Examples of renormalized SDEs

0.15 MB·

by  Yvain Bruned

Examples of renormalized SDEs
7
Y. Bruned, I. Chevyrev, and P. K. Friz
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2 January 6, 2017

n
a
J Abstract

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We demonstrate two examples of stochastic processes whose lifts to

] geometricroughpathsrequirearenormalisationproceduretoobtaincon-
R vergence in rough path topologies. Our first example involves a physical
P Brownian motion subject to a magnetic force which dominates over the
. frictionforcesinthesmallmasslimit. Oursecondexampleinvolvesalead-
h
lag process of discretised fractional Brownian motion with Hurst param-
t
a eter H ∈ (1/4,1/2), in which the stochastic area captures the quadratic
m
variationoftheprocess. Inbothexamples,arenormalisationofthesecond
[ iterated integralisneededtoensureconvergenceoftheprocesses, andwe

comment on how this procedure mimics negative renormalisation arising
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in thestudy of singular SPDEs and regularity structures.
v
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5 1 Introduction
1
1
0 Inrecentyears,the theoryofregularitystructures[7]hasbeenproposedtogive
. meaning to a wide class of singular SPDEs. A central feature of the theory is
1
0 the notionofrenormalisation,specifically “negativerenormalisation”[2], which
7 is required to obtain convergence of random models to a meaningful limit. It
1 is well-knownthat this procedure is inherit to the problem since naive approxi-
:
v mations of such equations typically fail to converge (with a number of notable
i exceptions, including a special variant of gKPZ [8]). The same feature thus
X
naturally appears in other solution theories which have been proposed to solve
r
suchequations,includingthetheoriesofparacontrolleddistributions[6]andthe
a
Wilsonian renormalisationgroup [9].
Viewingregularitystructuresasamultidimensionalgeneralisationofthethe-
oryof roughpaths [10], it is naturalto askhow renormalisationmanifests itself
inthe latter. As solutiontheoriesto singularS(P)DEs, bothsharethe common
feature that one must give meaning to often analytically ill-posed higher order
terms(iteratedintegrals)ofdistributions,whichistypicallydonethroughsome
stochastic means. However, a key difference in applications of rough paths to
SDEs and rough SPDEs [5] is that one can usually give meaning to such terms
as limits of iterated integralsof mollifications of the irregularnoise without the
need of renormalisation.
1
The purpose of this note is to demonstrate situations in roughpaths theory
whichfalloutsidethisusualsettingandforwhichrenormalisationisanecessary
feature. Specifically, we construct two stochastic processes whose lifts to geo-
metric rough paths fail to converge without a renormalisation procedure akin
to the one encountered in the theory of regularity structures.
The first is a physical Brownian motion subject to a magnetic force which
dominates over the friction forces in the small mass limit. This example builds
onthe work[4],where a similarsituationwasconsideringwith a constantmag-
netic field.
The second is a lead-lag process of a discretized path, which we take to
be fractional Brownian motion with Hurst parameter H ∈ (1/4,1/2]. The
stochastic area of this lead-lag process captures the quadratic variation of the
discretized path, and thus, as one can expect, the second iterated integral fails
toconvergeasthemeshofthediscretisationgoestozero(unlessH =1/2). This
exampleismotivatedfromasimilarHoffprocessconsideredforsemi-martinagles
in [3].
In both examples we demonstrate an explicit renormalisation procedure of
the second iterated integral under which the processes converge in rough path
topologies. These(diverging)counter-termsservepreciselythesamere-centring
role encountered in regularity structures (for a direct comparison, consider the
renormalisation of PAM [7, 6] where only one diverging term needs to be con-
sidered). In turn, rough differential equations driven by the renormalised and
unrenormalisedroughpathsarerelatedtooneanotherbytheadditionofdiverg-
ingterms,whichagainmimicsthesituationencounteredinsingularSPDEs. We
refer to the upcoming work [1] for a much more detailed study of this relation.
Acknowledgements. P.K.F. is partially supported by the European Re-
search Council through CoG-683166 and DFG research unit FOR2402. I.C.,
affiliated to TU Berlin when this project was commenced, was supported by
DFG researchunit FOR2402.
