Zeitschrift fur

Z. Wahrscheinlichkeitstheorie verw. Gebiete 70, 609-620 (1985)

W a h r scheinlichkeit s t h e o r i e und verwandte Gebiete

9 Springer-Verlag1985

On Isotropic Brownian Motions Yves Le Jan Universit6 Paris VI, Laboratoire de Probabilit6s, Tour 56, 4 Place Jussieu, F-75230 Paris Cedex 05, France

Summary. This paper includes a detailed study of isotropic brownian motion on matrices and brownian flows associated with isotropic gaussian fields. Characteristic exponents, stability, asymptotic behaviour, statistical equilibrium are the topics on which the main results are obtained.

The purpose of this paper is to investigate the qualitative behaviour of flows defined by stochastical differential equations associated with isotropic homogeneous velocity fields. Such velocity fields have been considered in models of isotropic turbulence, and the associated flows can be viewed as rescaled turbulent flows (cf. [51, [71, [81, [14]). Our results include the computation of Lyapunov exponents and the proof of existence, in the unstable case, of a non trivial statistical equilibrium. Intuitively one may consider these flows as representing the motion of a continuous distribution of particles moving slowly under the action of a turbulent stream. The statistical equilibrium is a stationnary r a n d o m distribution of particles. The proof involves a detailed study of isotropic Brownian motion on matrices (in particular, the computation of all characteristic exponents), using elementary Ito calculus which is of independent interest. Flows are stable in dimension d = l and unstable for d > 4 . Two phases occur in dimensions 2 and 3. Partial results were already given by the author in [111, [12], for some specific flows. These results have been announced in a note ([13]). Thanks are expressed to S. Kotani and F. Ledrappier for helpful conversations. See [3] for general results on ergodic theory of stochastic flows. Other examples have just been given in [11 . After completing this work, the author received a preprint by Baxendale and Harris ([2]) in which other results of interest can be found for the same class of flows.

610

Y.LeJan

1. Isotropic Brownian Motion on Matrices

In the following, we denote: - M(d) the ring of (d, d) matrices. - GI(d) the group of invertible (d, d) matrices. - O(d) the group of orthogonal (d, d) matrices. 1.a. We say that a gaussian measure on M(d) is isotropic iff it is invariant under all automorphisms z v of M(d): " c v ( A ) = U - 1 A U , U~O(d). Any gaussian measure on M(d) is characterized by its covariance, which is given by a tensor: C )ik__ , - 5 A~Akd#(A) 9 M (d)

# is isotropic if and only if, for all U~O(d), c i Jkl ~i TTi, vf jf j, U k Uz! = Ck,i'j"r .

Therefore, the tensor C has to be of the form ij

ij

i j

i j

Ckl = a • ' (~k,l -l- b t~k t~l -t- C (~l t~k

(cf. [3], Theorem 16.2). Conversely, a tensor of this form is the covariance tensor of an isotropic gaussian measure on M(d) if and only if a + c , a - c and a + c + d b are non negative. Proof. l ( l ~ i i . 4 _ C ~ i ) _-l> (Cii jj.C , . j j + ( C )ij~ + C ~ji) a +_c - - 2 t ~ j j ,__

=88S

+

d IA) >=O.

Conversely, if these conditions are satisfied: i) The matrix K~ = b + (a + c)b~ is the covariance matrix of a gaussian vector Indeed, V 2 1 , 2 2 . . . 2 d, ( a + c ) ( ~ 2 ~ ) + b ( ~ , ~ i ) 2 is non negative if b > 0 and larger than ( a + c + b d ) ( ~ 2 2 ) if b=<0. ii) Let liE.. 1 and W..t,j~ 2. O<_i
= 1 ~ --~-CW . 1 . _ ] ~ V

= W~

2

J"

[/

-C

2

W . 2.

a,~

for i = j

is a ganssian random matrix of covariance C~.

for i > j

for i < j

On Isotropic Brownian Motions

611

1.b. Given such a covariance matrix, we can also construct a M(d) valued additive b r o w n i a n m o t i o n Wj(t) of covariance

e(N'(t) W?(s))=c~t A s The solutions of the linear Stratonovitch S.D.E. in M(d)

dXd(t)=xd(t)odW(t)

~dXg(t)=~2g(t)odlTV(t)

