Current and density fluctuations for interacting particle systems with anomalous diffusive behavior

0.28 MB·English

by  M. Jara

CURRENT AND DENSITY FLUCTUATIONS FOR INTERACTING
PARTICLE SYSTEMS WITH ANOMALOUS DIFFUSIVE BEHAVIOR
9 M.JARA
0
0
2 ABSTRACT. We prove density and current fluctuations for two examples of
symmetric,interactingparticlesystemswithanomalousdiffusivebehavior: the
n
zero-rangeprocesswithlongjumpsandthezero-rangeprocesswithdegenerated
a
J bond disorder. Asanapplication, weobtainsubdiffusive behavior of atagged
particleinasimpleexclusionprocesswithvariablediffusioncoefficient.
2

]
R
P 1. INTRODUCTION
.
h
Since the work of Harris [15], it is known that the motion of a tagged or dis-
t
a
tinguished particle in symmetric, diffusive, one-dymensional systems of particles
m
that preserve the relative order of particles is subdiffusive, in the sense that the
[
mean square displacement E[x0(t)2] grows witht ast1/2, where x0(t) denotes the
1 positionofataggedparticle, initiallyattheorigin. Thisisamuchslowerratethan
v
thelinear growthobtained forusual diffusions. InHarris’ original work, asystem
9
2 of independent Brownian motions with reflection was considered. These kind of
2 systems are known in the physics literature as single-file diffusions (see [6] for a
0
recentdiscussion andfurtherreferences).
.
1 In this subdiffusive setting, it has been proved that the rescaled position of the
0
taggedparticleconverges,inthesenseoffinite-dimensionaldistributions, toafrac-
9
0 tional Brownian motion ([4], [10], [18] for the simple exclusion process; [27] for
: interacting Brownian motions). Recently, a functional central limit theorem has
v
i beenobtained[26]. Thissubdiffusivebehaviorischaracteristic ofsingle-filediffu-
X
sions;whenparticlescanpassoneovertheothersorindimensiond>1,thetagged
r
particleconvergestoaBrownianmotion[20]. Forbiasedparticles,thelimitisalso
a
diffusive[21].
From the work of Rost and Vares, we know that the displacement x (t) of the
0
tagged particle can be identified as the mass-current through the origin in the in-
crement process associated to the single-file diffusion. Let x(t);i Z be the
i
{ ∈ }
position of the particles at time t 0 in a single-file diffusion. We assume that
x(t) x (t). Define h (i)=x ≥(t) x(t) e , where e =0 in the case of par-
i i+1 t i+1 i
ticles≤evolving on the real line, and e =−1 in t−he case of particles evolving on the
lattice. The process h = h (i);i Z turns out to have a Markovian evolution
t t
{ ∈ }
with a local dynamics. Let J (t) be the accumulated current through the bond
0
2000MathematicsSubjectClassification. 65K35,60G20,60F17.
Key wordsand phrases. Densityfluctuations, zero-range process, current fluctuations, random
environment,fractionalLaplacian,simpleexclusion.
1
2 M.JARA
1,0 (of particles in the lattice, and of mass in the continuum). We have the
h− i
identity J (t)=x (t) x (0), andtherefore theasymptotic behavior ofthetagged
0 0 0
particleisgivenbyth−easymptotic behavior ofthecurrentJ (t)fortheprocess h .
0 t
A second identification, also known from the work of Rost and Vares, allows
us to obtain the current J (t) as a function of the empirical density associated to
0
the process h . At least on a heuristic level, J (t)=(cid:229) h (i) h (i) . When
t 0 i 0 t 0
the number of particles is finite, this relation is simply ≥an{integr−ated for}m of the
conservation of mass. When the number of particles is infinite, the sum is not
absolutely summable. We will see that when the process h is in equilibrium, the
t
truncated sums (cid:229) h (i) h (i) form a Cauchy sequence, and the limiting
0 i<n t 0
variableisprecisely≤the{curren−tJ (t).}
0
In this way, the asymptotic behavior of the tagged particle in single-file diffu-
sionscanbeobtained intermsoftheasymptotic behavior oftheempirical process
associated to the increments of the original process. This approach has been used
byvariousauthors[23],[27],[6],[14]inordertoobtainacentrallimittheoremfor
ataggedparticleinsingle-filediffusions.
