Download RIGOROUS BOUNDS ON THE FAST DYNAMO GROWTH RATE

RIGOROUS BOUNDS ON THE FAST DYNAMO
GROWTH RATE INVOLVING TOPOLOGICAL
ENTROPY
I. Klapper
Program in Applied Mathematics
University of Arizona
Tucson, AZ 85721
L.S. Youngy
Department of Mathematics
University of California
Los Angeles, CA 90024
Abstract
The fast dynamo growth rate for a C k+1 map or ow is bounded above by topological entropy plus a 1=k correction. The proof uses techniques of random maps
combined with a result of Yomdin relating curve growth to topological entropy. This
upper bound implies the following anti-dynamo theorem: in C 1 systems fast dynamo
action is not possible without the presence of chaos. In addition topological entropy is
used to construct a lower bound for the fast dynamo growth rate in the case Rm = 1.
This author is supported by an NSF postdoctoral fellowship.
y This
author is partially supported by an NSF grant.
1
1 The Kinematic Fast Dynamo Problem
Magnetic dynamo theory involves the study of the generation of magnetic eld in astrophysical objects such as planets and stars. One star of particular interest, of course, is the sun,
which exhibits vigorous magnetic eld activity on time scales much shorter than the magnetic diusive time. Kinematic fast dynamo theory attempts to gain some understanding of
the non-diusive processes that might be involved by addressing the question of what sort
of uid motions can induce exponential growth of magnetic eld at high magnetic Reynolds
number. This is one of a large class of singular problems with important physical implications
for which there is a need for a better understanding of the limiting behavior of complicated
processes. Natural questions that arise are: what if any relation holds between the limiting
and singular limit solutions, and what information about the limiting process can be gained
from the singular limit problem? In this paper we consider a conjectured growth rate bound
of Finn and Ott (1988) based on stretching properties of the uid ow and prove a slightly
generalized version of it.
The equations of dynamo theory are those of incompressible MHD (see Roberts (1967)).
Denoting the magnetic eld by B and the uid velocity by u the magnetic induction equation
@ B + u B = B u + 1 2B
B=0
(1)
@t
Rm
can be derived from Ohm's law and the Maxwell equations (using the standard MHD approximation of neglecting the displacement current @E=@t). Rm, the magnetic Reynolds
number, is a dimensionless parameter measuring the relative strength of advective to diusive processes. Requiring u = 0 and setting the uid density = 1 the uid momentum
equation is
@ u + u u = p + 1 ( B) B + 1 2u
(2)
@t
M
R
r
r
r
r
r
r
;r
A
r
r
where MA , the Alfven Mach number, is a dimensionless parameter measuring the typical
balance between uid and magnetic energy and R, the uid Reynolds number, is a dimensionless parameter measuring the typical balance between uid advection and diusion.
Equations (1) and (2) are supplemented by the appropriate boundary and initial conditions.
We will make the kinematic dynamo approximation of setting MA = (i.e., the magnetic
energy is assumed small). As the Lorentz force (
B) B is quadratic in B this may
perhaps be justied as a linearization of (2) around B = 0. Then the uid momentum
equation (2) is decoupled from the magnetic eld and so the uid velocity may be considered
to be prescribed independently of B (thus the term kinematic). Equation (1) is now a linear
equation for B. The object of kinematic dynamo theory is to nd linear dynamo instabilities,
that is, to determine for a given Rm which if any choices of u result in exponential growth
of B for some initial conditions B0. For our purposes, given u(x t) we dene kinematic
1
r 2
dynamo action to occur when
Z
1
p(Rm ) Tlim
(3)
!1 T ln V B(x T Rm) dx > 0
where V is the region containing magnetic eld.
This paper addresses the kinematic fast dynamo problem (Vainshtein and Zel'dovich
(1972)). A velocity eld is dened to be a kinematic fast dynamo if
j
j
lim p(Rm ) > 0:
(4)
Rm !1
(We assume that this limit exists in principle one might instead use lim inf or lim sup.) Motivation for this denition comes from the solar magnetic eld which shows activity (e.g. the
22 year solar cycle) on time scales much shorter than the diusive time scale ( 1010 years).
Hence the solar dynamo apparently functions in some sense independently of diusion. The
object of kinematic fast dynamo theory is to characterize velocity elds that support dynamo
action without direct use of diusion. Denition (4) states roughly that dynamo activity
should not cease in the limit of the diusive term becoming small compared to the advective
one.
2 Statement of Result and Discussion
Finn and Ott (1988) proposed that the kinematic fast dynamo growth rate of a steady or
periodic ow is bounded above by the topological entropy of the ow. A proof in the C 1
case was announced by Vishik (1992). In this paper we prove a more general result using in
a simple way standard techniques from dynamical systems.
Theorem 1 Let u and B0 be divergence free vector elds supported on a compact domain
Rd . We assume that u is of class C k+1 and B0 is C k for some k 2. Let f be the time
1 map of the ow generated by u, and let B(x t Rm ) be the solution of equation (1) with
initial conditions B0(x). Then
Z
1
lim sup lim sup n ln B(x n Rm) dx h(f ) + r(kf )
(5)
V
Rm !1 n!1
where h(f ) is the topological entropy of f (restricted to ) and
1
r(f ) nlim
Df n (x) :
!1 n ln max
x2D
This upper bound is also valid for Rm = .
D j
j
D
k
1
3
k
None of the arguments to be presented depend on dimension (of course the higher dimensiona are not physically relevant). We also note that a number of conditions in Theorem 1
are easily relaxed or modied. For example, u may be time periodic, and Rd can be replaced
by the d-dimensional torus (in which case, additionally, B0 need only be continuous), etc.
See section 7.
We state separately a version of Theorem 1 for maps, since maps have become popular
models for fast dynamos (e.g. Bayly and Childress (1988), Finn and Ott (1988), Gilbert
(1993),
(1994)). Let f be a dieomorphism of Rd and dene a velocity eld uf =
P1n=;1 (Soward
(t n)f ). Then at Rm = equation (1) has the formal solution
;
1
B(x n + 1) = (fB)(x n)
= f (f ;1(x)) B(f ;1(x) n):
r
This is the Cauchy solution, or in the language of dynamical systems, the pushforward of
B. Now let 0 be a family of probability densities on Rd with tending weakly to
0 = (0), the delta function at 0, as 0.
Theorem 10 Let f be a C k volume preserving dieomorphism of Rd with f = Id outside
a compact domain , and let B0 be as in Theorem 1. For n = 1 2 : : :, we de ne Bn+1
B(x n + 1 Rm ) to be the convolution of fBn with R;m1 . Then the conclusions of Theorem
1 hold.
f
g
!
D
The case of particular interest here is when is a d-dimensional Gaussian density with
isotropic variance . Then Bn is the formal solution to (1) with u uf as above.