X a (0) = I

and (

X (0) = I

are by definition the right and left multiplicative b r o w n i a n motions associated with C}~. They are isotropic in the sense that their joint law is invariant under z vXa(t) and Xg(t) are a.s. Gl(d)-valued processes. (Xa(t)) -1 is the left b r o w n i a n m o t i o n associated with - W ( t ) , and Xa(t) is the left b r o w n i a n m o t i o n associated with IYg. Therefore the processes (Xe(t)) -a Xd(t) and xg(t) have identical laws. Moreover, Xd(t)=xg(t) a.s. iff ITff(t)= W(t), i.e. a = c , and Xa(t) is orthogonal iff - W ( t ) = W ( t ) , i.e. iff a + c = b = O . Finally, using time reversal, we check that for fixed t, Xa(t) and Xg(t) have the same law v r We have the corresponding results for Xg(t). v t is a convolution semi g r o u p on Gl(d) stable under the transformations z v (i.e. isotropic), under transposition and inversion. In the following, we shall work on the right b r o w n i a n m o t i o n xa(t) denoted X(t) for the sake of simplification. L e m m a 1. X (t) satisfies the following Ito equation:

d X (t) = X (t)dW (t) + 89 + b + d c ) X (t). Proof. dX~(t) = E X[(t)odWJ(t)= ~ X[dW} +89y" X~d ( W f WjZ)(t) l

1 ~

k, l

/ , X lid W jI + ~1VA.~ y~ ki ~ctk jl l

k, 1

=2 X[dWJ+ ~(a+b+dc)Xj. l

1.c. X(t) acts naturally on covariant tensors r .... i., giving 0i .... ~.(t)=

9~b. X j* r J1 . . . j n il

X4"~n" r

Jl ...in

is always a diffusion. In the following, we shall be mostly interested in the case where r is a n-covector V (1) A V (z) A ... A V ~"). W e shall show that in this case L o g nO(t)ll is a b r o w n i a n m o t i o n with drift, and c o m p u t e it explicitly. The set of n-covectors is denoted by E, and the space of skew symmetric (0, n) tensors by F,. A scalar p r o d u c t is defined on F, using the canonical basis. F o r n-covectors, we have: """

( U m A ... A Ut")[ V tl) A ... A V ~")) = det ((U(~ Vu))). Let V (1), ..., V (") be n covectors and set O = Vm A ... A V C"). T h e n we have dO(t) = E ( - 1)k+l eU) A V(t)(t) A ... A v(k)(t) A ... A V(n)(t)odVS k) denoting e u) the k,j

612

Y. L e J a n

canonical basis of the space of covectors. Therefore, V f f ) ( - 1 ) k + l e ~ A V(1)A ... A V(k) A ... A V(")odWJ

d0(t)= ~ k,j,l

= ~ e(,i) A i(e(O)O(t)odW j j,l

with the interior p r o d u c t notation. If we set, for any tensor O~F,, Z~0=e(J) A i(e(l))O, O(t) verifies the following Stratonovitch equation:

dO(t) = Z zlO(t)~

9

j,l

L e m m a 2. O(t) verifies the following Ito equation: 9

n

dO(t) = ~ rJlO(t)dWJ(t)+~(a+nb +(d - n + 1)c)O(t)dt. Pro@ d O ( t ) = ~ ~iO(t)dW} + ~ j,l

1,i,,.j,l,t~;,i ~k ~l Y~' ~ j , k"

j,l,i,k

The last term can be written 89

with

j,l

In general P 1 JO ZqT

=bqz;O--e(P) J p

A

e~

A

i(e(q))i(e(t))l/A

In particular J J

r j; ~lj O = ~ } - e ~ A "

"

e(;)A i(ea))i(e(j))O.

By linearity, we can restrict our attention to the case where 0 is a tensor of the form e (il) A e (i2) A... A e (i"). Then z ] 0 = ~ @ ( - 1)p+ 1 eU) A el il) A... A e (ip) A... A e (i") and one checks that: v= 1

Z j0=n0 J

and e(J) A e(Oi(e(O)i(eO))O = n ( n - 1)~. j,l n

Therefore c~= ~ (a + n b + (d - n + 1) c) 0. L e m m a 3. For any ~, O~Fn,

d( O(t)lO(t))=- ~ ((zJ ~(t)lO(tlS + (v]O(tllO(tlS)dW}(t). l,j

+ n((d -- n + 2)(a + c) + 2nb) ( ~(t)l O (t) ) dt.

