Exponential asymptotics for intersection local times of stable processes and random walks
Xia Chen
a,1, Jay Rosen
b,∗,2aDepartment of Mathematics, University of Tennessee, Knoxville, TN 37996-1300, USA bDepartment of Mathematics, College of Staten Island, CUNY, Staten Island, NY 10314, USA Received 1 December 2003; received in revised form 4 August 2004; accepted 20 September 2004
Available online 5 February 2005
Abstract
We study large deviations for intersection local times ofpindependentd-dimensional symmetric stable processes of indexβ, under the conditionp(d−β) < d. Our approach is based on Feynman–Kac type large deviations, moment computations and some techniques from probability in Banach spaces.
2005 Elsevier SAS. All rights reserved.
Résumé
On étudie les temps locaux d’intersection de p processus β-stables d-dimensionnels indépendants, sous l’hypothèse p(d−β) < d. Notre approche est fondée sur les grandes déviations type Feynman Kac, des calculs de moments et quelques techniques de probabilités dans les espaces de Banach.
2005 Elsevier SAS. All rights reserved.
1. Introduction
LetX(t )be a non-degenerated-dimensional stable processes of indexβ. We assume thatX(t )is symmetric, i.e.X(t )= −d X(t ), but we do not assume it is spherically symmetric. Thus
E(eiλ·X(t ))=e−tψ (λ) (1.1)
* Corresponding author.
E-mail addresses: [email protected] (X. Chen), [email protected] (J. Rosen).
1 Research partially supported by NSF grant #DMS-0405188.
2 Research partially supported by grants from the NSF and from PSC-CUNY.
0246-0203/$ – see front matter 2005 Elsevier SAS. All rights reserved.
doi:10.1016/j.anihpb.2004.09.006
whereψ (λ)0 is continuous, positively homogeneous of degreeβ, i.e.ψ (rλ)=rβψ (λ)for eachr0,ψ (−λ)= ψ (λ), and for some 0< c < C <∞
c|λ|βψ (λ)C|λ|β. (1.2)
LetX1(t ), . . . , Xp(t )be independent copies ofX(t ). Their ranges will have a point in common aside from the initial point if and only ifp(d−β) < d, see [10]. Whenp(d−β) < d there is a random measureαp(ds1, . . . ,dsp) supported on
(t, . . . , tp)∈(R+)p; X1(t1)= · · · =Xp(tp)
. (1.3)
αp(ds1, . . . ,dsp)is called the intersection local time ofX1(t1), . . . , Xp(tp). Formally it can be written as αp(ds1, . . . ,dsp)=
Rd
p j=1
δ0
Xj(sj)−x dx
ds1· · ·dsp (1.4)
whereδ0(x)is the Dirac delta-‘function’ at 0.
In the cased=1 andβ >1, the intersection local time can be represented in terms of the spatialLp(R1)norms of the local timesLxt of the symmetric stable processes. In this case the large deviations and law of the iterated logarithm have been established for aαp(·)in recent work [4] for Brownian motion and [5] for the symmetric stable processes.
Whend=1 andβ1, or whend >1 for allβ, local time do not exist and we define the intersection local time αp(ds1, . . . ,dsp)as a limit. Lethbe a positive symmetric function in the Schwartz spaceS(Rd)with hdx=1.
Given >0, leth(x)=−dh(x/), and define the random measureαp,(·)on(Rp)+by αp,(ds1, . . . ,dsp)=
Rd
p j=1
h
Xj(sj)−x dx
ds1· · ·dsp. (1.5)
It can be shown that ifp(d−β) < dthe limit αp(B)= lim
→0+
αp,(B) (1.6)
exists a.s. and in allLm-norms for anym1 and any Borel set B⊂(Rp)+, andαp(·)is a measure supported on (1.3). We setαp,t =αp([0, t]p),αp,t,=αp,([0, t]p). For the convenience of the reader we show in Theo- rem 9 that a.s.αp,t =lim→0+αp,t, exists and is continuous int. Using the scaling property{X(t s); s0}=d t1/β{X(s); s0}of the stable process it is easy to check that
αp,t=tp−(p−1)d/βαp,1. (1.7)
We note that in the caseβ > d, where local times exist, we can also consider the analogue of (1.4) where we use a single process rather thanpindependent processes. Once again this is dealt with in [5]. However, in the case βd considered in this paper, where local times do not exist, if in (1.5) we use a single process rather thanp independent processes, the limit blows up. To get a non-trivial limit we must ‘renormalize’. Large deviations for the resulting limit in the casep=2 are discussed in [1,2].