2 Magnetic field blow-up
Consider a physical Brownian motion in a magnetic field with dynamics given
by
mx¨=−Ax˙ +Bx˙ +ξ, x(t)∈Rd,
where A is a symmetric matrix with strictly positive spectrum (representing
friction), B is an anti-symmetric matrix (representing the Lorentz force due to
a magnetic field), and ξ is an Rd-valued white noise in time. We shall consider
the situation that A is constant whereas B is a function of the mass m.
We rewrite these dynamics as
1
dX = P dt, X =0,
t t 0
m
1
dP =− MP dt+dW , P =0,
t t t 0
m
2
where M = A−B, and we have chosen the starting point as zero simply for
convenience. Wefurthermoreintroducetheparameterε2 =mandwriteXε,Pε,
t t
and Mε =A−Bε to denote the dependence on ε.
WeareinterestedintheconvergenceoftheprocessesPε andMεXεinrough
path topologies. Let G2(Rd) and g2(Rd) denote the step-2 free nilpotent Lie
groupandLiealgebrarespectively. Letusalsowriteg2(Rd)=Rd⊕g(2)(Rd)for
the decomposition of g2(Rd) into the first and second levels, where we identify
g(2)(Rd) with the space of anti-symmetric d×d matrices.
For every ε>0, define the matrix
∞
Cε = e−Mεse−(Mε)∗sds,
Z0
and the element
1
vε =− (MεCε−Cε(Mε)∗)∈g(2)(Rd).
2
For any v ∈ g(2)(Rd), p ∈ [1,3), and p-rough path (Z ,Z ) ∈ G2(Rd)
s,t s,t
(where we ignore zeroth component 1), we define the translated rough path
T (Z ,Z ) by
v s,t s,t
T (Z ,Z )=(Z ,Z +(t−s)vε). (1)
v s,t s,t s,t s,t
Consider the G2(Rd)-valued processes
t
(Pε ,Pε )= Pε , Pε ⊗◦dPε ,
s,t s,t s,t s,r r
(cid:18) Zs (cid:19)
t
(Zε ,Zε )= MεXε , MεX ⊗d(MεXε) ,
s,t s,t s,t s,r r
(cid:18) Zs (cid:19)
and the canonical lift of the Brownianmotion W
t
(W ,W )= W , W ⊗◦dW ,
s,t s,t s,t s,r r
(cid:18) Zs (cid:19)
where the integrals in the definition of Pε and W are in the Stratonovich
s,t s,t
sense.
The following propositionestablishes the convergenceof the “renormalised”
paths Tvε(Psε,t,Pεs,t) and Tvε(Zsε,t,Zεs,t).
Theorem 1. Suppose that
lim|Mε|εκ =0 for some κ∈[0,1]. (2)
ε→0
Thenforanyα∈[0,1/2−κ/4)andq <∞,itholdsthatTvε(Pε,Pε)→(0,0)and
Tvε(Zε,Zε) → (W,W) in Lq and α-Ho¨lder topology as ε → 0. More precisely,
as ε→0, in Lq
|Pε | |Pε +(t−s)vε|
sup s,t + sup s,t →0.
|t−s|α |t−s|2α
s,t∈[0,T] s,t∈[0,T]
3
and
|Zε −W | |Zε +(t−s)vε−W |
s,t s,t s,t s,t
sup + sup →0.
|t−s|α |t−s|2α
s,t∈[0,T] s,t∈[0,T]
The restofthe sectionis devotedtothe proofofTheorem1whichbuildson
the proof of [4] Theorem 1.
We set Yε =Pε/ε and obtain that
dYε =−ε2MεYεdt+ε−1dW
t t t
dXε =ε−1Yεdt.
t t
For fixed ε, we introduce the Brownian motion W˜·ε =εWε−2· and consider
dY˜ε =−MεY˜εdt+dW˜ε.
t t t
Observe that we have the pathwise equalities
Yε =Y˜ε , (3)
· ε−2·
and since Yε =0, we have
0
t
Y˜ε = e−Mε(t−s)dW˜ε. (4)
t s
Z0
The dependence of Mε on ε is, by construction, only though Bε, the anti-
symmetricpartofMε. Inparticular,sincethesymmetricpartAstaysconstant
andhasstrictlypositivespectrum,itfollowsthatforsomeλ>0,Re(σ(Mε))⊂
(λ,∞) for all ε>0. In particular,
|e−τMε|
supsup <∞. (5)
e−λτ
τ>0ε>0
We see then that
sup|Cε|<∞
ε>0
and
sup sup E |Y˜ε|2 <∞. (6)
t
ε>00≤t<∞
h i
Lemma 2. There exists C >0 such that for all ε∈(0,1] and s,t∈[0,T]
1
E Yε 2 1/2 ≤C min{ε−1|t−s|1/2,1}
s,t 1
h(cid:12) (cid:12) i
and (cid:12) (cid:12)
t 2 1/2
E Yε⊗Yεdr−(t−s)Cε ≤C min{ε|t−s|1/2,|t−s|}.
