A NNALES DE L ’I. H. P., SECTION B
S. B RASSESCO A. D E M ASI E. P RESUTTI
Brownian fluctuations of the interface in the D=1 Ginzburg-Landau equation with noise
Annales de l’I. H. P., section B, tome 31, n
o1 (1995), p. 81-118
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81
Brownian fluctuations of the interface in the D=1 Ginzburg-Landau equation with noise
S. BRASSESCO
Instituto Venezolano de Investigaciones Cientificas, Apartado Postal 21827, Caracas 1020-A, Venezuela
A. DE MASI and E. PRESUTTI
University of Cambridge, Isaac Newton Institute for Mathematical Sciences, U.K.
Vol. 31, n° 1, 1995, p. -118. Probabilités et Statistiques
ABSTRACT. - We consider the
Ginzburg-Landau equation
in an intervalof
R, perturbed by
a white noise and with Neumannboundary
conditions.The initial datum is close to the
stationary
solution(that
we callinstanton)
of the
equation
without noise. We provethat,
as the variance of the noisegoes to zero and the
length
of the interval isproportional
to the inverse of thisvariance, then,
the solutionapproaches
an instanton which moves as a Brownian motion.Key words: Stochastic PDE’s, interface dynamics, invariance principle
Nous considerons
1’ equation
deGinzburg-Landau
dans unintervalle de R avec bruit blanc additif et avec conditions de Neumann a la frontiere. La condition initiale est
proche
de la solution stationnaire(qu’ on appelle
« instanton»)
de1’equation
sans le bruit. Nous demontrons que,lorsque
la variance du bruit tend vers zero et lalongueur
de l’intervallespatial
estproportionnelle
a l’inverse de cettevariance,
la solution estproche
d’un « instanton »qui
sedeplace
comme un mouvement Brownien.1991 Mathematics Subject Classification : 45 M 10, 47 H 15, 82 C 24.
The research has been partially supported by CNR, GNFM, by grant S C 1-CT91-0695 of the Commission of European Communities.
Annales de l’lnstitut Henri Poincaré - Probabilités et Statistiques - 0246-0203
Vol. 31/95/01/$ 4.00/@ Gauthier-Villars
1. INTRODUCTION The semi-linear
parabolic equation
and the version with a stochastic force on its
right
handside,
appear in thephysical
literature as basic modelequations
forphase separation
andinterface
dynamics
insystems
with non conserved orderparameter,
seeHohenberg
andHalperin, 1973, [16],
and Allen andCahn, 1979, [2].
Westudy
in this paper the influence on the motion of the interface of a small white noise added to( 1.1 ).
We start
by recalling
a few results in the deterministic case when( 1.1 )
isdefined in the
interval 7~ ~= [-£-1, £-1),
£ >0,
with Neumannboundary
conditions. The solution of the
corresponding Cauchy problem
defines aflow >
0,
in the space of functions that are twicecontinuously
differentiable in the interior of
7§
and havevanishing
derivatives at theendpoints.
The flowT ~~~
hasobviously
twofixed,
stablepoints,
and an unstable one, Uo == 0. These are all the
stationary, spatially homogeneous
solutions of( 1.1 ),
but there are many others which are notspatially homogeneous.
For small £ the most relevant ones, in the senseof
stability,
are the two "instantons"J=~,
withu: (x)
astrictly increasing, antisymmetric
function of x,positive
for x > 0. In the limit as £ -0, u;
- mpointwise,
whereis a
stationary
solution of( 1.1 )
in the whole R.Fusco and
Hale, 1989, [15],
Carr andPego, 1989, [4],
andFusco, 1990, [14],
have studied thestability
of~u~
underT ~~~
and thecorresponding problem
for flowsgenerated by
semi-linearparabolic equations
with moregeneral
non linear terms. Theanalysis
shows that are saddlepoints
forthe flow with a one dimensional unstable
manifold,
that connects~
to the stablepoints
u+.ME
is invariant under the flowT ~~~
and itis
locally
stable in the sense that it attracts atexponential
rate(uniformly
as E -
0)
the orbits that start from aneighborhood
ofAnalogous
conclusions hold
The
points
in can beparametrized by x
E1;
and are denotedby
These are
essentially spatial
shifts of the instanton in,(1.2), namely
for any 0 R
1,
there is c > 0 so thatAnnales de l’lnstitut Henri Poincaré - Probabilités et Statistiques
The flow on
ME
isrepresented by
two orbits~~ (t), t
ER,
one,in
{x
>0~
and the other in{x 0~.