The idea of relating the position of the tagged particle to the current of parti-
cles through the origin can also be accomplished for the original process, without
considering the increment process. This approach was exploited in great gener-
ality in [11], where the authors obtain a functional central limit theorem for the
tagged particle in a system of particles with collisions. Besides the collision rule,
the evolution is independent. Considering different families of dynamics for the
motion of one particle, they obtain in the limit any exponent 0 < g < 2 for the
mean square displacement E[x (t)2] of the tagged particle. The main drawback
0
of the approach of [11] is that it does not generalize to systems of particles with
strongerinteraction.
Forfairlygeneraldiffusivesystems,theso-calledhydrodynamiclimitofthepro-
cessh isgivenbyanon-linearheatequationoftheform¶ u=¶ (D(u)¶ u),where
t t x x
D(u) isthe bulk diffusion coefficient, which isgiven by the Green-Kubo formula.
We say that the process h has an hydrodynamic limit if the rescaled empirical
t
process n−1(cid:229) x Zh tn2(x)d x/n(dx) converges in distribution to a deterministic limit
of the form u(t∈,x)dx, where u(t,x) is the solution of the hydrodynamic equation
¶ u=¶ (D(u)¶ u). From now on, we focus on lattice systems, so the empirical
t x x
processrepresents thedensityofparticles inthesystem. Noticethediffusivetime-
scaling. In particular, if we start the process with a fixed density of particles, at
a macroscopic level the density of particles does not change. When the invariant
measures of the process h have short-range correlations, the spatial fluctuations
t
ofthedensity ofparticles areofGaussian natureandtheyaregiven,inthemacro-
scopiclimit,byc (r )W,whereW isawhitenoiseinR,r isthedensityofparticles
andthequantityc (r )isthestaticcompressibilityofthesystem. Bythefluctuation-
dissipation relation, the density fluctuations around a fixed density r evolves in a
non-trivial wayunderadiffusivescaling, andtheysatisfy inthemacroscopic limit
theinfinite-dimensional Ornstein-Uhlenbeck equation
dY =D(r )D Ydt+ D(r )c (r )(cid:209) dW, (1)
t t t
p
ANOMALOUSFLUCTUATIONS 3
whereW isaspace-time whitenoise. Goingbacktotherepresentation ofthecur-
t
rentintermsoftheempiricaldensityofparticles, weseethatputting theGaussian
space scaling and the diffusive time scaling together, n 1/2J (tn2) should con-
− 0
verge to Y(H ) Y (H ), where H (x) = 1(x 0) is the Heaviside function.
t 0 0 0 0
− ≥
Since the process is Gaussian, a simple scaling argument allow us to conclude
that J (t) approaches a normal distribution of variance Q (r )t1/2 ast ¥ , where
0
→
Q (r )=c (r )/ D(r ).
Recently ([1], [6]), the following question has been posed. What happens with
p
thetaggedparticleifeachparticlehasitsown,differentdiffusionconstant? Itturns
outthat[14]undermildconditions, thebehavioristhesameasbefore: E[x (t)2]=
0
Q (r )t1/2, but now the diffusion coefficient D(r ) is given by an homogenization
formula.
In this article weare interested in systems on which the asymptotic variance of
the current (and therefore of the tagged particle) is of order tg for g =1/2. From
amathematical point ofview,theuniversality oftheg =1/2valueis6 pleasant and
satisfactory,butthisisnotagoodfactfromthepointofviewofmodeling. Thereis
alsoexperimentalandnumericevidencesupportingvaluesg <1/2forthevariance
growthofthetagged particleinsomeextremesituations. Fromaphysical pointof
view, only values of g in (0,1) are expected. In fact, if at a small time window
we observe a positive increment on the current J (t), this means that the density
0
ofparticles attherightoftheorigin islarger thanthedensity atleft, andtherefore
we expect the current to have negative increments in the near future. This means
that J (t) has negatively correlated increments, restricting ourselves to concave
0
functions forthevariance ofJ (t).