Certain remarks are in order here. First, positive topological entropy implies chaos in a
topological sense - this will be further discussed in the next section. Thus Theorems 1 and
10 are essentially anti-dynamo theorems if u(x) (or f (x)) is C 1 and not chaotic then u(x)
(or f (x)) is not a fast dynamo. Second, in the case k = the upper bound can be achieved
by Anosov systems (Arnol'd et al.(1981), Bayly (1986), Vishik (1989), Collet (1992), Gilbert
(1993), Oseledets (1993)). The two well known fast dynamo ows of Soward (1987) and
Gilbert (1988) are not C k , k > 2 and are not covered by Theorem 1 singularities in the eld
of Jacobian matrices are needed in an essential way. (In such cases the methods of this paper
cannot be applied.) Lastly, in addition to the result of Vishik mentioned at the beginning
of the section, we mention that Oseledets (1993) has derived an exact growth rate for C 1
maps on the 2 torus. His methods rely on special properties of 2-dimensional magnetic elds
(Zel'dovich (1957)).
The proofs of Theorems 1 and 10 are based on three basic ideas. First, magnetic eld lines
at Rm = are material curves, and the magnetic eld grows because of stretching of these
material curves. Thus the eld growth rate is bounded above by the material curve growth
rate, at least for Rm = . Here we emphasize that we refer to the growth rate of curves
of nite extent. As noted by Finn and Ott (1988) innitesimal line elements, i.e., tangent
1
1
1
4
vectors, grow more slowly in general, and their growth rate does not necessarily bound the
magnetic eld growth rate. The second idea, due to Yomdin (1987), is that when a map is
iterated the growth rate of a material curve is bounded above by topological entropy plus a
possible correction due to a lack of smoothness. This bound can be understood heuristically
by discretizing a section of curve with a large number of points. The growth rate of the
curve is approximated by the growth rate of the number of points needed to resolve the
curve to a given tolerance, which in turn is, roughly speaking, bounded by the topological
entropy (see section 3). But also, between the ne discretization, lack of smoothness in the
map or ow can cause the curve to locally \crinkle" in such a way as to increase the growth
rate by as much as r=k. The third basic idea is to exploit the fact that diusion of the
magnetic eld can be replaced by adding noise to the underlying dynamical system. This
allows us to consider random maps and to use a randomized version of Yomdin's arguments.
We show that a small amount of noise does not increase the topological entropy. Thus, in
the limit, the same upper bound on magnetic eld growth holds. (The introduction of noise
as an analytical tool for studying fast dynamos can be found in Molchanov et al. (1985) and
Klapper (1993) it has also found use as a numerical method (Klapper (1992)).)
To further highlight the relationship between topological entropy, the growth rate of
material curves, and the fast dynamo problem, we prove the following lower bound for
magnetic eld growth at Rm = :
Theorem 2 Let be a compact domain in Rd and let f :
be a C 1 ( > 0) volume
preserving dieomorphism of . Then q d 1 ( q d 2 if f is the time 1 map of a
ow) we have
1 ln Z B(x n ) qdx h(f )
sup lim
inf
B0 n!1 n D
where h(f ) is the topological entropy of f and the supremum is taken over all C 1 divergence
free vector elds B0 with compact support.
We remark again that ows are considered steady a periodic ow in dimension d can be
regarded as a steady ow in dimension d + 1.
The organization of the paper is as follows. Section 3 provides a brief introduction to
the relevant facts of topological entropy. In section 4 we prove Theorems 1 and 10 in the
special case that Rm = . While not of direct physical signicance this case provides an
opportunity to emphasize the rst two basic ideas of the proof. Section 5 extends the results
of section 4 for maps from Rm = to Rm
completing the proof of Theorem 10. Section
6 is concerned with those technical details necessary to extend our methods from maps to
ows and thus nish the proof of Theorem 1. Section 7 contains remarks concerning the
range of applications of the presented results to fast dynamos and related problems, and
section 8, which is independent of the rest of the paper, contains a proof of Theorem 2, the
Rm = lower bound.
1
D
D ! D
D
8
;
j
1 j
1
1
8
! 1
1
5
;
The authors thank J. Watkins for conversations related to section 6 and A. Gilbert for
helpful comments.
3 Topological Entropy
For the convenience of readers not familiar with this subject we present here two equivalent
denitions of topological entropy. For further details and background on this section see
Walters (1982). The rst denition, due to Adler et al. (1965), is based on open covers
(hence the adjective topological). Let be a compact topological space, and let be an
open cover of . Dene H (), the topological entropy of , by H () ln N () where N ()
is the smallest cardinality of any nite subcover of . Given f :
a continuous map,
h(f ), the entropy of f relative to , is dened to be
;1 ;i
1 H (n_
h(f ) = nlim
!1 n i=0 f ):
Here f ;i is the
open cover consisting of those sets f ;1A such that A is an element of ,
W
and theTjoin of two open covers and is the open cover consisting of all sets of the
form A B with A , B . Finally, the topological entropy of the map f is dened by
h(f ) = sup h(f ):
(6)
D
D
D ! D
2
2
The signicance of positive topological entropy can be intuitively understood in the
following manner.
Consider a cover with a nite
numberT of elements and let be an
W
T
n
;
1
;
i
;
1
element of i=0 f . Then is of the form Ai1 f Ai2 : : : f ;(n;1) Ain;1 for some Aij ,
i.e., x
i x Ai1 , f (x) Ai2 : : : f n;1 (x) Ain;1 . We think of the trajectories of x
and y as indistinguishable
with respect to if f i(x) and f i(y) fall in the same element of W
n
;
1
;
i. The number N ( 0 f i) then measures the number of trajectories of length n that are
pairwise distinguishable with respect to the labeling system given by , and positive entropy
tells us that this number grows exponentially with n. We take this to be an indicator of
chaos. Positive topological entropy in dierentiable dynamical systems is often detected
through the presence of horseshoes (e.g. using Melnikov methods). In dimension 2, it has
in fact been shown that essentially all of the entropy of a system is carried by horseshoes
(Katok (1980)).
Next we give a second denition of topological entropy due to Bowen (1971) for a continuous map f :
where ( d) is a metric space. We again assume that is compact
(although this denition can be adapted to non-compact situations). First we dene a new
metric dn dependent on f by
i
i
dn (x y) = 0max
(7)
in d(f (x) f (y)):
O
O
2 O
2
2
2
2
8
D ! D
D
D
6
Now, given > 0 and an integer n 0 we say that a set S
(n )-spans if y
x S such that dn (x y) . In other words for any y there is some x S such that
the trajectories of x and y over n iterations of f are indistinguishable to resolution . Let
n () be the smallest possible cardinality for any (n ) spanning set. Then n () measures
the number of dierent trajectories of length n to resolution . In terms of the metric dn,
n () is the minimum number of balls b(x n ) of radius dened by
b(x n ) = y : dn (x y) < needed to cover . We now dene the topological entropy to be
1 ln ():
h(f ) = lim
lim
sup
(8)
!0 n!1 n n
In the previous language, this is the growth rate in n of the number of distinct trajectories
measured at ner and ner resolution . For compact spaces it is easily shown that for any
metric that induces a given topology denition (8) is independent of metric and equivalent
to the open cover denition.