Theformer,
is anincreasing
function of t and
~cE,x, ~ -
converges to the stablepoint
u- ast - o and to
1~
as t - -oo.Analogous
behavior is found in the other half of where x 0. The motionalong
is very slow: thespeed
at x E is bounded
by
with r = EX andc(r)
> 0.Therefore the
points
in aneighborhood
of are "first" attractedby
and then move,
"very slowly", along ME,
In this paper we
study
thestability
of the manifoldME
under thesmall random
perturbations
of the flowTy
definedby
the stochasticpartial
differential
equation
in
7~
with Neumannboundary
conditions. 0152 is a white noise in space and time. Thetheory
of stochastic PDE’ sapplies
to(1.4),
asbriefly
recalledin the next
Section,
inparticular
we will refer to Faris andJona-Lasinio, 1982, [ 11 ], Walsh, 1984, [19],
and Da Prato andZabczyk, 1992, [8].
Hereit suffices to say that for any continuous initial datum mo and almost all the realizations of the
noise,
there is aunique,
continuous function mt, t >0,
that solves anintegral
version of(1.4)
with Neumannboundary
conditions in
7~.
The solution obtained in this way defines a continuous processmt, t
>0,
with values inC(7e),
hereafter referred to as "theGinzburg-Landau process".
As a consequence of our
analysis (but
forbrevity
we will not state atheorem)
it can be seen that is stable also for the stochastic flow definedby ( 1.4)
in the limit of small E. Inparticular
thepoints
in aneighborhood
ofif not "too close" to the
boundaries ±~-1
areagain
attractedby M~,
but
then,
at variance with the deterministic case,they
move"rapidly along"
In fact on times
E-1 t,
to becompared
to theexponential
times ofthe deterministic case, the process becomes
supported,
in the limit E -0, by
translates of the instanton m which thenperforms
a brownian motionon More
precisely
we consider an initial datum Esatisfying
N.b.c. in4
and such thatand call mt the process that solves
(1.4)
with initial datum We denoteby
PE theprobability
on the basic space, where the noise a and the process mt are constructed. Our main theorem in this paper is thefollowing:
Vol. 31, n° 1-1995.
1.1. THEOREM. - For any 0
(
1the following
holds. Given any E >0,
|x0| I ( 1 - 03BE)~-1,
mt and pé asabove,
there is a continuous process03BEt adapted
to mt such thatfor
any T > 0Moreover let be the law on
C(~0, T~, I~) of
the variable xo.Then converges
weakly
to a Brownian motionstarting from
0 withdiffusion coefficient
Analogous
results have been obtainedby Dell’ Antonio, 1988, [7],
forthe small random
perturbations
of a deterministic flow defined on a finite dimensional manifold and with an attractive submanifold thatplays
the roleof The
problem
in R with the noisestrength
a function of x thatsuitably
vanishes asIx I
- oo has been consideredby
one of us,S.B., 1993, [3], and,
morerecently by Funaki, 1993, [13].
In[3]
the process is studied for times much smaller than when thedisplacements
of theinstanton are still infinitesimal. In the limit as E - 0 and with a proper
normalization, they
are proven to converge to a Brownian motion.In
[ 13],
a paper that we have received whencompleting
the first draft ofthis one, the
analysis
concerns muchlonger
times. In ourlanguage
Funakistudies times t =
6’~~,
with 8positive
and small. Thedisplacement çt
of the instanton is renormalized as~ := Space
is renormalized in the same way so that the instanton becomes in the new coordinates astep
function. Funaki proves that thisstep
function moves in the limit asE - 0 as a Brownian motion with a
drift,
which comes from thespatial dependence
of the noisestrength.
The differentscalings
and the differentnorms used make this paper
quite
different from ours. We are indebted to T. Funaki for his observations on our paper. In Section 3 we make some more comments on hisapproach.