0
Notice that the question about current fluctuations can be posed for any one-
dimensional system, related or not to a single-file diffusion. Therefore, in this
article we pose the question about the asymptotic behavior of current and density
fluctuations for general symmetric, one-dimensional particle systems. In particu-
lar, we propose two classes of models which will allow us to find a central limit
theorem for J (t) in the full range of scales g (0,1). We recall now that our
0
heuristic derivation of the g =1/2 law is very ro∈bust, we have only assumed that
thestaticfluctuationsarenormal,thatthehydrodynamiclimitisdiffusive,andthat
thefluctuation-dissipation relationholds. Followingthesamescheme,wewillob-
tain a different value of g if the hydrodynamic limit of the process h holds in a
t
non-diffusive scaling. If this is the case, we say that we are in prensence of an
anomalousdiffusion. Inrecentworks,twosymmetricmodelsonwhichanomalous
diffusionoccurshavebeenintroduced. In[12],asimpleexclusionprocesswithde-
generated bonddisorder hasbeenintroduced. Thesimpleexclusion process isjust
a system of simple, symmetric random walks in Z, conditioned to never overlap.
Thisisprobablythemoststudiedexampleofasingle-filediffusion. Bonddisorder
isintroduced inthefollowingway. Let x ;x Z beasequence ofi.i.d.,positive
x
{ ∈ }
randomvariables. Assumethatthecommondistributionisonthedomainofattrac-
tionofana -stable law,a (0,1). Forsimplicity, take x 1foranyx Z. This
x
doesnotalterthetailbehav∈iorofx ,whichturnsouttobet≥herelevantpar∈t. Weput
x
4 M.JARA
awallofsizex betweensitesxandx+1,meaningthateachtimeaparticletriesto
x
jumpfromxtox+1orviceversa(andifthejumpisallowedbytheexclusionrule),
the jump is accomplished with probability x 1. Due to the heavy tails of x , the
x− x
dynamics isdramatically sloweddown. Infact, thecorrect timescaling isn1+1/a ,
which is always bigger than n2, and the hydrodynamic equation is ¶ u=¶ ¶ u,
t x W
whereW isana -stablesubordinatorcorrespondingtothescalinglimitofthewalls
and ¶ (also denoted by d/dW in the sequel) denotes the inverse of the Stieltjes
W
integral with respect toW. As we can see, the randomness of the environment is
sostrongthatitsurvivesinthelimit,andeventhehydrodynamicequationdepends
on the corresponding realization of the environment. This scaling is robust in the
sense that stronger interaction between particles does not lead to a different time
scaling[13].
Looking back to the heuristic formula for g , we obtain g = a /(1+a )<1/2
forany a (0,1), so thecurrent in this modelshould satisfy E[J (t)2] ta /(1+a )
0
∈ ∼
forlarget. Insteadofconsideringthesimpleexclusionprocesswithbonddisorder,
inthis article westudy the zero-range process with degenerated bond disorder. In
thiswayweemphasizethatthesubdiffusivebehaviorholdsregardlessofthedetails
of the local interaction. For the simple exclusion process with variable diffusion
coefficientdefinedin[14],theincrement process isprecisely azero-range process
with bond disorder. Therefore, a central limit theorem for the current through the
origin leads to a central limit theorem for the tagged particle in this last model,
whichfallsintothecategoryofsingle-filediffusions.
A second example of a symmetric system with non-diffusive hydrodynamic
limit is the zero-range process with long jumps [17]. In this system particles in-
teract between them only when they share positions, and the jump probability of
theunderlying random walksatisfies apowerlaw: p(z)=c/z1+a ,a (0,2). In
this case the correct scaling limit is na , which is always sma|ll|er than∈n2, and the
hydrodynamicequationisoftheform¶ tu=D a j (u),whereD a = ( D )a /2isthe
fractional Laplacian and j ()is a function encoding the interaction−b−etween parti-
cles. Inthiscase,theheurist·icformulagivesg =1/a ,whichisalwaysbiggerthan
1/2. Noticethatfora 1,thisformulagivesg 1. Itturnsoutthatfora 1,the
≤ ≥ ≤
current through the origin is not well defined, since in any time window [t,t+h]
there is an infinite number of particles crossing from one side of the origin to the
otherinthattimewindow.