We now state without proof (see Walters (1982)) some properties of topological entropy
that will be of use later:
Property 1 h(f m ) = mh(f ).
The quantity diam(), the diameter of , is dened to be the supremum of the diameters
of the elements of .
Property 2 If n is a sequence of open covers of with diam(n) 0 as n
,then
h(f ) = nlim
!1 h(f n ):
9
2
D
D
2 D
f
8
2 D
2
g
D
D
Property 3
!
! 1
n () card (Wni=0 f ;i)
where diam() < .
The notion of entropy generalizes to random maps. Let C ( ) be the set of continuous
maps from to and let be a probability measure on C ( ). Dene g(n) = gn : : : g1 g0
where g0 = Id and gi C ( ) (and dene g;(n) = g1;1 g2;1 : : : gn;1). We note
that the denitions (6) and (8) generalize to sequences ! = g0 g1 g2 : : : chosen iid with
distribution where f i is replaced by g(i) and n is calculated using balls b(x n ) dened
by
b(x n ) = y : d(g(i)(x) g(i)(y)) < 0 i n :
(9)
D D
D
D
D D
2
D D
f
f
g
7
g
In particular the limits exist and are nonrandom (i.e., independent of !) (Kifer (1986)).
When working with random maps of Rd problems could arise with the use of topological
entropy since Rd is not compact. Note however that Bowen's denition continues to make
sense if we limit our attention to orbits that originate from a compact set . When considering (non-random) maps f dened on all of Rd but with f (x) = x x , we denote the
topological entropy of f restricted to by h(f ).
D
8
D
62 D
jD
4 Growth of Magnetic Field at Rm = 1
We begin with the Rm = case of Theorem 1 (which as will become evident is identical
to the Rm = case of Theorem 10). Let f be the time 1 map of the ow u, and let the
initial conditions B0 be a C k vector eld supported on . When Rm = the solution to
equation (1) is given by
B(x n) = J(f ;n (x) n)B0(f ;n (x))
the Cauchy solution, where J(f ;n (x) n) is the Jacobian matrix of the time n map of the
ow evaluated along the trajectory of the point f ;n (x). We will abbreviate J( n) by Df n ( )
(and in general Dl denotes the tensor of `th order derivatives). Since for Rm = the
magnetic eld is a material vector eld, it is reasonable to suggest that the eld growth rate
p(Rm = ) is related to the growth rate of material curves. This is the subject of Claim 1.
Dene a C k curve in to be a C k map : 0 1]
and let k max1lk Dl .
We will often confuse with its image and use `() to denote the length of the image.
1
1
D
1
1
1
D
Claim 1 If Rm =
! D
k k
then for every C k initial condition B0,
1 ln Z jB(x n)jdx lim sup 1 lnsup `(f n )]
lim
sup
n!1 n
n!1 n
D
where the supremum is taken over C k curves : 0 1] ! D with kkk 1.
k
k
1
(10)
Proof: Using the Cauchy solution this is equivalent to showing that
Z n
1
1 lnsup `(f n )]:
lim
sup
ln
jD f B0 (x)jdx lim sup
n!1 n
n!1 n
D
Choose a C k vector eld B0 with support contained in D. The natural way to demonstrate
this inequality would be as follows: divide D into a number of ux tubes (or ow boxes) of
B0 and track the stretching rate of these tubes. Having done this we can bound the growth
rate of the magnetic eld by the maximum stretching rate of a ux tube, which is in turn
bounded by the maximum growth rate of a material curve. Unfortunately, this argument
is problematic because dividing B0 into ux tubes becomes dicult near null points of B0.
Instead we will use the following trick to accomplish much the same plan: Let X be an
8
arbitrary constant vector eld on Rd with X 100 max B0(x) . Then neither X nor
B0 X has null points and
Z n
Z n
Z n
1
1
Df X(x) dx + Df (B0 X)(x) dx
lim sup ln Df B0(x) dx
lim sup ln
n!1 n
n!1 n
D Z
D
D
1
= max lim
sup ln Df n X(x) dx
n!1 n
ZD n
1
lim
sup ln D Df (B0 X)(x) dx :
n!1 n
In fact, both X and B0 X point so uniformly in one direction that we may think of the
active region as being contained in a single ux tube. More precisely, let B~ 0 = X or
B0 X, and let P be any hyperplane roughly perpendicular to X. For x P , let x be
the integral curve of B~ 0 with x(0) = x. Choose r0 large enough so that is contained in
; y = x (s) for some x P and s r0 . Then
1 ln Z Df n B~ (x) dx
1 ln Z Df n B~ (x) dx
lim
sup
lim
sup
0
0
n!1 n
n!1 n
D
;
Z
1 ln
= lim
sup
T `(f nx )dF (x)
n!1 n
j
j j
j
j
j
;
j
j
j
j
j
j
;
;
j
j
;
D
;
2
D
f
2
j j j
j
g
j
=;
j
!
P
1 ln sup `(f n )
lim
sup
x
n!1 n
x2
where dF is the ux (or transverse measure) of B~ 0 across and x is understood to be
dened on r0 r0]. We here have used the fact that B~ 0 is divergence free and thus the total
ux is independent of the choice of cross-section.
To nish it remains to observe that the right side of inequality (10) remains unchanged if
the supremum is taken over curves dened on intervals of length C and with k C
for a xed C . This is Strue Sbecause by subdividing and reparametrizing, each can be
decomposed into = 1 : : : p where each i is normalized according to the requirements
of claim 1.
Next we relate the growth rate of material curves to topological entropy following Yomdin.
;
Claim 2
k k
1 ln(sup `(f n)) h(f ) + r(f )
lim
sup
k
n!1 n
where is as in Claim 1 and r is de ned as in the statement of Theorem 1.
Proof: This is a slightly stronger statement than that made by Yomdin (1987) but it follows
immediately from Yomdin's proof. The idea is as follows: let
b(x n ) = y : d(f i (x) f i(y)) < 0 i n
f
9
g
so that n () is the smallest number of sets of this form needed to cover . Given a curve ,
let
h \
i
V (f ) = sup ` f n b(x n ) :
x2D
Then
`(f n) n() V (f )
In other words we distinguish between 2 types of growth. The growth rate of n (), i.e.,
topological entropy, bounds the growth rate of `(f n) as detected up to resolution (uniformly
in ), whereas local crinkling - crinkling that occurs in smaller scales but which can in
principle pile up exponentially fast - is measured by the growth rate of V (f ). Yomdin
proved that
1 lnsup V (f )] r(f )
lim
lim
sup
!0 n!1 n k
if f is C k . Appendix A contains an outline of his argument. 2
Claims (1) and (2) together prove the Rm = cases of Theorems 1 and 10.