2. THE GINZBURG-LANDAU PROCESS
In this section we state the
Cauchy problem
for(1.4)
in a form that isparticularly
convenient for theproof
of Theorem 1.1. The standard way, see Annales de l’lnstitut Henri Poincaré - Probabilités et StatistiquesWalsh, 1981, [19],
and1984, [18],
togive
a sense to theCauchy problem
for
(1.4)
is to consider thecorresponding integral equation
where mo E
C(T )
is the initialdatum,
is the Greenoperator
forthe heat
equation
with Neumannboundary
conditions(N.b.c.)
in4
andxA is the characteristic function of the set A. The stochastic
integral
in(2.2)
defines a Gaussian process in space and time for which thefollowing properties
hold:2.1 LEMMA. - For any E > 0 the process is continuous in both variables and there are constants
bo
andbl positive
such thatfor
any a > 0The
proof
of Lemma 2.1 isgiven
in Section 5. It uses some Gaussian processesresults,
that can be found in[ 1 ] .
Some useful estimates on the covariance are taken from[18].
By
Lemma 2.1 we can restrict to continuous in(2.1 ),
in which casean existence and
uniqueness
theoremholds,
as astraightforward
extensionof the
proof
in[ 11 ], 1982,
for the case of Dirichletboundary
conditions.In the
sequel
we will consider anequivalent representation
of the process obtainedby using
the reflectionsymmetry
associated to N.b.c.2.2 DEFINITION. - For any continuous
function
m in4
wedefine
its .extension m to I~
by reflecting
mthrough
and thenextending
to I~ withperiod 4E-1.
That isReciprocally,
we say thata function
m Edefined
on the whole linesatisfies
N. b. c.if
it is the extension in the above senseof
afunction
inT .
Vol. 31, n° 1-1995.
We define
and
refer toZt
as the "freeprocess".
We denoteby Ht
the Greenoperator
for the heatequation
on the wholeline,
so that for any m EC(~ ),
2.3 PROPOSITION. - For any E >
0, for
any mo E thatsatisfies
N. b. c. in
4
andfor
anyZt(x)
continuous in both variables andsatisfying N. b. c.,
there is aunique
continuous solution mtof
theintegral equation
with
Ht
as in(2.6).
Moreover mt = where solves(2.1 )
withand obtained
by restricting Zt
and mo toTE.
Proof. -
We know from[ 10]
that(2.7)
has aunique
continuous solution which therefore satisfies N.b.c. in R. Thenby (2.6)
its restriction to4
solves
(2.1 ),
whose solution isunique.
DWe shall hereafter call the solution mt of
(2.7)
theGinzburg-Landau
process.
By Proposition 2.3,
mt is a processadapted
to the basic processZt.
We
study
theGinzburg-Landau
processby following
theapproach proposed
in
[3]
which is based on aperturbative analysis
of theGinzburg-Landau
process around the instanton solution
(1.2).
We writethen the linearized
Ginzburg-Landau
evolutionoperator
aroundmxo
iswhere
V"
is the derivative ofObserve that if mo satisfies N.b.c.
then limo -
is notsmall,
even ifmo were
equal
to an instanton in7~ However, by proving
a barrierlemma, Proposition 5.3,
we will be able to "localize" theanalysis
and the relevantAnnales de l’lnstitut Henri Poincaré - Probabilités et Statistiques
norm will be a supremum restricted to
7~.
In this way ourperturbative
scheme will work also for initial data that
satisfy
N.b.c. inT .
Lxo
is aself-adjoint operator
inR)
andm’x0
aneigenvector
ofLxo
witheigenvalue
0. Theremaining part
of thespectrum
ofLxo
is inthe
negative
axis at non zero distance from theorigin, [12].
Therefore thesemigroup
gt,xo =t,
is a contractionsemigroup
whose norm restrictedto the
subspace orthogonal
todecays exponentially
in t. This is truealso in as it can be seen
by using
a Perron-Frobeniusargument,
based on the strict
positivity
of theeigenvector
Theanalysis
has beencarried
through
in a paperinvolving
two of thepresent authors, (A.
DMand E.
P),
and T.Gobron, 1993, [9],
for a different case, where( 1.1 )
isreplaced by
a non local evolutionequation.