On Isotropic Brownian Motions

613

Proof F r o m the preceding lemma, we have d( ~(t) lO(t)) = ~ ((z~(t) lO(t)) + (z]t)(t)l(o(t)))d W} (t) j, 1

+ n ( a + n b + ( d - n + l))(O(t)[O(t))dt + ~ (z~O(t)lzq4(t)) Clq ~Pdt. The last term can be written" (a(z~0lr~ qS) + b (z*~0Iz~ qS) + c(z~0Lz~0)) l,k

(we omit the t's for convenience). The second term of this sum equals n2(~,lq~). To compute the first term, take 0 = e ( ~ ) A . - - / ~ e (i~), with i s < i 2 . . . < i . , and ~b=e (3~ ~), with Jl
We compute the last term, in a similar way: ( z ~ 0 L z ~ b ) = l if l=k, iq=jq for all q, and ip=l for some p, and (z~OIz~r = 0 otherwise. It implies that ~ (ZkOlZl4)=n(r z k Assume now that 4 = 0 l,k

=V(~)A...AV ("). Denote by M(t) the martingale part of (~b(t)]qS(t)). Then (M,M)t=4n(a+c+bn)t. Indeed, using isotropy, we can assume that V (q) belongs to the subspace spanned by {e(~), ..., e(q)}, for all q
2 +b(z~O[O ) ( z I 0 [ 0 > + c ( z ~ 0 1 0 ) (v~0]0)) ,

k,l

the results follows. Applying Ito's formula we get the following:

Proposition 1. Log (110(t) ll) = t ~ (d - n)(a + c) + B t with

Bt =~[ ( ~~ ( t ) ] O ( t ) ) dWlk(t) and (B, B)t=n(a+c +nb ). Remark. ~ (d-n)(a+c) represents the sum of the n first Lyapunov exponents a i of Oseledetz ne{1, ..., d}.

theorem

(cf. r91). Therefore

for

all

2. Application to Stochastic Flows

2.a. Stationnary and Isotropic Gaussian Fields A gaussian random vector field Wi(x) on IRa is determined by a family of covariance matrices BiJ(x,y)=E(Wi(x)WJ(y)). A smooth version of the field

614

Y. LeJan

can be constructed if and only if B Ij is smooth in (x, y), which we shall assume in the following. Wi(x) is stationary if and only if B~J(x, y) depends only on x - y . It is isotropic if and only if, for any UeO(d), W(Ux) and UW(x) are identically distributed, i.e. if and only if for any z e R ~, U~U/Bkl(z)=BiJ(Uz). The general structure of such covariance matrices is known and has been used in the theory of turbulence (cf. [-14]). We assume d=>2. If FiJ(dk) denotes the Fourier transform of BZJ(z), it admits a representation of the following form:

Fia(dk) = uluJ o~(du)(FL - FN) (dp) + 6iJ cn(du) Fu(d p), ki

where p = ][kl[, u i =--, co(du) is the normalized surface measure on S e-l, and F N P and F L positive measures on ~ + with m o m e n t s of all order. Conversely, given two such measures, this formula gives the covariance of a smooth isotropic stationary gaussian field. We shall assume that F N and F L do not charge {0}. BiJ(z) can be written in the following form: i "

i "

Z~ Z J

B a(z)= (5 aBN(llzll) q - ~

(BL(llzll) -BN(IIzlI))

with

BL (r) = B1 l(r el) = S S cos (p u 1 r)u 2 oo(du)(FL (d p) - FN(d p) ) + ~ ~ cos (p u 1r)co(du)FN(dp) (e i denotes the canonical basis on Re) and BN(r ) = BZ2(r eO = j"j" cos (pu~ r)u2 co(du)(FL(dP) -Fu(dp) ) +~ ~ cos (pu 1 r)co(du)FN(dp). N o t e that B L and B L are positive and that"

Bs(r ) = o~--fir 2 + O(r 4) and BL(r) --BN(r) = --2 r2 + O(r 4) with

u oo(du)

+ 5p

F (dp)

2 fl = ~ u 2 u 2 ~o(du) ~ p2 (FL _ FN)(dp ) + ~ u~ oo(du) ~ FN(dp) 2 7 =~ (u~ -u2u2)o~(du) ~ p2(F L-FN)(dp). Elementary calculations using spherical coordinates show that:

5 u2o,(au)=

'

5 u u o,(du) - d ( d +12 )

and

3 5u~oo(dU)-d(d+2).