Sincetheparticlejumpsarenotrestrictedtonearest-neighbors, itisnotpossible
to find a single-file diffusion for which the increment process falls into this class.
Therefore, for any g (0,1), we have a model for which the current of particles
should satisfy E[J (t)∈2] tg ,but forthetagged particle problem wehave amodel
0
forwhichE[x (t)2] tg∼onlywheng 1/2.
0
∼ ≤
Although the heuristic plan looks simple and it has been accomplished in the
diffusive case for many examples, anomalous diffusive behavior poses difficulties
thatareabsent fordiffusive systems. Themaintechnical difficulty reliesonestab-
lishingthefluctuation-dissipation theoremfortheprocessh . Inthesuperdiffusive
t
case, themainobstacle isthatthefractional Laplacian D a doesnotleaveinvariant
ANOMALOUSFLUCTUATIONS 5
the Schwartz space S(R) of test functions, and therefore the classical construc-
tion of generalized Ornstein-Uhlenbeck processes due to Holley and Stroock [16]
doesnot apply. Solutions of(1)canbeconstructed using theformalism ofGauss-
ianprocessforfairlygeneraldriving, non-positive operators L [3]. However,this
construction isnotsuitableforproving convergence when(asitisthecasehere)it
is not easy to show that the limiting process has Gaussian distributions regardless
of initial conditions. Section 2is basically arecall of [8] and [9], where powerful
methods have been developed to prove such convergence theorem, very much in
thespirit of[16]. Weinclude thissection withnonewresultsforthereader’s con-
venience, sincewearenotawareofprevious resultsapplying theseideastolattice
systems. In particular a notion for uniqueness of equation (1) is stated, and the
notionofintermediate spacesisintroduced.
In the subdiffusive case, even the definition of the corresponding Ornstein-
Uhlenbeck process poses a challenge. In fact, besides constant functions, there
are no smooth functions on the domain of the operator L = ¶ ¶ . Moreover,
W x W
for any two realizations of the subordinator W, the domains of the correspond-
ing operators L have in common only constant functions. The new material
W
on this article starts at Section 3. In Section 3 we construct a nuclear Fre´chet
space F = F which will serve as a test space in order to define the general-
W
ized Ornstein-Uhlenbeck process associated to the operator L . Our key input
W
is a compactness result for weighted-Sobolev spaces, very much in the spirit of
the definition ofthe usual weighted-Sobolev spaces in R. Ourconstruction works
for any increasing, unbounded function W, and could be of independent interest.
In Section 4 we give detailed definitions of the zero-range process wiht random
environment and with long jumps. We also state our main results concerning the
asymptotic behavior of the density and the current of particles. In Section 5 we
obtain the fluctuations of the density in the superdiffusive case. In Section 6 we
obtain the fluctuations of the density in the subdiffusive case. A key intermediate
resultistheso-calledenergyestimate,whichroughlysaysthatthespace-timefluc-
tuationsofagivenfunctioncanbeestimatedbytheDirichletformassociatedtothe
underlying random walk. This result holds true for any reversible system, regard-
less of the super or subdiffusive behavior of the system. The universality of this
estimate is more evident in Section 8, where weprove fluctuations for the current
ofparticles through theorigin inboth super and subdiffusive cases. Wefinishthis
article inSection 8by obtaining the fluctuations of atagged particle in the simple
exclusion process with variable diffusion coefficient, as a direct consequence of
the results in Section 7. We point out that all our results in the subdiffusive case,
appliesforanyprocessW(x),stochasticornot,suchthatlimn ¥ W(x)= ¥ and
→± ±
suchthattheenvironment hasaversionconverging almostsurelytoW(x).