Theproofs
however extend to thepresent
case and for the sake ofbrevity
are omitted. We thus statewithout
proof
thefollowing
theorem.THEOREM 2.4. - There > 0 and c so that
for any 03C6
E andxo e R
where is the
unitary
vector inL2 (dx, (~) parallel
toWe will next
give
a heuristicproof
of Theorem 1.1. Asthe
initial state ismxo, (forgetting
for the timebeing
of the finitiness of7~).
themain contribution to mt at the very
beginning
comes from the noise.We introduce then an orthonormal basis whose first vector is
mxo
anddecompose
mt -mxo
in this basis. Its coordinates are at short timesapproximately independent
brownian motions. At some later time the drift in theequation
becomessignificant,
but the deviations that are still smallare
ruled,
toleading orders, by
thesemigroup
gt,xogenerated by Lxo .
Sinceall the directions are
exponentially damped except along
after a timeTE,
in the successiveanalysis TE
=(1/10
should beregarded
asmuch smaller than
1/2)
we willhave,
seeProposition 5.4,
with
b~E
the value of a standard brownian motion at timeTE .
We thenwrite,
to firstorders,
Vol. 31, n° 1-1995.
By iterating
thisprocedure
weeventually
reach a time when thedisplacement
from the initial instanton is finite.By (2.14)
its law is that of the limit Brownian motion of Theorem 1.1.There are several
points
in theargument
to befixed,
one for all the errorwhen
going
from(2.13)
to(2.14):
the first termneglected
isproportional
to
~b2T~m’’x0.
If wesimply
sum up all these errors and iterate till timewe
get
a finite contribution. Infact,
sincebTE according
to(2.14)
we need
NE
iterations to have a finitedisplacement
of theinstanton,
where(having supposed
that the increments areindependent).
Since the sum of the errors is
NEETE,
the sum is also of theorder of 1.
The circumstance that
helps
us in this case is thatmo
is normal toin
L2 (IR, dx),
so that the above error is killedby
theexponential decay
inTheorem 2.4 at the next
step
of the iteration. We deal with this and the otherproblems implicit
in the above heuristicargument
in some unifiedway,
asexplained
in the nextSection,
and conclude this oneby writing
theequation
for mt in terms of thesemigroup generated by Lxo .
We denote
by ut
:= mt -mxo
and weexpand
aroundmxo .
Then
using
thatmxo
is astationary
solution to the deterministicequation
in
R,
we rewrite(2.7)
asRecalling
that gt,xo = we havewhere
The
proof
of(2.16),
that isomitted,
makes formal thefollowing
heuristicargument:
we rewrite(2.15)
in differential formand then observe that it is "solved"
by (2.16).
Annales de l’lnstitut Henri Poincaré - Probabilités et Statistiques
The
representation (2.17)
shows thatt,x0 is
bounded and continuous in anycompact
time interval in the same set where this holds forZt,
henceby
Lemma2.1,
withprobability
1. Some estimates on the distribution of like those inProposition 5.4,
are moreeasily
establishedby writing
an almost
surely equal
version oft,x0 as
the stochasticintegral:
We omit the
proof
of(2.19)
that makes formal thefollowing argument,
where we rewrite(2.17)
in differential form:which then becomes
that
yields (2.19).
We summarize the above discussion in the
following Proposition.
2.5 PROPOSITION. - Let E > 0 and mo
and Zt as in Proposition
2.3.Then, given
any Xo E7;.,
mt solves(2.7) if
andonly if
ut = mt -mx0
solves(2.16)
with as in(2.17),
or,alternatively,
as in(2.19).
3. FLUCTUATIONS OF THE INSTANTON
In this section we reduce the
proof
of Theorem 1.1 to severalpropositions,
that will be proven in the next sections.
An
important
role in the wholeproof
isplayed by
the notion of center: the center of an instanton is theposition
where the instanton has the value 0.For a
general
function we set:3.1 DEFINITION. - The
function
m E has a center x°if, using
thenotation
(2.8),
Vol. 31, n° 1-1995.
In the next
Proposition
we will see that if a function is close to aninstanton then it has a center. We will then
proceed by showing
thatthe
Ginzburg-Landau
process satisfies withlarge probability
this closeness condition at all the times involved in Theorem 1.1. The notion of center will then be crucial in theproof
of Theorem 1.1 as it is in theproof
ofFunaki, [13].