r r 1
"(cid:12)Zs (cid:12) #
(cid:12) (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
4
Proof. The first inequality is clear from (3), (4) and (6). For the second,
from (4), we see that for every r>0, Y˜ε has distribution N(0,Cε) where
r r
r r
Cε = e−Mε(r−u)e−(Mε)∗(r−u)du= e−Mεue−(Mε)∗udu.
r
Z0 Z0
Hence Yrε =Y˜ε−2r has distribution N(0,Cεε−2r). Thus
t t t ε−2r
E Yε⊗Yεdr = Cε dr = e−Mεue−(Mε)∗ududr =:µε .
r r ε−2r s,t
(cid:20)Zs (cid:21) Zs Zs Z0
Observe that from (5)
t ∞
|µε −(t−s)Cε|≤ |e−Mεue−(Mε)∗u|dudr
s,t
Zs Zε−2r
t ∞
≤C e−2λududr
2
Zs Zε−2r
t
≤C e−2λε−2rdr
3
Zs
≤C min{ε2,|t−s|}
4
≤C min{ε|t−s|1/2,|t−s|}.
4
We now claim that
t 2
E Yε⊗Yεdr−µε ≤C min{ε2|t−s|,|t−s|2},
r r s,t 5
"(cid:12)Zs (cid:12) #
(cid:12) (cid:12)
(cid:12) (cid:12)
from which the(cid:12) conclusion follows. I(cid:12)ndeed, by Fubini and Wick’s formula
t 2
E Yε,iYε,jdr = E Yε,iYε,jYε,iYε,j drdu
r r r r u u
"(cid:18)Zs (cid:19) # Z[s,t]2
(cid:2) (cid:3)
= E Yε,iYε,j E Yε,iYε,j drdu
r r u u
Z[s,t]2
(cid:2) (cid:3) (cid:2) (cid:3)
+ E Yε,iYε,i E Yε,jYε,j drdu
r u r u
Z[s,t]2
(cid:2) (cid:3) (cid:2) (cid:3)
+ E Yε,iYε,j E Yε,jYε,i drdu
r u r u
Z[s,t]2
(cid:2) (cid:3) (cid:2) (cid:3)
≤(µε )2 +4 |E[Yε⊗Yε]|21{r≤u}drdu.
i,j s,t u r
Z[s,t]2
Observe that for r ≤u
E[Yε |Yε]=e−ε−2Mε(u−r)Yε.
u r r
5
and so
|E[Yε⊗Yε]|21{r≤u}≤C e−ε−22λ(u−r)|Cε |≤C e−ε−22λ(u−r).
u r 6 ε−2r 7
Thus
t 2 t 2
E Yε,iYε,jdr−(µε ) =E Yε,iYε,jdr −(µε )2
r r i,j s,t r r i,j s,t
"(cid:18)Zs (cid:19) # "(cid:18)Zs (cid:19) #
t t
≤C e−ε−22λ(u−r)dudr
8
Zs Zr
≤C min{ε2|t−s|,|t−s|2}
9
as claimed.
Lemma 3. There exists C >0 such that for all ε∈(0,1] and s,t∈[0,T]
10
Pε ≤C min{ε,|t−s|1/2}
s,t L2 10
and (cid:12)(cid:12) (cid:12)(cid:12)
(cid:12)(cid:12) (cid:12)(cid:12)
Pε +(t−s)vε ≤C |Mε|min{ε|t−s|1/2,|t−s|}
s,t L2 10
Proof. The (cid:12)fi(cid:12)rst inequality i(cid:12)s(cid:12)immediate from Lemma 2. For the second, we
(cid:12)(cid:12) (cid:12)(cid:12)
have
t
Pε =ε2 Yε ⊗◦dYε
s,t s,r r
Zs
t t 1
=− Yε ⊗MεYεdr+ε Yε ⊗dW + (t−s)I.