2. GENERALIZED ORNSTEIN-UHLENBECK PROCESSES
In this section we give a precise definition to what we mean by a generalized,
orinfinite-dimensional, Ornstein-Uhlenbeck processandwestatesomeconditions
for uniqueness of such processes. All the material in this section has been taken
6 M.JARA
from [8] and [9]. The interested reader can find a more detailed exposition and
furtherapplications inthosearticles.
Remark 1. Throughout thisarticle, weusethedenomination “Proposition” forre-
sultthathavebeenprovedelsewhere. Wereservethedenomination“Theorem”for
originalresults.
2.1. Preliminary definitions. Let L2(Rd) be the Hilbert space of square inte-
grable functions j :Rd R. Let L :D(L) L2(Rd) L2(Rd) be the gen-
erator ofa strongly conti→nuous contraction sem⊆igroup S ;→t 0 inL2(Rd). Let
t
beanincreasingfamilyof(notnecessarilyfinit{e)nor≥ms}inL2(Rd)witha
n n
{co||m·|m| }onkernelF suchthat j 2= j (x)2dx. LetF L2(Rd)betheFre´chet
spacegenerated by0 ,|t|hat||i0s,thecompletion ofF⊆underthemetric
{||·||n}n R 0
d(j ,y )= (cid:229) 2 n j y 1 .
− n
|| − || ∧
n 0
≥ (cid:0) (cid:1)
Let us denote by F the topological dual of F. We can construct F in such
′ ′
a way that F L2(Rd) F and such that the inner product , in L2(Rd)
′
restricted toF⊆ F can⊆be continuously extended toa continuohu·s·biilinear form
0 0
, :F F ×R. WeassumethatthespaceF isnuclear, thatis,foranyn 0
′
h· ·i × → ≥
there exists m > n such that any -bounded set is a -compact set. In
m n
||·|| ||·||
what follows we will always consider a family of norms for which the set
n
j L2(Rd); j <+¥ is aHilbert space under ||,·a|l|though this point is
n n
{ ∈ || || } ||·||
not essential. Our objective is to describe some conditions under which existence
anduniqueness ofsolutions canbeestablished forthestochastic equation
dY =L Y dt+dZ , (2)
t ∗ t t
where Z is a given semimartingale in F and L is a given, maybe unbounded,
t ′
operator in L2(Rd). The canonical example of a nuclear, Fre´chet space is the
SchwartzspaceS(Rd)oftestfunctions inRd. Inthiscase,
1/2
j = (cid:229) (1+x2)n ¶ kj (x) 2dx ,
n
|| || Rd
(cid:26) k nZ (cid:27)
| |≤ (cid:0) (cid:1)
wherekdenotesamulti-index(k1,...,kd)with|k|=k1+···+kd and¶ k =(cid:213) i¶ xkii.
Thedual ofS(Rd)isthespace S (Rd)oftempered distributions and acommon
′
kernel for each norm is the set C¥ (Rd)of infinitely differentiable functions
n c
inRd ofcompactsupp||o·r|t|.
Foragiventopological space E,wedenote byD([0,T],E)thespace ofca`dla`g
trajectories in E. For simplicity, we consider a finite time interval [0,T]; results
for[0,¥ )willfollowfromstandard extension arguments. Fora<b,wedenoteby
C¥ (a,b) the space of C¥ functions in [a,b] with support contained in (a,b). The
space C¥ (a,b) is anuclear Fre´chet space withrespect to the topology ofuniform
convergence on compacts of the function and its derivatives of any order. Notice
thatfunctions inC¥ (a,b)vanishatt =a,b.
Letusdenote byF thetensor product F C¥ (0,T). Foraclearexposition
0,T
⊗
about tensor products, see [28]. This space is also aFre´chet space. We denote by
ANOMALOUSFLUCTUATIONS 7
F the topological dual of F . Now we state some technical lemmas which
0′,T 0,T
willbeusefulintheidentification ofsolutions of(2).