This is therefore agood point
for some comments on thenotion of center and on the Funaki’s
approach.
Recalling
Theorem 2.4 the center xo of m can beinterpreted
as the centerof the instanton
mxo
to which the orbitstarting
from m converges when the evolutionTt,
definedby ( 1.1 )
in the wholeR,
is linearized aroundmxo .
From this
perspective
it looks more natural to define the centerdirectly
interms of the flow
Tt :
the new center yo,generally
different from xo, is such thatmyo .
Theadvantage
ofworking
with yo is that is amartingale
when mt is the theGinzburg-Landau
process(in
the wholeR),
as observed
by
Funaki. Infact, by
itsdefinition,
does not vary whenmoving
in the direction of the drift. Thissuggests
aproof
of Theorem 1.1 based on amartingale
convergencetheorem,
which is indeedaccomplished
in
[ 13] .
Thestarting points
in[ 13]
and in thepresent
paper arequite similar,
because the process will be proven tostay always
very close to aninstanton,
hence the two notions of centeryield
variables very close to each other.It is therefore
amazing
to see how such a small differencedevelops
ratherdifferent
strategies
and in the endproofs
that areessentially
different.3.2 PROPOSITION. - There are 8 > 0
and, given
any~’ and (
such that0
1’ ( 1,
there are c and EO sothat for
any 0 E EO and any] ( 1 - 0~
thefollowing
holds. Let m E2,
andThen
( 1 )
m has acenter 03BE
inT ,
and 03BE
isunique
in(1 - ~’~-1}.
Let
then
Annales de l’lnstitut Henri Poincaré - Probabilités et Statistiques
Then m* has a
unique
center~* ( 1 - (’)£-1}
and(4) If m satisfies
N. b. c. in4,
then m has aunique
center in~ .
In
Proposition
5.2 we will prove that withlarge probability
theGinzburg-
Landau process that starts from an instanton has sup norm bounded
by
2at all the times t that are involved in our
analysis,
hence the condition 2 inProposition
3.2.3.3 DEFINITION. - We
define
thefunction
as the centerof
m wheneverm
satisfies
the conditionsof Proposition
3.2 withsome (
> 0 and we then say that is"proper".
In the other cases we set = 0 and say thatis not proper. We will also use the shorthand notation
A set where is proper that will often appear in the
sequel
is theset
C~, with (
and Epositive,
whereThen, according
toProposition 3.2,
for any 0(’ (
there is EO such thatfor all E::; EO, any m E has a center in
(1 - ~’)E-1~.
We prove next that if m E
CE,(, then,
after a "shorttime", (that only
grows as b >
0,
that may be chosenarbitrarily small)
itgets
"much closer thanE1~4"
to the instanton with center3.4 PROPOSITION. - For any 0
(
1 and 0 a1/4
there arepositive
constants C andb,
b1 / 10,
andgiven
n, cn, so that thefollowing
holds.
Suppose
that ma E and that itsatisfies
N. b. c. in4.
Let xo be the numbercorresponding
to mo, see(3.9).
Denoteby
mt theGinzburg-Landau
process
starting from
mo andcall ~ -
thenVol. 31, n° 1-1995.
Using
the above result we aregoing
to prove thefollowing
Lemma:3.5 LEMMA. - Let
(,
a, band mo be as inProposition
3.4. Then there is c’and, given
n, cn sothat, setting
s~ =Proof. -
Let k* be the firstinteger
such that > andC’
and
c~
the values of C and cn inProposition
3.4when (
isreplaced by 1’
=10 - 2 ~, (a,
band ~
as in the statement of theLemma).
We are
going
to proveby
induction on k1~*,
that there isc’
so thatthat
implies (3.11 ).
By Proposition
3.4By (3.3)
we thenget (3.13)
with k =1, by choosing properly c’.
We next suppose
(3.13)
true for 1 ~k*
and we prove it for k + 1.We call m* =
?T~, ~ = ç(m*)
and suppose that m* is in the set onthe left hand side of
(3.13).
Then we canapply Proposition
3.4with (’
instead
of (
becausefor all E small
enough.