On Isotropic Brownian Motions

615

Therefore

1 FL(dp)+d_~ 1

d+l

2fi=~-+~A+~B with

and

A=~p2FL(dp)

and

1 y=~(A-B)

B=~[. p2FN(dp).

Note that A=2fi+(d+l)7 and B = 2 f l - 7 . Remark that c~, A and B can take arbitrary positive values. Note that simplified expressions can be given for B N and B L using Bessel functions (cf. [15]): Report the formulas cos (xuOco(du) =

F(d/2) Jd~2 (x)

and (d - 1 ) 5 cos(xuOu2co(du)=5 cos (XUl)(1 --U12) o)(du)

F(d/2)(d-21) =

Jd (x)

in the expressions of B N and B R. Asymptotic expressions can be obtained by integrating those of Bessel func1--d

tions. In particular B N and B R are O ( r ~ .

Wj(x)=~x Wi(x ). We have E(Wj(x)Wzk(y))= Cjilk+O(llx _yl12) with C)~ J i k i k =2fi6 i k 63z+y(6j8z+6z6j) and E(VVj(x)Wk(y))=O(I]x-yll). For any x d R a, Wj(x) is an isotropic gaussian matrix independent of Wi(x). The conditions B=2fl-y>=O, A=2fl+(d+l)y>_O, coincide with the con-

Define

ditions under which C~ is positive definite ( 1 - a ) .

2.b. Brownian flows Given any such gaussian vector field, we can associate with it a family Wi(x, dr) of white noises on ~ , such that: - For any fel~lR, dt), ~ f(t)Wi(x, dt) is a gaussian variable.

- E(~ f(t)Wi(x, dt) ~f(t) WJ(y, dr))=BiJ(x -y) ~f2(t)dt. Using a generalized Ito integral, this vector field-valued white noise can be integrated into a flow ~ of stochastic diffeomorphisms with independent increments, called Brownian flow, (cf. [10] and its bibliography)

Vs<=tr

= i w'(~s,~(x), s

du),

616

Y. LeJan

with

~,~=Id. If s<_t<_u, cbt,uO~s,,Oeb,, a.s. and ~, u and cb~ are independent. ~ t and q~-i have identical laws depending only on t - s . For each x, ebs,s+h(x)=B h is a Brownian motion. The jacobian matrices ,

,

,

s,t

Yj(x, s, t)-~x~ ~,~(x) satisfy the lto equation: __

i

t

V(~,s,t)=S ~k(x,s,u) w~'(#s,.(x), du). 0

Let x and s be fixed and set: s+t

wj(t)= ~ Ws(~s,,(x),du ' ) and

Yj(t)= yji(x,s,s-kt).

s

Writ) is an additive isotropic Brownian motion on matrices. The results of the first paragraph can be applied to W.i and Y/ Since we have dY.i= Y](t)dW.k(t), by Lemma 1, Y](t)=e-At/Zx~(t) where X~(t) is the left multiplicative Brownian motion solution of the associated Stratonovitch S.D.E." 9

.

J

.

J'

j

J

dX}(t)=XIj(t)od wki (t). 1

(2fi+(d+ 1 ) 7 ) = A = 0 if and only if FL=0 and also since A=d.~. E(Wi i, rW/), if r

and only if ~ - ~ 7 Wi(x) =0- Moreover, in that case, Proposition 1 shows that det (Yj) is then constant. Therefore, the flow preserves Lebesgue measure if and only if FL=O. From the results of the first paragraph, it appears that the sum of the n largest Lyapunov exponents of the flow equals:

~r,=~(d-n)(2fl+ 7)-

A

n(d-n) ( 2 d + d B ]

n

The nTM Lyapunov exponent equals:

75 !-A/2=2(U+~ ((d-4n)A +d(d-2n+ l)B). More precisely, Proposition 1 implies the following. Corollary 1. For any V (1) ... V(")e~a, n<=d, if V/~

J Log [[V~ ) A v(2) ^ v") II=t~ 9 (t) ..... A --(t)H

YS(t)Vff)

On Isotropic Brownian Motions

617

B t being a Brownian motion of quadratic variation c,t, with c , = n ( 2 f l + ( n + l ) 7 ) :