Lemma1. Foranytrajectory x inD([0,T],F ),letx˜ F bedefinedby
· ′ ∈ 0′,T
T
x˜,y = x ,y dt
t t
h i 0 h i
Z
for any y F . Then the mapping x x˜ is a continuous mapping from the
0,T
spaceD([0∈,T],F )intoF . · 7→
′ 0′,T
Lemma2. Letx beaprocessinD([0,T],F ),a.s.continuous att=T. Thenthe
′
·
distributions ofx,x˜determine eachother.
·
Definition1. Wesaythat afamily X ;f F ofintegrable random variables is
f
{ ∈ }
alinearrandom functional ifX islinearandcontinuous asafunction of f.
f
Lemma 3. Let X be a linear random functional. Then, there exists a unique
f
randomvariable X inF suchthat X,f =X a.s.forevery f F.
′ f
h i ∈
2.2. Weak formulation. When the operators L and S have the good taste of
t
leaving the space F invariant, there is a very intuitive notion of solutions for (2).
This is the case, for example, when L =D . We start doing some formal manip-
ulations. Here and below, the initial condition Y will be a random variable with
0
values in F . Apply (2) to a test function j , multiply by another test function
′
f C¥ ( d ,T), integrate the equation over time and perform an integration by
∈ −
partstoobtain
T T
Y ,j f (t)+ f(t)Lj dt = Y ,j f(0) + Z ,j f (t) dt.
t ′ 0 t ′
0 h i h i 0 h i
Z Z
Since wewant tocapture the initial distribution, wewillextend thespace F
0,T
a little bit. Notice that a function f F can be thought as a trajectory in F
0,T
whichvanishesatt=0,T. Takeanyd∈>0anddefineF d ,T asthetensorproduct
F C¥ ( d ,T). WedefineFT asthesetoftrajectories−inF d ,T restrictedtothe
inte⊗rval[0−,T]. Thesetrajectoriesalwaysvanishatt=T,andof−courseF F .
0,T T
⊆
Bylinearity and an approximation procedure, wecan extend the previous identity
totestfunctions y F . Fory F ,thisidentityreads
T T
∈ ∈
T ¶ T ¶
Y , y +Ly dt = Y ,y + Z , y dt. (3)
0 h t ¶ t t ti h 0 0i 0 h t ¶ t ti
Z Z
Definition 2. Wesay that aprocess Y in D([0,T],F )is a solution of (2) if L :
′
F F iscontinuous and(3)holdsfo·ranyy F .
T
→ ∈
Let us recall now the variation of parameters method to solve linear evolution
equations. The equation dY = L Y has as a solution the process Y = S Y ,
t ∗ t t t∗ 0
which means that Y is defined via the relation Y ,j = Y ,S j . Notice that
t t 0 t
Y is well defined in D([0,T],F ) if S : F hF isiconthinuous. iThe variation
t ′ t
of parameters method suggests to search for→a solution to (3) of the form Y =
t
S X ,whereX isasemimartingalesatisfying X =Y . Inthatcase,Y formally
t∗ t t 0 0 t
satisfies (3) if dX = S dZ . Since S is not well defined, the usual trick is
t ∗t t ∗t
− −
8 M.JARA
to multiply this expression by S to obtain a well defined relation for 0 t t :
S dX = S dZ . Integratingt∗′this expression, we obtain that S X =≤S Y≤ +′
t∗ t t∗ t t t∗ t t∗ 0
t′S dZ . ′−
0 t∗ s t
In−termsoftestfunctions, thisexpression gives
R
t
Y ,j = Y ,S j + dZ ,S j .
t 0 t s t s
h i h i 0 h − i
Z
Foratestfunctionoftheformj (x)f(t)withj F and f C¥ (0,T),thesame
∈ ∈
manipulations yield
T T T t
Y ,j f(t) dt = Y , f(t)S j dt + dt dZ ,S j f(t).
t 0 t s t s
0 h i h 0 i 0 0 h − i
Z Z Z Z
Thelasttermcanbewrittenas
T T T T
dZ , f(t)S j dt = Z , j f(s) f(t)S Lj dt ds
s t s s t s
0 h s − i − 0 h − i− s − i
Z Z Z Z
T T
= Z , f (t)S j dt ds.