Therefore(3.14)
holds withm;
the processstarting
from m* and we have
Annales de l’Institut Henri Poincaré - Probabilités et Statistiques
By (3.13),
that holds for k(by
the inductionassumption), (3.15)
and(3.3)
we then
get
(3.13)
is thusproved
for allRecalling
the statement below(3.13)
we areonly
left with theproof
of
(3.12). Using (3.13)
andProposition
3.4with (’
insteadof (
we havefor
We then derive
(3.12)
from(3.13), (3.17)
and(3.3)
so that the Lemma isproved.
DBy
Lemma 3.5 theGinzburg-Landau
process converges to an instantonas £ -
0,
thusbecoming
a one-dimensional process which describes the motion of the center,(this
under theassumption
that the process starts closeenough
to an instanton and till times muchlonger
thanE-1). (3.12)
alsogives
bounds on the valuesof 03BEt
butthey
are far fromsharp,
as shownin the next Theorem:
3.6 THEOREM. - Given
any (
> 0 and e >0,
letmt be the
process that startsfrom
see( 1.5),
with|x0] ( 1 - ()£-1. Define
and let PE be the law on
f~) of Xt .
Then pE convergesweakly
asE -t 0 to P the law
of
the Brownian motion withdiffusion
D= 3 4
thatstarts
from
0.Theorem
3.6, (3 .10)
and(3.11 )
prove Theorem 1.1. To prove Theorem 3.6 it is convenient to work in discrete times. With b > 0 as inProposition 3.4,
we define
and
Vol. 31, n° 1-1995.
The reason of this choice of
TE
will become clear in theproof
ofPropo-
sition 3.8 below. We set
Of course for all T =
Etn.
We call ~°E the law on0~)
of
Y;
and we will first provethat,
as E -0,
it convergesweakly by subsequences.
We will then show that the limitpoints
aresupported by R)
andfinally
thatthey
are allequal
to the same measure P. We will also prove that the increments ofXt
in thesingle
intervals]
vanish as E -
0,
thuscompleting
theproof
of Theorem 3.6.We call
0t
thea-algebra generated by
the processZs,
st, recalling
that mt is
adapted
to0t,
and state thefollowing
twokey Propositions.
Inthe first one we state criteria for
tightness
andsupport properties
of thelimit measures and in the second one that
they
are satisfied.3.7 PROPOSITION. - Given any T >
0,
thefamily of
laws E >0,
ontight if
there is c so thatfor
all Ewhere
with
If (3.22)
holds andif
then any limit
point P of
pE issupported by C(~0, T~, IR).
Finally, if (3.22)
and(3.26)
holdand if
Annales de l’lnstitut Henri Poincaré - Probabilités et Statistiques
D = 3 ,
then any limitpoint P
isequal
toP,
the lawof
the Brownian motion withdiffusion
D that startsfrom
0.3.8 PROPOSITION. - Under the same
assumptions
as in Theorem3.6,
the conditions(3.22), (3.26), (3.27)
and(3.28)
aresatisfied for
all E smallenough.
Thus
Propositions
3.7 and 3.8 prove that !pE convergesweakly
to P. Atthis
point
very little ismissing
forproving
Theorem3.6, namely
that theincrements of
X:
in any of the intervals areinfinitesimal,
see the remarksfollowing
Theorem 3.6.By (3.11 )
and(3.12), if I ( 1 -
there is
(’
> 0 sothat,
for any T >0,
We then fix any interval
[tk, tk
take mtk in the set on the left hand side of(3.29)
and consider the process for a timeTE starting
fromsuch
By (3.12),
theprobability
that thecorresponding
increments of~t
are boundedby c’ El/4-1/10
islarger
than 1 - The intersection of all these eventswith tk
has alsoprobability larger
than 1 -(with
differentcoefficients).
Theorem 3.6 and Theorem 1.1 are thusproved,
once all the above
Propositions
are alsoproved.
4. PROOFS OF THE PROPOSITIONS OF SECTION 3
In this Section we prove the
Propositions
stated in Section 3using properties
of theGinzburg-Landau
and of the free processes that will be proven in Section 5.Proof of Proposition
3.2. -By assumption
the functionis continuous in R and
Vol. 31, n° 1-1995.