D
1
We close this section with a discussion of growth rates of nite material curves versus
growth rates of innitesimal line elements. Let : 0 1]
be such a curve, and suppose
for the sake of argument that for almost every s 0 1],
1
n 0
nlim
!1 n ln Df (s) = :
Integrating, we obtain
1 Z 1 ln Df n0(s) ds
= nlim
!1 n 0 Z
1 ln 1 Df n0(s) ds
lim
inf
n!1 n
0
1
= lim
inf n ln `(f n):
n!1
If the vectors Df n0(s) converge to their eventual growth rate nonuniformly for dierent s,
then the above inequality could be strict. This is in fact generally the case: metric entropy,
which for volume preserving dieomorphisms is the sum of the positive Lyapunov exponents,
is almost always strictly smaller than the topological entropy. This means that the growth
rate of a curve is often dominated by stretching at exceptional points.
As far as the dynamo problem goes, it has been observed that Lyapunov exponents do
not necessarily bound the growth rates of magnetic elds. The proof of Theorem 2 will shed
further light on how eld growth rates are tied to growth rates of curves, which in turn are
tied to the growth rates of innitesimal line elements at certain exceptional points.
! D
2
j
j
j
j
j
10
j
5 Upper Bound on the Growth Rate for Maps in the
Limit Rm ! 1
In this section we complete the proof of Theorem 10. The main point of consideration is the
growth rate of curves under the action of random maps most of which are small perturbations
of f. We show that in the zero-noise limit this growth rate has the same upper bound as
that for growth of curves under the action of f.
Denote by the set of C k volume preserving dieomorphisms on Rd and let f be
such that f is equal to the identity outside of the compact domain . We consider a one
parameter family of probability measures on with 0 such that
1. Dlg C , 1 l k, for a.s. g, where C is a constant independent of 2. > 0 we have g : g f Ck < 1 as 0.
(The C k norm of h, h Ck , is dened to be max Ds h , 0 s k, and should not be
confused with the norm h k max Ds h , 1 s k.) For a given B0 dene Bn by
2
D
jj
jj 8
f
jj
jj
;
jj
g !
!
jj
jj
jj
jj
jj
Bn =
=
jj
jj
Z
(Dg Bn;1 ) (dg)
Z (n) n
n
(Dg B0) (dg1 : : : dgn )
where g(n) = gn : : : g1, n = : : : . The aim of this section is to prove the following:
Proposition 1 Let and Bn be de ned as above. Then
1 ln Z B dx h(f ) + r(f )
lim sup lim
sup
n
k
Rd
!0 n!1 n
where h(f ) is the topological entropy of f restricted to .
To prove Theorem 10, let be the distribution of f +! where ! is a d-dimensional vector
distributed according to . Then
j
j
D
Z
Bn (x) = (Df Bn;1 )(f ;1(y)) (y x)dy:
;
If in addition gives a distribution consisting of d dimensional vectors with independent
gaussian entries of variance 2, then Bn (x) formally solves the magnetic induction equation (1) with the -function ow dened in section 2.
11
We begin the proof of Proposition 1 with the following observation:
Z
Rd
Z Z (n) n Dg B0d dx
ZD Zn (n) Bn dx =
j
j
Dg B0 dx dn
D
and, by identical reasoning to that used in the previous section,
Z Z (n)
Z Z (n)
1
1
n
lim
sup ln n Dg B~ 0 dxdn
lim
sup ln n Dg B0 dxd
n!1 n
n!1 n
D
n
Z Z; n
1
= lim
sup ln n
`(g x )dF (x) d
n!1 n
j
j
n
j
j
j
j
~ 0 (that depends only on B0) and is a transverse
where ; is a ux tube of some vector eld B
section of ;. Moreover we can interchange limits again to obtain
Z Z
n
Z Z
`(g(n)x )dF (x) dn =
`(g(n)x )dn dF (x)
n
Z
F () sup n `(g(n)x)dn
x2
where F () is the ux of B~ 0 through (a constant). For the rest of the proof let us assume
that is enlarged to contain ; and that f = Id in a neighborhood of the boundary of .
The proof is then reduced to the following claim:
D
D
Claim 3 Given 0 > 0, 0 > 0 such that < 0 N such that n N and : 0 1]
with k 1
Z n n n n(h(f ) + r(f )=k + 50)
`(g ) (dg ) e
:
9
jj
jj
8
9
8
8
! D
Proof of Claim 3:
Step 1. Preliminary choices.
(a) For each > 0, let be a covering of Rd by (round) balls of diameter . We assume
that is uniformly spaced so that
(i) no element of intersects more than Cd other elements of , and
(ii) 9`() > 0 such that 8x 2 Rd , the `()-ball centered at x is contained in some
element of (i.e., `() is a Lebesgue number for ).
The subcollection A : A T =
(b) We choose > 0 such that
f
2
D 6
g
will be denoted by 12
jD
.
(i) h(f jD ) < h(f jD) + 0, and
(ii) if g (x) (1=)g(x=), then for -a.e. g, kDl g k kDg k for l = 1 : : : k
((i) is possible by property 2 in section 3, and (ii) is possible because of our rst
requirement on (see also appendix A)).
(c) For a sequence (g0 : : : gn ) with g0 = Id, we will estimate `(g(n)) by
`(g(n)) n (g(n) ) V (g(n) )
where n (g(n) ) is the minimum number of sets of the form
b(x g(n) ) = fy 2 Rd : d(g(i)(x) g(i)(y)) < 0 i ng
needed to cover D and
\
V (g(n) ) = sup `(g(n) b(x g(n) )):
x2D
(d) There are 2 types of g's that will result in dierent estimates for and V . They are
those \near" f and those \far away" we distinguish between them through the use of
the parameter . For > 0, let
= fg 2 : kf ; gkCk < g:
A suitable choice for will be determined later.
Step 2. Estimation of .
The main point here is that suciently small noise does not increase the growth rate of
distinct trajectories (and large noise is too rare to be of importance).
(a) Let n0 be chosen so that
0 ;1
1 ln N (n_
f ;i ( jD)) < h(f jD) + 2
0
n0
0
and let 0 be a subcollection of Wn0 0 ;1 f ;i ( ) that covers and achieves the minimum
cardinality. The number n0 will be xed throughout, and we will be thinking in terms
of blocks of maps g1 : : : gn of length n0. A block is called a good block if gi i it is called a bad block if at least one gi .