s ′ t s
− 0 h s − i
Z Z
As before, using linearity and an approximation procedure we can extend this
identityforarbitrary testfunctions y F :
T
∈
T T T T ¶
Y ,y dt = Y , S y dt Z , S y dt ds. (4)
0 h t ti h 0 0 t t i− 0 h s s t−s¶ t t i
Z Z Z Z
Definition 3. A process Y in D([0,T],F ) is said to be an evolution solution of
′
(2)ifS :F F iscontin·uous, forj F thecurvet Y ,j iscontinuous in
t t
F,and(4)is→satisfiedforanyy F .∈ 7→h i
T
∈
NoticethatbyLemma2,relation(4)determinesthedistribution ofY. Nowwe
explainonwhichsensewehaveauniquesolution of(2): ·
Proposition1. LetY beadistributioninF . ThereexistsauniquesolutionY of
0 ′
(2)with initial condition Y if the conditions on L and S stated inDefinitions· 2,
0 t
3arefulfilled. ThissolutionisgivenbytheevolutionsolutiondefinedinDefinition
3.
Uniqueness follows from (4) and Lemma 2. Existence follows from standard
methodsforevolution equations. Wecalltheprocess Y thegeneralized Ornstein-
Uhlenbeck process of characteristics (L,Z ) and initia·l condition Y . The most
t 0
well known example for which this Proposition applies is the operator L = D .
In that case, it is well known that F =S(Rd) is left invariant by L and by S .
t
However, an important example, relevant for our purposes, which falls out of this
setting is the operator L = ( D )a /2, a (0,2), i.e., when L is the fractional
− − ∈
Laplacian in Rd. The fractional Laplacian is an integral operator which can be
writtenintheform
dy
Lj (x)=ca Rd yd+a j (x+y)−j (x)
Z | |
for a properly chosen constant ca >0. Due(cid:8)to the long-range(cid:9)integration, for any
positive function j S(Rd), the function Lj / S(Rd): decay at infinity fails,
∈ ∈
ANOMALOUSFLUCTUATIONS 9
as it is easily shown taking Fourier transforms. Therefore, for the case on which
L = ( D )a /2 a new interpretation of (2) is needed. This is the content of the
− −
followingsection.
2.3. The intermediate spaces. Throughout this section, we take F = S(Rd),
and therefore F =S (Rd). For p>0 and j C(Rd), the space of continuous
′ ′
functions inRd,define ∈
j p,¥ = sup j (x) (1+x2)p
|| ||
x Rd
∈ (cid:12) (cid:12)
andCp= j C(Rd); j p,¥ <+¥ . D(cid:12) efine(cid:12) also
{ ∈ || || }
C = j C(Rd); lim j (x)(1+x2)p=0 .
p,0
{ ∈ x ¥ }
| |→
Clearly, Cp, Cp,0 are Banach spaces with respect to the norm p,¥ . We also
||·||
have the continuous embeddings F ֒ Cp,¥ for p> 0 and Cp,0 ֒ L2(Rd) for
p>d/2. LetCp′,¥ bethetopologicald→ualofCp,¥ . Wehavethecha→inofinclusions
F ֒ C ֒ L2(Rd)֒ C ֒ F .
→ p,0 → → p′,0 → ′
RaDdoennomteeabsyur|e|s·|m|−pin,¥ Rthdesudcuhalthnaotrm(i1n+Cxp′2,¥). pmLe(dtxM) <p+ +be¥ t.heWseetgoivfeptoosiMtive+,
− p
the p-vague topology, which is the weakest topology that makes the mappings
R
j m ,j =: j dm continuousforeveryj C(Rd)ofcompactsupportandfor
Nj (o7→xti)ch=ea(l1so+itxh2a)tR−Lpeabsewsgeulel.mNeoatsicuerethiastinfoMrmp+∈f∈Mor pp+,>||dm /||2−.pD,¥ e=finRe(D1a+=x2)−(pmD()dax/)2.
and let St be the semigroup generated by D a . The following propositio−ns−can be
proved taking the representation of D a in Fourier space and the representation of
S inFourier-Laplace space.
