# zbMATH — the first resource for mathematics

On isotropic Brownian motions. (English) Zbl 0576.60072
A Gaussian measure $$\mu$$ on the vector space M(d) of real $$d\times d$$ matrices is called isotropic if it is invariant under all automorphisms $$\tau_ u: M(d)\to M(d)$$, $$A\mapsto U^{-1}AU$$, where $$U\in O(d)$$, the group of orthogonal matrices. $$\mu$$ is characterized by its covariance C. Let W(t) be an M(d) valued (”additive”) Brownian motion with covariance C(t$$\wedge s)$$. The solution of the Stratonovich SDE $$dX(t)=X(t)\circ dW(t)$$ is a (right multiplicative) Brownian motion in Gl(d), whose law is isotropic again. The Lyapunov exponents of X(t) can be calculated in terms of the covariance C, using elementary Itô calculus.
Consider now a stationary Gaussian (vector) field V(x) on $${\mathbb{R}}^ d$$. There is an associated family of white noises W(x,dt), which can be integrated into a stochastic flow $$\phi_ t$$ of diffeomorphisms. If V(x) is isotropic (i.e., V(Ux) and UV(x) have the same distribution for any $$U\in O(d))$$ $$\phi_ t$$ is called Brownian flow, its derivative flow is a multiplicative Brownian motion. The Lyapunov exponents $$\alpha_ 1\geq...\geq \alpha_ d$$ can be expressed in terms of the covariance of V(x), they are equidistant and $$\alpha_ 1<0$$ if $$d=1$$, $$\alpha_ 1\geq 0$$ if $$d\geq 4$$ and $$\alpha_ 1>0$$ if $$d\geq 5$$. For $$d=2,3$$ the sign of $$\alpha_ 1$$ depends on V(x).
Finally, assuming instability (i.e., $$\alpha_ 1>0)$$, weak convergence of $$\phi_ t^{-1}(dx)$$ to a (random) distribution on $${\mathbb{R}}^ d$$ is proved, where dx is Lebesgue measure on $${\mathbb{R}}^ d$$. This stationary distribution, the statistical equilibrium, is singular with respect to dx iff the flow does not preserve dx.
The paper is not self contained.
Reviewer: H.Crauel

##### MSC:
 60J65 Brownian motion 60G15 Gaussian processes 58J65 Diffusion processes and stochastic analysis on manifolds 60J99 Markov processes 60B15 Probability measures on groups or semigroups, Fourier transforms, factorization
Full Text:
##### References:
 [1] Baxendale, P.: Asymptotic behaviour of stochastic flows of diffeomorphisms. Two case studies. Preprint. Aberdeen University. · Zbl 0581.60046 [2] Baxendale, P., Harris, T.E.: Isotropic stochastic flows. Preprint. · Zbl 0606.60014 [3] Carverhill, A.P.: Flows of stochastic dynamical systems: ergodic theory. Preprint. Warwick University. · Zbl 0536.58019 [4] Gurevitch, G.B.: Foundations of the theory of algebraic invariants. Groningen: Noordhoff 1964 [5] Harris, T.E.: Brownian motions on the homeomorphism of the plane. Ann. Probab. 9, 232-254 (1981) · Zbl 0457.60013 · doi:10.1214/aop/1176994465 [6] Ikeda, N., Watanabe, S.: Stochastic differential equations and diffusion processes. Amsterdam Oxford New York Tokyo: North Holland/Kodansha 1981 · Zbl 0495.60005 [7] Kesten, H., Papanicolaou, G.: A limit theorem for turbulent diffusion. Commun. Math. Phys. 65, 97-128 (1979) · Zbl 0399.60049 · doi:10.1007/BF01225144 [8] Kunita, H.: Convergence of stochastic flows connected with stochastic ordinary differential equations. Preprint · Zbl 0612.60051 [9] Ledrappier, F.: Cours à l’Ecole d’Eté de Saint-Flour. Lecture Notes, n? 1097 1982. Berlin Heidelberg New York: Springer 1985 [10] Le Jan, Y., Watanabe, S.: Stochastic flows of diffeormorphisms. Proceedings of the Taniguchi Symposium 1982. Amsterdam New York Oxford: North-Holland 1984 [11] Le Jan, Y.: Equilibrium state for turbulent flows of diffusion. Proceedings ?Stochastic processes and infinite dimensional Analysis?. Bielefeld 1983. [To appear in Pitman, Lecture Notes, London] [12] Le Jan, Y.: Equilibre et exposants de Lyapunov de certains flots browniens. Comptes Rendus Acad. Sci. Paris, t. 398, Série I., 361-364 (1984). [13] Le Jan, Y.: Exposants de Lyapunov des mouvements browniens isotropes. Comptes Rendus. Acad. Sci. Paris t. 299, Série I, 947-949 (1984) [14] Monin, A.S., Yaglow, A.M.: Statistical Fluid mechanics (Vol. 2). Cambridge: MIT Press 1975 [15] Yadrenko, M.I.: Spectral theory of random fields. New York: Software Inc. 1983 · Zbl 0539.60048
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.