(b) If Wnecessary we move theTelements of slightly so that for any 2 elements A and B of
n0 ;1 f ;i ( ), either A B = or their closures do not intersect (i.e., they do not touch
0
along the boundary). We assume is suciently small so that if g1 :W: : gn0 is a good
block, then
there is a 1-1 correspondence between the elements
of n0 0;1 f ;i ( ) and
W
W
n
;
1
n
;
1
those of 0 0 g;(i)( ). Moreover, the subcollection of 0 0 g;(i)( ) corresponding
to 0 covers .
C
D
f
g
2
8
62
6
f
C
D
13
g
(c) Let g0 g1 g2 : : : be given (and
xed for the rest of this step). For j = 1 2 : : : we will
W
jn
0 ;1 ;(i)
pick a subcollection Cj of 0 g ( ) which covers D. The rules for constructing
Cj from Cj ;1 are as follows: let A0 : : : A(j ;1)n0 ;1 ] represent an element of Cj ;1 , i.e.,
Ai 2 and A0 T g;(1)A1 T : : : T g;((j;1)n0;1)A(j;1)n0;1 is in Cj;1 . Then
Case 1. fg(j;1)n0 : : : gjn0 ;1 g is a good block and A(j;1)n0 ;1 D. In this case we
may attach any A(j;1)n0 : : : Ajn0;1 ] in C0 to the given sequence and consider the
resulting sequence as an element of Cj . (Of course, many of these elements may
be empty.) In this case the random maps are acting very much like f.
Case 2. fg(j;1)n0 : : : gjn0 ;1 g is good and A(j;1)n0;1 6 D (i.e., previous noise has
caused aTdrift out of D). In this case we attach a sequence of the form A : : : A]
where A A(j;1)n0;1 6= . The number of such elements is Cd (see Step 1(a)(i)),
and since f = Id outside of D, we know from Step 1(a)(ii) that if is suciently
small, then for every x 2 A(j;1)n0;1 , 9A 2 such that g(j;1)n0+i : : : g(j;1)n0 x 2
A 8i = 0 : : : n0 ; 1.
Case 3. fg(j;1)n0 : : : gjn0 ;1 g is a bad block (i.e., a large noise event occurs in this
block). In this case we
may attach sequencesT of the form B0 : : : BnT0 ;1] where
T
g((j;1)n0;1)A(j;1)n0;1 B0 6= , g((j;1)n0)B0 B1 6= , g((j;1)n0+1)B1 B2 6= ,
etc. The number of these new elements is dicult to control however large noise
events are rare.
Note that the Cj so obtained is a cover for D, i.e., every possible trajectory has been
accounted for.
(d) We view the sequence of maps g0 g1 : : : gjn0;1 as a concatenation of n0-blocks.
Claim 4 For some C0 independent of j we have for almost every sequence g1 g2 : : :
+
;
card Cj en0(h + 20);j en0C0;j
where ;+j is the number of good blocks and ;;j the number of bad blocks among the rst
j blocks.
Proof. We proceed by induction, counting at the jth stage the maximum number of
sequences that can be attached to each sequence in Cj;1 . In case 1, this number is
n (h+2 0 ) by 2(a). In case 2, this number is Cd (see step 1(a)), which we may
e 0
assume is en0(h+2 0 ) if n0 is chosen suciently large. In the last case, since there is a
uniform Lipschitz constant a.s., we have that for every A 2 , giA is contained in
a ball of xed radius. By step 1(a), this ball intersects eC0 elements of for some
C0. 2
We conclude step 2 by by noting that, using property 3 of section 3, jn0 (g(jn0) ) card Cj the number in claim 4.
14
Step 3. Estimation of V.
(a) The crucial observation is that Yomdin's proof (see appendix A) works equally well for
compositions of random maps, and that the estimates involved depend only on the `th
derivatives, 1 ` k, of these maps. Let g0 g1 : : : be such that i, Dl gi
C,
1 l k (condition 1 on ). Assuming the scaling condition in step 1(b) and that
n0 is suciently large, we see from the argument in appendix A that
1 ln V (g(jn0) )
1 ln Dg(jn0 ) + 0
jn0
kjn0
j some j0.
(b) Claim 5
+
;
V (g(jn0) ) en0;j (r(f )=k + 20)en0C1;j
where C1 is a constant independent of j and gi , and ;+j and ;;j are the number of
good and bad blocks respectively among the rst j blocks of length n0 .
Proof. Write
j 1
1 ln Dg(jn0 ) 1 X
(in0 )
ln
D
g
((
i;1)n0 +1)
jn
j n
8
k 8
k
k
k
f
k
0
k g
k
i=1 0
k
where g(((ini;0)1)n0+1) denotes the composition gin0 : : : g(i;1)n0+1. Assuming that is
0)
suciently small, we have (1=n0) ln Dg(((jnj+1)
(1=n0) ln Df n0 + 0 r(f )+20
n0 +1)
if g(i;1)n0 +1 : : : gin0 ;1 is a good block. In general all we can say for a bad block is
that Dg(((ini;0 )1)n0) C1n0 for some C1. 2
Step 4. The count
Summarizing, we have shown that for N -a.e. g1 g2 : : : and suciently small , if we
partition this sequence into blocks of length n0, then the contribution to the -term of the
jth block is en0 (h+2 0) for a good block and eC0n0 for a bad one. Also the jth block
contributes en0 (r(f )=k+2 0) to the V-term for a good block and eC1n0 for a bad one. Since
the blocks are i.i.d., we conclude that
k
f
k k
k
g
k
k E (`(g(jn0 )))
Pg en0(h + r=k + 40) + Pb eC2n0
j
where E (`(g(jn0 ))) is the expected value of `(g(jn0 )), Pg is the probability of a good block,
Pb is the probability of a bad block, and C2 = C0 + C1. This bound is clearly ejn0(h+r=k+5 0 )
if Pb is smaller than some , and such a circumstance can be guaranteed by choosing small
enough so that 1 ( ( ))n0 . 2
;
15
6 Upper Bound on the Growth Rate for Flows in the
Limit Rm ! 1
In this section we adapt the methods of section 5 to ows, thereby completing the proof of
Theorem 1. Consider again the magnetic induction equation
@ B = B u u B + 1 2B
(11)
@t
Rm
where u is a C k+1, k 2 divergence free vector eld with compact support on Rd . For
simplicity of notation write = 1=Rm . In order to use the methods of section 5, we need to
produce a family of probability measures 0 on , the space of C k volume preserving
dieomorphisms of Rd , so that R satises the conditions at the beginning of section 5 and
also such that given B0, Bn
Dg(n) B0dn is the solution of (11) with initial conditions
B0.