Let
then ç
is a center of m if andonly
ifC(~ - xo)
= 0. Since forall y
we have
F and H are in F is
odd, increasing
andMoreover,
there is c sothat,
for all y such thatIxo
+yl (1 - ~’)W1,
and, analogously,
We fix
arbitrarily 7
>0,
then there arex, 6
and 60positive
so that forall 6 60 and all 28 the
following
holds.Firstly
ifIyl
~y, thenIxo
+(1 - ~’)E-1, (because (1 - ~)E 1 and (’ ().
MoreoverWe are
going
to show that for all E smallenough C(y)
has a zero, yo, inIyl
’7 and no other onein Ixo + ~/! ~ ( 1 - ~’ ) E -1.
Annales de l’lnstitut Henri Poincaré - Probabilités et Statistiques
By (4.8),
=F(’)’) + H(-y)
> 0 andsince, analogously, C( -’)’) 0, by
thecontinuity
ofC(y)
there is a zero inIyl
~.By (4.9)
it isunique
in
Iyl ~y
andby (4.8)
inIxo
+yl (1 - (’)£-1.
To refine the location of the zero we observe thatF(0)
= 0 sothat, by (4.9),
Thus, by (4.6),
hence y+, which proves
(3.3).
MoreoverWe then have
hence
(3.5).
Calling h*(~) _ m*(x) -
we have26,
hence(4.8)
and(4.9)
and the conclusions thereafter are also valid for h*. SinceC(yo)
=0,
we have
By taking 8
smallenough
weget
We can then
apply (4.9)
andusing (4.10)
so that
(3.7)
is proven.The last statement in
Proposition
3.2(about
theuniqueness
of the centerin the
whole ?e)
follows afterreplacing
the firstinequality
in(4.6) by
Vol. 31, n° 1-1995.
that we need to bound
uniformly
in xo + yE 4. By
theassumption
thatm satisfies N.b.c. in
7~,
when x =E-1 + x’,
0x’ ~’E-1,
so that
Same bound holds when x
= -~E-1 ~ x’~,
0x’ ~’E-1,
so thatusing (4.11 )
and(4.13 )
we prove that the last bound in(4.6)
holds in the whole7~
so that theuniqueness
in thewhole 4
follows and theProposition
is
proved.
DProof of Proposition
3.4. -Setting ço
=by (3.3)
we have thatlço -
xoI c~1/4
andby (1.2)
Then
We define
A linear
interpolation
in themissing
intervalscompletes
the definition ofmo.
Let
mt
and mt be the solutions of(2.7)
with initial datarespectively mo
and mo(and
samenoise). By Propositions
5.2 and5.3,
for any n there is cn so thatWe will prove that there is & a
and,
for any n, cn so thatAnnales de l’Institut Henri Poincaré - Probabilités et Statistiques
where
Xo
is the center ofmo.
Observe thatby (3.7),
for any n there is cn so thatThen from
(4.17), (4.18)
and(4.19)
After
proving (4.18)
andconsequently (4.20)
we will extend the resultby lifting
the condition on x in(4.20)
thushaving
a sup over the whole7~.
Then,
with thehelp
ofProposition
3.2 we willeasily
conclude theproof
ofthe
proposition.
We next prove(4.18).
Given a1/4,
letand let
with YE
as in(5.27). Then, by Propositions
5.2 and5.4,
for any n there is cn so thatLet
then, by (2.11 )
and(2.16),
there are constantsCi
andC2
suchthat,
inBE,
for all t
E-b. By (4.21)
the last termi~
boundedby 6~. By (4.19)
Vol. 31, n° 1-1995.
where
C3
> 1 is a suitable constant.Then from
(4.25)
weget that,
inBE,
Let
We next prove
by
contradiction thatE-b T,
in We thus suppose that TE-b, then,
for all t T
and, by
thecontinuity
ofHence
that is
which cannot hold for all E small
enough because, by (4.21 ),
b1 /4.
Wehave thus proven that
E-b
T.By (4.25)
and(4.26)
we thenget
for all tE-b,
inBE
hence
setting t
=E - b :
By (4.23)
we have thuscompleted
theproof
of(4.18)
hence also that of(4.20).
In
{~ 2014 çol ]
>10’~(6~}
we useProposition
5.3 to reduce to the casewith the initial datum close to a function
identically equal
either to 1 or-1.
