Abstract
It is well known that every stochastic flow on Rd, whose one-point motion has an invariant measure m, gives rise to a measure-valued process (v t, t≥0) with v 0=m, which converges almost surely to a random measure ℝd, called the statistical equilibrium. We prove here that if the flow is spatially homogeneous and isotropic, and if either the covariance is smooth and the top Lyapounov exponent is strictly negative, or if the flow is “of coalescing type” (these phenomena can only occur when d≤3), then v ∞=0 a.s.
The first author thanks Université Paris VI for its hospitality, and the National Science Foundation (Grant DMS 8502802) and the CNRS for partial financial support for this work.
Produced by R.W.R. Darling on a Macintosh Plus, using the Maine fonts.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
P.H. Baxendale and T.E. Harris, Isotropic stochastic flows, Annals of Probability 14, 1155–1179 (1986).
R.W.R. Darling, Constructing nonhomeomorphic stochastic flows, Memoirs of the AMS 376, (1987)
R.W.R. Darling, Rate of growth of the coalescent set in a coalescing stochastic flow, Stochastics (1988).
Y.Le Jan, On isotropic Brownian motions, Z.f. Wahrsch. 70, 609–620 (1985).
Y.Le Jan, Equilibre statistique pour les produits de difféomorphismes aléatoires indépendants. Ann. Inst. H. Poincaré 23, 111–120 (1987).
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 1988 Springer-Verlag
About this chapter
Cite this chapter
Darling, R.W.R., Le Jan, Y. (1988). The statistical equilibrium of an isotropic stochastic flow with negative lyapounov exponents is trivial. In: Azéma, J., Yor, M., Meyer, P.A. (eds) Séminaire de Probabilités XXII. Lecture Notes in Mathematics, vol 1321. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0084135
Download citation
DOI: https://doi.org/10.1007/BFb0084135
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-19351-7
Online ISBN: 978-3-540-39228-6
eBook Packages: Springer Book Archive