Cn

It follows that NV(1)(t) A .../X V l ( t ) = e (an+--2) t Mr, where

n((n+2)A+(d-n)B) d+2

M t

is an exponential martingale. N o t e that a , + c, 2 - 2 n( d( d+ -2n~) (A+(d+I)B). In particular det (Yj(t)) is a m a r -

tingale (with respect to the natural filtration ~tt (~)= a{ W(x, du), u e [st + s]}. It degenerates into 1 when A = 0 . Since M, depends m e a s u r a b l y on x, we obtain: Corollary 2. I f V, is a n dimensionnal compact submanifold of IR a, denote IVy[ the

total mass of the induced riemannian measure. --

o"

co) Ir

+ - - t

is a ~t(~)-martingale.

e ( " 2

3. Statistical Equilibrium 3.a. Two Points Diffusion Given any pair of distinct points x , y ~ N d, and stiR, the process (x(t),y(t)) =(4~s,s+t(x), ~bs,s+t(y))sN d x 1t d - { d i a g o n a l } with generator:

2.

r '

[~2 F

~2 F \

~Y22 -t-~"2

..

~2f

i,j

oxioy~ !

An elementary calculation shows that: IIx(t)-y(t)ll is also a diffusion process on N + - {0} with generator: Ag (r) = (c~- n t (r)) g" (r) + (d - 1) (~ - BN (r)) g' (r). r

This diffusion on IR + admits an invariant m e a s u r e O(r)dr and its semi group Qt is symmetric with respect to this measure. ~b(r) is determined, up to a multiplicative constant, by the identity: B~(r) + (d - 1)(~ --BN(r)r -1

~b'

(r)-If we take L o g q S ( r ) = L o g

e_BL(~

a --SL(r ) +(d-1)Logr-(d-1)

r BL(X)--BN(X)x(e_BL(X)) dx

(the integrand is smaller than 0(x-312)) qS(r)~r d-1 as r ~ o o and qS(r)~ Cr u as 9 r ' (d-1)fi 24 ( d - 1 ) ( A + ( d + l ) B ) r ~ 0 , with # = l r l m o @ ( r ) = - 2 4 fl+~3A+(d-1)S . Since / x + l =

(d-4)A+d(d-1)B 3A+(d-1)B

-

d)o1 , ~b is integrable on [0,1-1 if and only if 2 1 > 0 -2 d

(since 2 d is negative). Moreover, the scale measure of dr= ][x(t)-y(t)H, denoted s'(r)dr equals: e -(d-l) ! ~dXdrC~-BN(x)

618

Y. LeJan

s'(r)~cr -u-2 as r--,0 and s'(r)~r 1-d as r ~ o o . Therefore, s([1 oo]) is finite for d_> 3 and infinite for d = 1 or 2. s([0 1]) is finite if and only if/~+ 1 < 0 i.e. 21 <0. Therefore (cf. [5], VI-3) for d > 3 d t ~ + oo a.s. if 2a __>0. If 21 <0, dt-*O or + oo and Pl(dt~oo)= s((O 1]) For d = l d t ~ 0 a.s. For d = 2 , d ~ 0 a.s. if "~1<0. s((0+ 0o])' 1

It is recurrent for 21 > 0 but d t ~ o o in probability for 21 > 0 (since 5 Off)dr< oo o

O(r)dr = + oo).

and 0

We shall assume the unstability condition 2, > 0 in the followin G. It is verified for d > 5 and not verified for d = l . In dimension 2(3) it is verified for A < B (A <6B). In dimension 4, it is verified except for the case B = 0 in which 2, =0. Set O(r)=rl-e~(r). O ( r ) ~ l as r--*oo and O(r)~cr u+l-d as r--*0. Let Pt(2) be the semi-group associated with (x(t), y(t)). Lemma. For any bounded function with compact support g,

~ Pt(Z)(g|

~ ~ ~ g(x)g(y)tp([Ix -y]])dxdy.

Note that P~(2)~(x, y)=Q,@(]lx-y]] ) (with ~(x, y)=O(llx-y][)). Since ~ + 1 - d

-d(21+2e)-d(2d+2)A
Qt(~)~last~oo.

2d 2a Moreover it follows from isotropy considerations (or from a direct calculation showing thet e 8@ O(]lx -YI[) + ~. BjJ(x -y)O(][x -YII) = 0, (cf. [5] Chap. V J

3-4), that Pt(2) is symmetric with respect to the measure O(]lx-yN)dxdy. Then

(ld)(llx-Yll)O(llx-Yll)dxdy

f SdxdyP,(~'(g|

which converges

towards the limit.