t
Proposition2. Fort 0andd/2<p<(d+a )/2,S isaboundedlinearoperator
t
fromC intoitself,an≥dalsofromM+ intoitself.
p p
Proposition3. Letj Cpbesuchthatthelimitlimx ¥ j (x)(1+x2)exists. Then
t S j isacontinuou∈strajectoryinC . Foranym | |→M+,t S m isa p-vaguely
t p p t
7→ ∈ 7→
continuous trajectory inM+. Moreover, thereisapositive constantC suchthat
p T
Stj p,¥ CT j p,¥
|| || ≤ || ||
foranyj C andany0 t T.
p
∈ ≤ ≤
Proposition4. Themappingj F D a j Cp,0 iscontinuous.
∈ 7→ ∈
The idea now is to define a generalized version of what we mean by a solution
of (2). Notice that a priori expresions of the type Yt,D a j are not well defined,
since D a j does not belong to F. In the followinghtwo defiinitions, F denotes an
arbitraryFre´chetspace.
Definition4. WesaythataprocessY inD([0,T],F )isageneralizedsolutionof
′
the(2)ifthereexistsaBanachspaceV· offunctions inRd suchthat
i) F ֒ V ֒ L2(Rd)
→ →
10 M.JARA
ii) ThelinearmappingL :F ֒ V iscontinuous
iii) T Y ,Ly dt iswelldefin→edforanyy (D(L) V) C¥ (0,T)
iv) F0orhantyy atsiiniii),theidentity (3)holds·∈a.s. ∩ ⊗
R
·
Definition 5. Aprocess Y in D([0,T],F )is ageneralized evolution solution of
′
(2)ifthereexistsaBanach·spaceV suchthat
i) F ֒ V ֒ L2(Rd)
ii) S :→F →V iscontinuous foranyt [0,T]
t
iii) t S j→isacontinuous trajectory in∈V foranyj F
t
iv) T7→heright-hand sideof(4)iswelldefinedforanyy∈ V C¥ (0,T)
v) Foranyy asiniv),theidentity (4)holdsa.s. · ∈ ⊗
·
Proposition5. UndertheconditionsofDefinitions4,5,foranyinitialdistribution
Y in F there exists a unique generalized solution of (2), which is given by the
0 ′
generalized evolution solution definedinDefinition 5.
Notice that in particular, the stochastic equation dYt = D a∗Ytdt+dZt has a
uniquesolution foranyinitialdistribution Y inF .
0 ′
3. GENERALIZED DIFFERENTIAL OPERATORS
3.1. Thederivatived/dW. InSection2wehavediscussedexistenceandunique-
ness of generalized Ornstein-Uhlenbeck processes defined indual Fre´chet spaces.
Up to this point, the examples discussed so far (corresponding to the fractional
Laplacians D a ) do not require to take an abstract Fre´chet space, since the only
space considered was the space S (Rd) of tempered distributions. In this section
′
wegiveanexamplewheread-hocnuclearspacesneedtobeconstructed.
LetW :R Rbeastrictlyincreasingfunction. Byconvention, weassumethat
W isca`dla`g a→ndthatW(0)=0. Fortwogivenfunctions f,g:R R,wesaythat
→
g=df/dW iftheidentity
x
f(x)= f(0+)+ g(y)W(dy)
0
Z
holds for any x R, where the integral is understood as a Stieltjes integral, and
x means integra∈tion over the interval (0,x]. In the same spirit, we say that g=
0
d/dxdf/dW if
R
df x z
f(x)= f(0)+ (0) W(x) W(0) + W(dz) dyg(y)
dW − 0 0
Z Z
(cid:8) (cid:9)
for any x R. Wewill denote the operator d/dxd/dW by L . Notice that when
W
∈
the function W is differentiable andW (x)=0 for any x (that, W is a diffeomor-
′
phism),L =¶ (W (x) 1¶ ). Wesaythata6 function f isW-differentiableifthere
W x ′ − x
exists afunction g:R R such that g=df/dW. WhenW is adiffeomorphism,
→
thisnotionreducestousualdifferentiability.