We will describe two dierent ways of obtaining , one using stochastic ows and the
other a product formula for operators. Since the techniques involved are fairly standard, we
will state the results here and give the proofs in the appendices.
r
;
r
r
f
f
g
g
f
g
6.1 The stochastic dierential equations approach
Let u be as above. (For technical reasons this method requires that u be of class C k+1+ for
k 2, > 0.) Consider the SDE
q
(12)
dXt = udt + 2 dbt
where bt is standard Brownian motion on Rd. >From the theory of stochastic ows (see e.g.
Kunita (1990)) it follows that X(x t !) can be chosen in such a way that for each (xed)
!, t Xt! is a continuous path in the space of C k dieomorphisms of Rd.
7!
Lemma 1 Given an initial condition B0, de ne
h
i
B(x t) = E DXt! (X;t!1 (x))B0(X;t!1 (x)) :
Then B(x t) is the solution of (11) with initial condition B0.
(Remember that D( ) here as elsewhere in this paper refers to spatial derivative, i.e., derivative with respect to x.) See Appendix B for a proof.
Let be the distribution of Xt=1! . It follows from the Markovian property of the
solution of (12) that the distribution of Xn! is the same as that of gn
g1 where the
gi's are iid with law . Condition 1 at the beginning of section 5 follows from the fact that
the Brownian term has constant coecient. Condition 2 is a standard fact about stochastic
ows (see e.g. Kunita (1990)). Theorem 1 now follows by the arguments of section 5.
16
6.2 The product formula approach
We now consider the product formula construction. Let Nt be the time t propagator of the
equation
@ = u u @t
(the Cauchy solution) and Mt the time t propagator of the diusion equation
@ = 2:
@t
We approximate the solution of (11) after time t by the product formula B()( t) = (M N )t=]B0
where B0 is the initial magnetic eld and is the size of each time step. The convergence of
this scheme, a standard result, is the subject of the following lemma:
Lemma 2 For xed t and r
;
r
r
lim
Z
!0 Rd
B(x t) B()(x t) dx = 0:
j
;
j
where B is the solution to equation (11).
The proof is sketched in Appendix C.
Let us now return to calculating with maps. For xed = 1=m, m a large integer, let
be the distribution of
g = ( + !m ())
( + !1())
where s is the ow generated by u and the !i () are d-dimensional random vectors with
i.i.d. Gaussian entries of mean 0 and variance 2. An easy calculation shows that
B()( n) =
Z
n
n (dg : : : dg ):
(Dg(n) B0)
1
n
We need the following lemma controlling the uniform dependence on of as to nish the proof:
Lemma 3 For given > 0
lim lim g : g 1 Ck < = 1:
!0 !0 f
jj
;
jj
!
0
g
This statement takes the place of the previous Condition 2. The proof can be found in
Appendix C.
We further make the following claim:
17
Claim 6 Given 0 > 0, 0 > 0 such that for any xed < 0, N Z + and > 0 such
9
that 8n N , , if : 0 1] ! D is a
Z
Ck
9
2
curve with kkk 1, then
n (dgn ) en(h(f ) + r(f )=k + 50 ) :
`(g(n))
The proof is identical to that for maps provided 0 is chosen small enough to assure that the
probability of occurence of a \bad block" is suciently small. The existence of such an 0 is
guaranteed by Lemma 3.
To complete the proof of Theorem 1, we rst replace a given initial condition B0 by B~ 0
as in section 4. Then we argue as in the paragraphs preceding Claim 3 in section 5 that for
arbitrary , , and n,
Z
Rd
B()(x n) dx C sup
j
j
Z
n
n
`(gn)d
where C depends on B0 but not on , or n. Let
0 > 0 be given and let < 0 be as
R
in Claim 6. We now estimate the growth rate of B(x n) dx as n
for this xed .
Temporarily x n N where N is as in Claim 6. We choose < such that the dierence
in Lemma 2 is less than 1. (Note that must be allowed to depend on n since there is no
uniformity inR this approximation as time tends to .) It then follows from Lemma 2, our
estimate on B()(x n) dx above, and Claim 6, that
Z
B(x n) dx 1 + Cen(h + r=k + 50):
j
j
! 1
1
j
j
Rd
j
j
7 Further Remarks
The range of applicability of Theorems 1 and 10 can be extended in several important ways.
We note rst that while we have been assuming that the map f or ow u is deterministic, the
results also hold under mild restrictions for random maps and ows such as the \renovating
ow" dynamos (Zel'dovich et al. (1988), Gilbert and Bayly (1992)). The extension to random
systems of the key steps of the Rm = results, namely denitions of topological entropy
and line stretching and the Yomdin upper bound can be found in Kifer (1986) and Kifer and
Yomdin (1988). The arguments contained in this paper also extend to random distributions
in a straightforward manner. Furthermore the bound of Theorem 1 applies for any Rm with
h(f ) replaced by h(g) where the distribution of g consists of Gaussian perturbations of f
with variance 2R;m1 . Secondly the proof is easily modied to apply to periodic boundary
conditions { the main dierence is that case 2 of step 2(c) in section 5 is no longer necessary.
Furthermore for periodic boundary conditions B0 need only be continuous diusion will
smooth the eld after an arbitrarily small time. (The same argument cannot be used for
the original case because of apparent technical diculties arising from unboundedness of
1
18
the domain.) We note also that u may be time periodic. A periodic ow in Rd can be
embedded as a steady ow in Rd S Rd+1 . In this case diusion acts only in the Rd
directions. In principle bounds involving topological entropy can also be given for more
general time-dependent ows.
Certain restrictions on Theorems 1 and 10 should be noted as well. As has already been
seen, for technical reasons we assume the initial eld is contained in a bounded region. In
addition we assume that the diusivity is constant in space. From the point of view of this
paper, a non-constant diusivity would result in spatially varying noise. Thus diusion could
stretch the magnetic eld and in principle have a direct role in the dynamo process. For the
same reason issues concerned with fast dynamo action surrounded by an insulating region
(Hollerbach et al. (1994)) are not addressed here.
The techniques of this paper can also be applied to the evolution of a passive scalar
gradient eld. The equation for the evolution fo a passive scalar quantity is
@ + u = 1 2
(13)
@t
Pe
where the Peclet number Pe is a dimensionless parameter measuring the relative strengths
of advective and diusive processes. Taking the gradient of (13) we get
@ ( ) + u ( ) = ( u)T + 1 2( )
@t
Pe
T
where ( u) is the transpose of u. When Pe = this equation is identical in form to
the time evolution equation for a eld of material area elements. To see this, let b c be a
material area element. Abbreviating d=dt = @=@t + u , by volume preservation we have
for any material vector a
d (a (b c))
0 = dt
"
#
d
T
= a ( u) (b c) + dt (b c) :
r
r
r
r r
r
; r
r
r
r
r
1
r
r
Since a was arbitrary we obtain
d (b c) = ( u)T (b c):
dt
As a Yomdin inequality holds for area elements as well as line elements, it can be shown
that at Pe =
Z
1
2r(f )
nlim
!1 n ln D dx h(f ) + k
where, as before, f is a C k map (or time 1 integration of a steady or period 1 ow). This
result might be interpreted as requiring h(f ) + 2r(f )=k > 0 as a necessary (although not
sucient) condition for ecient mixing at large Peclet number. If k = then in fact chaos
becomes a necessary condition for ecient mixing.
