliptic operators and its probabilistic characterization. Z. Wahrscheinlichkeitstheor. Verw. Geb. 30, 235–254 (1974)
Kifer, Y.: Ergodic theory of random transformations. Boston: Birkhäuser 1986
Norris, J.: Simplified Malliavin calculus. To appear in Seminaire de Probabilities, Strasbourg
Oseledec, V.I.: A multiplicative ergodic theorem. Lyapunov characteristic numbers for dynamical systems. Trudy Moskov. Mat. Obsc. 19, 179–210 (1968)
Royer, G.: Croissance exponentielle de produits markoviens de matrices aléatoires. Ann. Inst. Henri Poincare 1, 49–62 (1980)
Ruelle, D.: Ergodic theory of differentiable dynamical systems. Publ. Math. IHES 50, 275–305 (1979)
Virtser, A.D.: On products of random matrices and operators. Theor. Probab. Appl. 24, 367–377 (1979)
Yosida, K.: Functional analysis, 3rd edn. Berlin, Heidelberg, New York: Springer 1971
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Carverhill, A. Furstenberg's theorem for nonlinear stochastic systems. Probab. Th. Rel. Fields 74, 529–534 (1987). https://doi.org/10.1007/BF00363514
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DOI: https://doi.org/10.1007/BF00363514
Keywords
- Differential Equation
- Manifold
- Stochastic Process
- Probability Measure
- Probability Theory