s,r r s,r r 2
Zs Zs
Since Yε ⊗MεYε = (Yε ⊗Yε)(Mε)∗ and we can directly verify that vε =
s,r r s,r r
Cε(Mε)∗− 1I, we have
2
t t
Pε +(t−s)vε =− Yε ⊗Yεdr−(t−s)Cε (Mε)∗+ε Yε ⊗dW .
s,t s,r r s,r r
(cid:18)Zs (cid:19) Zs
From Lemma 2, we see that
t
ε Yε ⊗dW ≤C min{ε|t−s|1/2,|t−s|}.
s,r r 11
(cid:12)(cid:12) Zs (cid:12)(cid:12)L2
(cid:12)(cid:12) (cid:12)(cid:12)
Furthermore,(cid:12)b(cid:12)y Fubini and W(cid:12)ic(cid:12)k’s formula, we can readily show
(cid:12)(cid:12) (cid:12)(cid:12)
t
Yε⊗Yεdr ≤C min{ε|t−s|1/2,|t−s|}.
s r 12
(cid:12)(cid:12)Zs (cid:12)(cid:12)L2
(cid:12)(cid:12) (cid:12)(cid:12)
It now follows(cid:12)f(cid:12)rom Lemma 2(cid:12)t(cid:12)hat
(cid:12)(cid:12) (cid:12)(cid:12)
Pε +(t−s)vε ≤C |Mε|min{ε|t−s|1/2,|t−s|}.
s,t L2 13
(cid:12)(cid:12) (cid:12)(cid:12)
(cid:12)(cid:12) (cid:12)(cid:12)
6
Lemma 4. There exists C >0 such that for all ε∈(0,1] and s,t∈[0,T]
14
E |Zε −W |2 1/2 ≤C min{ε,|t−s|1/2}
s,t s,t 14
and (cid:2) (cid:3)
E Zε +(t−s)vε−W 2 1/2 ≤C |Mε|min{ε|t−s|1/2,|t−s|}.
s,t s,t 14
h(cid:12) (cid:12) i
Proof. T(cid:12)he first inequality follow(cid:12)s from Zε = W −εYε and Lemma 2. For
s,t s,t s,t
the second, we have
t t t
Zε ⊗dZε = Zε ⊗dW −ε Zε⊗dYε−Zε⊗Yε
s,r r s,r r r r s s,t
Zs Zs (cid:18)Zs (cid:19)
t t
= Zε ⊗dW −ε Zε⊗Yε− dZ ⊗Yε−Zε⊗Yε
s,r r t t r r s t
Zs (cid:18) Zs (cid:19)
t t
= Zε ⊗dW −εZε ⊗Yε+ MεYε⊗Yεdr.
s,r r s,t t r r
Zs Zs
We see that
t t 2
Zε ⊗dW − W ⊗dW ≤C min{ε2|t−s|,|t−s|2}.
s,r r s,r r 15
(cid:12)(cid:12)Zs Zs (cid:12)(cid:12)L2
(cid:12)(cid:12) (cid:12)(cid:12)
Furth(cid:12)e(cid:12)rmore, by Fubini and Wick’s form(cid:12)(cid:12)ula, we can readily show
(cid:12)(cid:12) (cid:12)(cid:12)
t 2
εZε ⊗Yε 2 = MεYε⊗Yεdr ≤C |Mε|min{ε2|t−s|,|t−s|2}.
s,t t L2 r t 16
(cid:12)(cid:12)Zs (cid:12)(cid:12)L2
(cid:12)(cid:12) (cid:12)(cid:12) (cid:12)(cid:12) (cid:12)(cid:12)
F(cid:12)i(cid:12)nally, by Le(cid:12)m(cid:12) ma 2(cid:12)(cid:12) (cid:12)(cid:12)
(cid:12)(cid:12) (cid:12)(cid:12)
t
MεYε⊗Yεdr−(t−s)MεCε| ≤C |Mε|min{ε|t−s|1/2,|t−s|}.
r r 17
(cid:12)(cid:12)Zs (cid:12)(cid:12)L2
(cid:12)(cid:12) (cid:12)(cid:12)
It(cid:12)f(cid:12)ollows that (cid:12)(cid:12)
(cid:12)(cid:12) (cid:12)(cid:12)
1
Zε −W −(t−s)(MεCε− I) ≤C |Mε|min{ε|t−s|1/2,|t−s|}.
s,r s,t 2 18
(cid:12)(cid:12) (cid:12)(cid:12)L2
(cid:12)(cid:12) (cid:12)(cid:12)
W(cid:12)(cid:12)e(cid:12)(cid:12)candirectlyverifyvε =−MεCε+1(cid:12)(cid:12)(cid:12)(cid:12)I,fromwhichtheconclusionfollows.