; r
1
jr j
1
19
8 Proof of the Lower Bound
The proof of Theorem 2 uses some basic techniques from smooth ergodic theory. These
techniques are standard to workers in the subject, but without a substantial amount of
notation and preliminaries it is dicult to make precise statements. Instead of giving a
formal proof we will elucidate the main geometric ideas, referring the reader interested in
technical details to an expository article (Young (1995)) or Pesin's original paper (Pesin
(1978)). An argument similar to ours has been used by Newhouse (1988). Throughout this
section let f be as in Theorem 2.
Idea 1. Measures maximizing entropy
If is an f -invariant Borel probability measure on , let h(f ) denote the metric
entropy of f with respect to (see Walters (1982) for a denition). In much the same
way that topological entropy measures chaos in a topological sense, metric entropy
measures randomness in the sense of probability. The following variational principle is
well known:
h(f ) = sup
h (f ):
The supremum is taken over all f -invariant probability measures it is also adequate
to only consider ergodic measures.
Our rst step is to pick (and x throughout) an ergodic measure with h(f ) >
h(f ) 0. More likely than not, is a singular measure, i.e., it lives on a set of
0 volume. The purpose of this proof is to show that points that are typical with
respect to , exceptional as they may be from
the point of view of volume, may have
R
a signicant inuence on the growth rate of Bn q dx.
Idea 2. Lyapunov exponents and special coordinates
According to a theorem of Oseledets (1968), there exists a set of numbers 1 > : : : > r
with multiplicities m1 : : : mr respectively such that at -a.e. x the tangent space Tx
splits into Tx = E1(x)
Er (x) in such a way that v Ei (x)
1 ln Df n v = :
lim
i
x
n!1 n
These numbers are called the Lyapunov exponents of (f ). The subspaces Ei(x) vary
measurably with x { they vary continuously if we are willing to disregard a set of
arbitrarily small measure.
In general Lyapunov exponents are asymptotic growth rates, but there are point dependent changes of coordinates due to Pesin (1978) that allow us to see f locally as
small perturbations of linear maps with eigenvalues exp(i ). More precisely, at -a.e.
x there exists a dierentiable change of coordinates "x : Nx Ux , where Nx is a small
neighborhood of 0 in Rd and Ux is a small neighborhood of x, with the properties that
D
;
j
j
D
D
8
j
2
j
!
20
"x (0) = x and D"x (0) carries the splitting Rm1
Rmr to E1(x)
Er (x)
if ~fx : Nx : Nfx is the representation of f in the new coordinates, then ~fx is C 1
near D~fx (0) and D~fx (0) is a linear map with D~fx (0)v exp(i ) v v Rmi .
It should be noted that these change of coordinates distort distances by arbitrarily large
amounts (depending on x) and that there are various technical problems of which one
ought to be aware when working with them (see e.g. Young (1995)).
Idea 3. The inuence of -typical points on dynamo growth
We will assume in this paragraph that we are working in our special coordinates so
that locally f resembles a linear map with multipliers exp(i) of multiplicity mi. In
fact for a typical x let us think of Ux as a product of the form Uxu Uxs , where Uxu is
a disk contained in Ei(x), i 0, and Uxs is a disk contained in Ei(x), i < 0. As
before let b(y n) = z : d(f i y f iz) < 0 i < n . Consider y very near the
center of Ux and assume that diam
Ux . Then b(y n) C Uxs for some C with
P
+
cross-sectional area exp n( i mi), +i max(i 0). (This is true exactly if f is
locally linear in our coordinates the nonlinearities are mild.)
Next let h = h(f ). A property of metric entropy is that given any set U of positive
measure, for n suciently large there exists O(exp(nh)) points in UTthat are (n )separated, i.e., x1 : : : xk U , k exp(nh), such that b(xi n) Sb(xj n) =
whenever i = j . We pick these points in Ux very near x and let An = i b(xi n).
To complete the proof we now exhibit a vector eld B0 with the desired growth rate.
All that we really require of B0 is that in Ux it points roughly in the direction of E1(x).
Then if q = dim Uxu, we will have
1.
2.
!
j
j j
j8
2
f
8
g
;
9
2
6
Z
An
Df n B0 q dx
j
j
eq
1n (Lebesgue measure of An)
P
eq
1n e;( +i mi)nenh
enh
Z q
1
lim inf n ln Bn dx h:
(14)
D
If h = 0 there is nothing to prove. If h > 0 then h P +i mi by Ruelle's inequality
(1978) so 1 > 0. The same reasoning tells us that (f ) must have a strictly negative
exponent hence any q d 1 will work in (14). If f is the time 1 map of a ow then
in addition to the negative exponent there is a zero exponent and so q d 2 suces.
and so
j
j
;
2
21
;
Appendices
A Outline of the Proof of Yomdin's Theorem
We present here a proof of Claim 2 following Yomdin (1987). The proof uses the following
approximation lemma which allows one to renormalize after each iteration:
Lemma 4 Let b(x 1) and b(x 2) be the balls of radius 1 and 2 around a point x in a Rd .
Let f : b(x 2) Rd be a C k map with Dsf
M , 1 s k, and let : 0 1] b(x 2)
!
jj
jj !
form = M 1=k
be a curve with jj jjk 1. Then there exist a constant of the
depending only on k and d) disjoint intervals I1 : : : I 0 1] such that
1.
(f );1b(f (x) 1) Ij (f );1 b(f (x) 2)
Ck
(with j
2. If j is an a
ne contraction mapping 0 1] onto Ij then
Ds (f j )
jj
jj 1 1 s k:
Lemma 4 deals with the approximation of C k maps by polynomials and has nothing to do
with dynamics. We will omit its proof.
Consider now f as in Theorems 1 or 1'. We dene a map ^f by blowing up f to ^f =
(1=)f (x). For suciently small,
max Ds^f = D^f = Df :
(15)
s=1:::k
jj
jj
jj
jj
jj
jj
Choose such that (15) holds. Given a C k curve dene a C k curve ^ by blowing up to
^ = (1=) and assume for the moment that ^ k 1. We now apply the lemma to ^f and ^
to get the maps j and thus see that for any x
\
`(^f^ ^b(x 1 1)) jj jj
where = (k d) Df 1=k . We now repeat this process again and again, dening a new
curve ^ 0 from the previous curve ^ by ^ 0 = ^f ^ j (summing over all the j 's). Then for
any x
\
`(^fn^ ^b(x n 1))) n:
This implies that
h \
i
lim sup 1 ln `(f n b(x n )) ln (k) + 1 ln Df (16)
n!1 n
k
jj
jj
22
jj
jj
which is in the direction of the desired result.