3.b. Statistical Equilibrium Let g be a bounded function with compact support in IRd. s being fixed in R, set

M[S~g = S g ( < L t ( x ) ) d x =5 g(x) det Y (x, s, t + s)dx. .Mff)g is a martingale. This in general follows from the fact that the flow is symmetric, has independent increments and that g((b2,~+t) is integrated by the invariant measure of the associated Markov process:

V u > O, E(M (~) ~,1 ~t (s)],' = Err g(4)~-Jt~ s~4)7+1t+, s+,(x)dxl ~(s)) \ t+uO \d = Y eu(g~W,,s)(x)d~ = M[S~g (Pu denotes the semi group of the one point process).

On Isotropic Brownian Motions

619

The martingale M}~)g always converges since it in the difference of two positive martingales, and its limit define a random measure IX+(dx) such that E(IX+ (dx))
0(LIx-yll)dxdy. Thus we have the following:

Proposition 2. If 21 >0, there exists a random measure IX+ on IRd, such that for any bounded function g with compact support, g(x)ix+ (dx)= lim ~ g(eb[,,J+,(x))dx t ~ o~

a.s. and in I~. E(ix+ (dx))=dx and

e(ix~ (dx) ix~ (dy)) = ~,(llx -yll)dxdy. s i~--1 s+t We have the chain rule Ix+=-s,~+tix+ 9 Since the flow is stationary, all these random measures have the same law. Using the backward flow ~bs_,,~, instead of ~b~,s+t we can define in the same way a measure Ix~_-independent of IX%, but identical in law. It describes the "statistical equilibrium" at time s associated with the flow. If A = 0 , #~+(dx)=dx since the flow preserves Lebesgue measure.

Proposition 3. I f A is non zero, IX~ is

a.s. singular.

Proof. We know that det(Y(t)) tends to 0 as t tends to + oe by Corollary 1. Set h~(x)=limsupe-dix+(B(x,e)), h~(x) is a version of dix+(x) ~o dx . Let v be the law of hS(x), which is clearly independent of (x,s). By the chain rule, if kS(y) =hs+~(~bs,~+,(y)) d e t ( ~ , s + t ( y ) ) , k~(y)=hS(y) a.e. P a.s. and the law of kS(y) (which is clearly independent of y) has to be v. For any r/> 0, choose M such that v ( [ 0 M ) ) > l - t / and t such that P(det Y(t) < tl/M ) > 1 -tl. Then

v((0 ~]) = P(kS(y)__< ~) P({h

(q~s,s+,(y))
_-> p ( hs+t (~b~,~+z(y)) 1-2t/(q)s,s+ t being independent of h s+~) and since t/ is arbitrary, v=5 o.

Remark. The Hausdorf dimension of g% is a.s. larger than g + 1 < d (cf. [1]).

620

Y. LeJan

References 1. Baxendale, P.: Asymptotic behaviour of stochastic flows of diffeomorphisms. Two case studies. Preprint. Aberdeen University. 2. Baxendale, P., Harris, T.E.: Isotropic stochastic flows. Preprint. 3. Carverhill, A.R: Flows of stochastic dynamical systems: ergodic theory. Preprint. Warwick University. 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, 232254 (1981) 6. Ikeda, N., Watanabe, S.: Stochastic differential equations and diffusion processes. Amsterdam Oxford New York Tokyo: North Holland/Kodansha 1981 7. Kesten, H., Papanicolaou, G.: A limit theorem for turbulent diffusion. Commun. Math. Phys. 65, 97-128 (1979) 8. Kunita, H.: Convergence of stochastic flows connected with stochastic ordinary differential equations. Preprint 9. Ledrappier, F.: Cours /t l'Ecole d'Et6 de Saint-Flour. Lecture Notes, n~ 1982. Berlin Heidelberg New York: Springer 1985 lo. LeJan, Y., Watanabe, S.: Stochastic flows of diffeormorphisms. Proceedings of the Taniguchi Symposium 1982. Amsterdam New York Oxford: North-Holland 1984 11. LeJan, 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. LeJan, Y.: Equilibre et exposants de Lyapunov de certains flots browniens. Comptes Rendus Acad. Sci. Paris, t. 398, S6rie I., 361-364 (1984). 13. LeJan, Y.: Exposants de Lyapunov des mouvements browniens isotropes. Comptes Rendus. Acad. Sci. Paris t. 299, Serie 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

Received October 15, 1984; in revised form April 28, 1985