2
Proof of Theorem 1. Observe that condition (2) implies that
lim|Mε|ε=0.
ε→0
FromLemmas3and4,alongwithGaussianchaos,wethusobtainthepointwise
convergence as ε→0 for any q <∞ and s,t∈[0,T] in Lq
|Pε |+|Pε +(t−s)vε|1/2 →0
s,t s,t
7
and
|Zε −W |+|Zε +(t−s)vε−W |1/2 →0.
s,t s,t s,t s,t
Furthermore, since min{ε|t−s|1/2,|t−s|} ≤ εκ|t−s|1−κ/2 for all κ ∈ [0,1],
condition (2), Lemmas 3 and 4, and Gaussian chaos imply that for any q <∞
there exists C >0 such that for all s,t∈[0,T] and
q
sup E |Pε |q ≤C |t−s|q/2,
s,t q
ε∈(0,1]
(cid:2) (cid:3)
sup E |Zε −W |q ≤C |t−s|q/2
s,t s,t q
ε∈(0,1]
(cid:2) (cid:3)
and
sup E |Pε +(t−s)vε|q ≤C |t−s|q(1−κ/2),
s,t q
ε∈(0,1]
(cid:2) (cid:3)
sup E |Zε +(t−s)vε−W |q ≤C |t−s|q(1−κ/2).
s,t s,t q
ε∈(0,1]
(cid:2) (cid:3)
Applying Theorem A.13 of [5] completes the proof.
3 Rough lead-lag process
Consider a path X :[0,1]7→Rd. Let n ≥1 be an integer and write for brevity
Xn =X . Consider the piecewise linear path X˜n :[0,1]7→R2d defined by
i i/n
X˜n =(Xn,Xn),
2i/2n i i
X˜n =(Xn,Xn ),
(2i+1)/2n i i+1
and linear on the intervals 2i,2i+1 and 2i+1,2i+2 for all i = 0,...,n−1.
2n 2n 2n 2n
Note that this is a variant of the Hoff process considered in [3].
Denote by X˜n = exp(X˜(cid:2)n +An(cid:3)) the(cid:2)level-2 lift(cid:3)of X˜n, where An is the
s,t s,t s,t s,t
(2d)×(2d) anti-symmetric L´evy area matrix given by
1 t t
An = X˜n ⊗dX˜n− X˜n ⊗dX˜n .
s,t 2 s,r r s,r r
(cid:18)Zs Zs (cid:19)
Let H ∈ (0,1) and consider a fractional Brownian motion BH with covari-
ance R(s,t) = 1(t2H +s2H −|t−s|2H). Let X : [0,1]7→ Rd be d independent
2
copies of BH.
Recallthe definition of T from(1). We areinterestedin the convergencein
v
roughpathtopologiesofTv˜n(X˜n)wherev˜n ∈g(2)(R2d)isappropriatelychosen.
Define the (diagonal) d×d matrix
1 n−1 n1−2H
vn = E (Xn −Xn)⊗(Xn −Xn) = I,
2 k+1 k k+1 k 2
" #
k=0
X
8
and the anti-symmetric (2d)×(2d) matrix
0 −vn
v˜n = ∈g(2)(R2d).
vn 0
(cid:18) (cid:19)
Finally, consider the path X˜ = (X,X) : [0,1] 7→ R2d, its canonically defined
L´evyareaA(whichexistsfor1/4<H ≤1),anditslevel-2liftX˜ =exp(X˜+A).
The following is the main result of this subsection.
Theorem 5. Suppose 1/4 < H ≤ 1/2. Then for all α ∈ [0,H) and q < ∞,
it holds that Tv˜n(X˜n) → X˜ in Lq and α-Ho¨lder topology. More precisely, as
n→∞, in Lq
|X˜n −X˜ | |An +(t−s)v˜n−A |
s,t s,t s,t s,t
sup + sup →0.
|t−s|α |t−s|2α
s,t∈[0,T] s,t∈[0,T]
The rest of the section is devoted to the proof of Theorem 5. We first state
two lemmas which are purely deterministic.