We remark rst that in the process of blowing up , Ds^ k increases for s > 1. We may
however subdivide ^ into shorter curves ^ j in such a way that, after reparametrizing, each
^ j satises ^j k 1. The number of subdivisions depends only on k and (and not on ).
To go from (16) to the desired result, we use the standard trick of working with a power
of f . Let 0 be given. Select > 0 and an integer p so that
1. (1=p) ln (k) < (1=3)0
2. (1=kp0 ) ln Df p0 < r(f ) + (1=3)0 for p0 p
3. ^fp = f p blown up by 1= has property (15)
4. (1=p)h(f p ) < h(f ) + (0=3). Here the in h(f p ) refers to the in denition (8).
Then for n 1 (with b(x n ) calculated using the map f p )
1 ln `(f np) = 1 lim sup 1 ln `(f np)
lim
sup
p n!1 n
n!1 np
1 h(f p ) + 1 lim sup 1 ln h`((f p)n \ b(x n ))i
p
p n!1 n
h(f ) + r(kf ) + 0
independent of choice of , as was to be shown.
k
k
k
jj
k
jj
B Details on Stochastic Flows
Proof of Lemma 1. Since Xt! is in fact C 3, B(x t) as dened in Lemma 1 is at least C 2 in
x so the right side of (11) is well-dened. For (x0 t0) 2 Rd 0 1) let
@ B (x t ) = lim B(x0 t0 + t) B(x0 t0) :
@t 0 0 t!0+
t
We will show that (x0 t0) this limit exists and is equal to the right side of (11). (To
conclude that @ B=@t really exists we will observe that as t 0+ the convergence is in fact
uniform in t0 for t0 in any nite time interval. This fact together with the continuity of
t0 @ B=@t(x0 t0) proves that @ B=@t+ = @ B=@t; t0 > 0.)
To study @ B=@t+, consider the backward derivative process associated with (12). More
precisely, consider
( dY = udt + 2db Y(0) = x
t
t
0
(17)
dJ = Du
J(0) = Id
;
8
!
7!
8
;
t
p
;
23
on Rd GL(R d), and introduce the function
f (Y J) = J;1B(Y t0):
Then using the fact that X;t 1 and Yt have the same distributions, we have for t > 0
h
i
B(x0 t0 + t) = E (DXt! )X;t ! (x )B(X;t! (x0) t0) = E (f (Y(t) J(t))
0
so that
@ B = lim 1 E f (Y(t) J(t)) f (x Id)]:
0
@t+ t!0+ t
Note that f is C 2. Ito's formula tells us that
Z t @ 2f Z t " @ f @ f ! u !# q Z t @ f
f (Y(t) J(t)) f (x0 Id) = 0 @ Y2 ds + 0 @ Y @ J
Du ds + 2 0 @ Y dbs :
Let us call the three terms in this equation (i), (ii), and (iii). Then E (iii) = 0. Since (i) and
(ii) are ordinary integrals, one veries (using the uniform boundedness of B(x t0), u, and
Du, and standard large deviation estimates for the trajectories of (Y(t) J(t))) that
1 E (i) = ( 2B)(x t )
lim
0 0
+
t!0 t
and
1 E (ii) = (B u B u)(x t ):
lim
0 0
t!0+ t
2
;
;
;
;
r
r
;r
C Details on Operator Splitting
Proof of Lemma 2. This is a standard and well-known result of operator splitting arising in
slightly dierent forms in many dierent contexts. For completeness we outline here a proof
using a particularly simple argument found in Chorin et al. (1978) p. 209.
Dene the operator Kt = Mt Nt where Nt is the time t propagator of the evolution
equation
@ = ru ; u r
@t
(we assume u is C k+1, k 2) and Mt is the time t propagator of the evolution equation
@ = 2:
@t
r
24
Let Ft be the time t propagator for
@ = u u + 2:
@t
The object is to show that in the limit 0, Kt=] converges to Ft for any t. Two basic
properties are necessary to demonstrate convergence. First consistency requires that
dK = dF :
dt t t=0 dt t t=0
Consistency follows immediately from the limit
1 (K ) = 1 M (N ) + 1 (M )
t t
t t t
t t
( dtd Nt t=0 + dtd Mt t=0)
r
;
r
r
!
j
j
;
;
!
;
j
j
as t 0. The second property required, stability, is boundedness of Kt=] as 0 for
small t. Stability can be shown, for example, by expanding the Green's function for Kt=]
for small t. We will not carry out the expansion here although, in fact, this argument can
be found as part of the proof of Lemma 3.
Using consistency and stability it is then easily shown that (Ft Kt=]) converges to 0
for small t. The convergence for arbitrary T then follows from a compactness argument.
!
!
;
Proof of Lemma 3: We will show for suciently small and large m that the two equations
x_ = u x(0) = x0
m
X
x_ = u + aj (t ; j=m) x (0) = x0
j =1
where aj are iid d-dimensional Gaussian random variables with mean 0 and variance 2=m,
have solutions that remain uniformly close pointwise in x0 as m
with probability
approaching 1 as 0. The higher derivatives are left to the reader.
First we consider the linearization of x_ x_ around the trajectory x(t), x(0) = x0. That
is, we consider the equation
Xm
_ = ( u) + aj (t j=m):
(18)
! 1
!
;
r
j =1
;
The rst task is to estimate the rst passage time for , i.e., to estimate the probability
that c rst occurs rst at time t. Now equation (18) can be integrated to yield the
solution
X
tm]
(t) = J(x(j=m) + (j=m) t j=m)aj (19)
j j
j j ;
j =1
25
(showing that is uniformly bounded with high probability). Let = sup u . Then
J(x(j=m) + (j=m) t j=m) e. The probability of rst passage of the sum (19) of
Gaussian random variables is less than or equal to the rst passage probability of the sum
of Gaussian random variables
X
tm]
0
(t) = e
aj :
j j
kr
k
In the limit of m
(1957))
;
k
k j =1
! 1
the probability f (z t) that 0 exceeds z > 0 is given by (Feller
j
(ze)2=4t :
f (z t) = ze
e
3
t
;
p
In particular for > 0
j
2
f (1=2; 1) = 2 e where = e; =2, which goes to zero as 0.
Now a straightforward computation (using the rst 3 derivatives of the ow generated
by u) gives the result
x (t) x(t) = (t) + O( 2))
which we have argued is small with high probability in the limit 0.
p
;
!
;
j j
!
26
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