By symmetry
we mayjust
consider the intervalAnnales de l’lnstitut Henri Poincaré - Probabilités et Statistiques
Recalling (4.15),
weget
for all E small
enough.
We define a newmo
asThen, by Proposition 5.3,
for any n there is cn sothat, calling
x* :== xo +
10-5(E-l,
To estimate
mE-b
we shouldstudy
theequation
linearized around m = 1.The
analysis
is very similar to theprevious
one since m = 1 islinearly
stable for the deterministic evolution
( 1.1 ).
We then obtain the same estimatewe had
before,
details are omitted. We have thusproved
that(4.20)
holdswith the sup over x
E 4.
We now recall what stated
right
at thebeginning
of thisproof, namely
thatWe then use this
inequality
toreplace by mxo
in the first sup in theimproved
version of(4.20).
For the second sup we useProposition
3.2to conclude that
We then
get
for theimproved
version of(4.20),
the one with the supover x E
?~.
hence
(3.10). Proposition
3.4 is thereforeproved.
DProof of Proposition
3.7. - We callmT
= m~-103C4 and7§
thea-algebra generated by
for T’ T.0t
is instead thea-algebra
associated to mt, as usual. We will first prove thatVol. 31, n° 1-1995.
and that
with
where i =
1, 2 .
Proof of (4.30)
and(4.31 ). -
We shorthandThen, obviously,
Moreover
where
Recall the formula ~2 =
L f 2 - 2 f L f
for the"operateur
carre duchamp"
valid for the "second
compensator"
in time-continuous processes,[17].
To prove
(4.35)
it isenough
to check thatwhose
proof
is omitted. Then(4.30)
and(4.31)
are proven sincethey
arejust (4.34)
and(4.35)
with different notation.Tightness. - Having
now therepresentation (4.30)
and(4.31),
we canuse the
following
sufficient condition fortightness,
see for instance§
2.7.6of
[6] :
which is
implied by (3.22).
Annales de l’lnstitut Henri Poincaré - Probabilités et Statistiques
Support properties. -
A sufficient condition(see
for instance theproof
ofTheorem 2.7.8 in
[6])
for thesupport property
stated inProposition
3.7 iswith
Y +
andYT_ respectively
theright
and left limits ofY;,
at T.Recalling
the definition of
Y;, (4.39)
isequivalent
towhich follows from
Since
(4.39)
follows from(3.26).
Identification of
the limit law. - For whatalready
proven, any limit law P of P~ issupported by C([0, T] , R).
It is thenenough
to show thatYr
and
Y2 - DT,
D= 4,
are Pmartingales, because, by
theLevy’s theorem,
P is then
equal
to 7~.Let ~
be a bounded continuous function onC([0,T],R+),
measurablewith
respect
to the coordinate process till time a, 0 T. We need to show thatalong
aconverging subsequence,
forany ~
as above and anyT such that a T T
In
fact,
since the functions in theexpectation
are P- a.s. continuous inD(~0, T] , IR+),
because of thesupport properties
ofIP,
thenwith E the
expectation
withrespect
to P.By (4.30)
Vol. 31, n° 1-1995.
Using (4.32)
and(3.27)
we then obtain(4.43).
Analogously,
to prove thatY2 -
DT is a Pmartingale
wewrite, shorthanding
Yn forYrn’
which is a PE
martingale
withrespect
toAnalogously
to(4.32)
wedefine
It is then
enough
to prove that for all~,
a, T asbefore,
which is
implied by (3.28).
TheProposition
is thereforeproved.
DProof of Proposition
3.8. -By (3.11 ), (3.12)
and the definition ofTE,
we
get
By definition,
see Definition3.3, ~t ~ E-1.
It thus suffices to prove thatwhere the process
mt
starts from mo and the latter satisfies N.b.c in7§
andmoreover
mx0 liE
:S;E1/2-a
with(1 - ~/2)~-1.
c in(4.47)
and the
convergence
in(4.48)
must be uniform in mo(provided
it satisfies the aboveconditions).
Given mo,ço
=ç(mo),
letm;
be the process that starts from and denoteby ç}e
the variables and~TE
inthis new process. We will prove below that there are
c’ and,
for any n, cn so that for any mo as aboveAnnales de l’lnstitut Henri Poincaré - Probabilités et Statistiques