Let Yn : [0,1] 7→ Rd be the piecewise linear interpolation of X over the
partition 0, 1,...,n−1,1 , let Y˜n =(Yn,Yn):[0,1]7→R2d, andlet Yn be the
n n
L´evy area of Y˜n.
(cid:0) (cid:1)
Lemma 6. Let s∈[m,m+1] and t∈[k,k+1] with s<t, and define
n n n n
m+1
∆ =n ∧t−s |Xn −Xn|,
1 n m+1 m
(cid:18) (cid:19)
∆ =|Xn−Xn | if k >m, 0 if k =m
2 k m+1
k
∆ =n t− ∨s |Xn −Xn|.
3 n k+1 k
(cid:18) (cid:19)
There exists a constant C > 0 such that for all n ≥ 1 and 0 ≤ s < t ≤ 1, it
1
holds that
|X˜n −Y˜n|≤C (∆ +∆ ).
s,t s,t 1 1 3
and, if k >m, |An −An | and |Yn −Yn | are bounded above by
s,t mn+1,nk s,t mn+1,nk
C ∆2+(∆ +∆ +∆ )∆ +∆ ∆ .
1 1 1 2 3 3 1 2
(cid:0) (cid:1)
Proof. Direct calculation and triangle inequality.
The second part of the above lemma essentially allows us to work over the
partition 0, 1,...,n−1,1 , on which computations are easier.
n n
Lemma 7(cid:0). Suppose 0≤m(cid:1) ≤k ≤n.
1) For all pairs 1≤i,j ≤d and d+1≤i,j ≤2d
i,j i,j
An = Yn .
mn,nk mn,nk
(cid:16) (cid:17) (cid:16) (cid:17)
9
2) For all 1≤i≤d<j ≤2d
k−1
i,j i,j 1
An = Yn − (Xn,i −Xn,i)(Xn,j −Xn,j)
mn,nk mn,nk 2 r+1 r r+1 r
(cid:16) (cid:17) (cid:16) (cid:17) rX=m
Proof. DenoteX˜n =(Mn,Nn),sothatMnisthelagcomponent,andNnisthe
lead. ThefirstequalityisclearsinceMn andNn aresimplyreparametrisations
of Yn over the interval [m, k]. For the second, observe that
n n
k/n k−1
Mn,i dNn,j = (Xn,i−Xn,i)(Xn,j −Xn,j)
m/n,r r r m r+1 r
Zm/n r=m
X
and
k/n k−1
Nn,j dMn,i = (Xn,j −Xn,j)(Xn,i −Xn,i).
m/n,r r r+1 m r+1 r
Zm/n r=m
X
Remark now that the signature of Yn over [m, k] is
n n
eXm+1−Xm...eXk−Xk−1,
so that a calculation with the CBH formula gives
k−1
i,j 1
Yn = (Xn,i−Xn,i)(Xn,j −Xn,j)−(Xn,j−Xn,j)(Xn,i −Xn,i).
m/n,k/n 2 r m r+1 r r m r+1 r
(cid:16) (cid:17) rX=m
Using the fact that
i,j 1 k/n k/n
An = Mn,i dNn,j − Nn,j dMn,i ,
m/n,k/n 2 m/n,r r m/n,r r
(cid:16) (cid:17) Zm/n Zm/n !
the conclusion readily follows.
We now return to the specific case that X : [0,1] 7→ Rd is given by d-
independent copies of a fractionalBrownianmotion with Hurst parameter H ∈
(0,1). In particular, this implies that for all s,t,a,b∈[0,1]
1
E (Xi−Xi)(Xj −Xj) =δ (|t−a|2H+|s−b|2H−|t−b|2H−|s−a|2H). (7)
t s b a i,j2
h i
Consider the (2d)×(2d) anti-symmetric matrix
Pn =An −Yn +(t−s)v˜n.
s,t s,t s,t
Lemma 8. There exists C >0 such that for all H ≤1/2, n≥1, and 0≤m≤
2
k ≤n
(k−m)1/2
Pn ≤C .
m/n,k/n L2 2 n2H
(cid:12)(cid:12) (cid:12)(cid:12)
(cid:12)(cid:12) (cid:12)(cid:12)
(cid:12)(cid:12) (cid:12)(cid:12)
10