To describe our results we need some further notation. For any functionf ∈L2(Rd)set Eψ(f, f )=:(2π )−d/2
Rd
ψ (λ)f (λ)ˆ 2dλ (1.8)
wheref (λ)ˆ denotes the Fourier transform off.Eψ(f, f )is the Dirichlet form of{X(t ); t0}. Let Fψ=
f ∈L2(Rd)| f2=1, Eψ(f, f ) <∞
. (1.9)
and
Mψ,p= sup
f∈Fψ Rd
f (x)2pdx 1/p
−Eψ(f, f )
. (1.10)
In the next section we show thatMψ,p<∞whenp(d−β) < d, and thatMψ,p can be expressed in terms of the best possible constant in a Gagliardo–Nirenberg type inequality.
We can now present our theorem describing the exponential asymptotics and large deviations forαp,t. Theorem 1. Assume thatp(d−β) < d.Then for anyλ >0
tlim→∞t−1logE(eλα
1/p p,t )=λ
pβ
pβ−d(p−1)p−pβd(p−1)−d(p−1)Mψ,p. (1.11)
Equivalently for anyh >0
tlim→∞t−1logP{α1/pp,t ht} = −hpβ/d(p−1)Aψ,p (1.12) where
Aψ,p=d(p−1) β
βp−d(p−1) βpMψ,p
βp−d(p−1)
d(p−1)
. (1.13)
Using the scaling (1.7) our theorem is equivalent to the fact that for anyh >0
tlim→∞t−1logP{αβ/d(pp,1 −1)ht} = −hAψ,p. (1.14)
Thus E(eλα
β/d(p−1)
p,1 )
<∞ ifλ < A−ψ,p1 ,
= ∞ ifλ > A−ψ,p1 . (1.15)
We next describe the law of the iterated logarithm forαp,t. Theorem 2. Assume thatp(d−β) < d. Then
lim sup
t→∞ t−(p−(p−1)d/β)(log logt )−(p−1)d/βαp,t=A−ψ,p(p−1)d/β a.s. (1.16) For the case of Brownian motion, i.e.β=2, these results were obtained by the first author in [3]. The methods of that paper depended heavily on the continuity of the Brownian path and the fact that the generator of Brownian motion, the Laplacian, is a local operator. In the course of overcoming the various problems associated with the stable process we have developed a new approach which greatly simplifies the proofs even for the case of Brownian motion.
We have also developed analogous results for random walks. Thus, considerS1(n), . . . , Sp(n) independent copies of ad-dimensional symmetric random walkS(n). We will assume that our random walks are in the domain of attraction of our nondegenerate symmetric stable processX(t )of indexβ, i.e.
S(n)
b(n)→X(1) (1.17)
withb(x)a function of regular variaton of index 1/β. Set Ip,n=
n
n1,...,nk=0 x∈Zd
p j=1
δ
Sj(nj), x
(1.18)
where
δ(y, x)=1 ifx=y,
0 otherwise (1.19)
is the usual Kroenecker delta.
Let{νn}represents a positive sequence satisfying
νn→ ∞ and νn/n→0. (1.20)
Here is our analogue of Theorem 1 describing the exponential asymptotics and moderate deviations forIp,n. Theorem 3. For anyλ >0
nlim→∞
1 νn
logEexp
λνn
nb(nνn−1)
d(p−1) p Ip,n1/p
=λ
pβ−d(p−1)pβ p−
d(p−1)
pβ−d(p−1)Mψ,p (1.21)
and for anyh >0
nlim→∞
1 νnlogP
Inhnpb(nν−n1)−d(p−1)
= −h
β
d(p−1)Aψ,p. (1.22)
This gives rise to the following LIL forIp,n. Theorem 4.
lim sup
n→∞ n−pb n
log logn d(p−1)
Ip,n=A−ψ,p(p−1)d/β a.s. (1.23)
2. Sobolev inequalities and Feynman–Kac formulae
Lemma 1. Ifp >1 andβ > d(p−1)/pthenFψ⊆L2p(Rd), and for anyδ >0
f22pCδf22+δEψ(f, f ) (2.1)
for someCδ<∞. In particular for anyλ >0 Mψ,p(λ)=: sup
f∈Fβ
λf22p−Eψ(f, f )
<∞. (2.2)
Proof of Lemma 1. Whenψ (λ)= |λ|β we write Fβ,Eβ, Mβ,p,d for Fψ,Eψ, Mψ,p. Because of (1.1) we have EψCEβ and hence it suffices to prove (2.1) whenψ (λ)= |λ|β. By the Hausdorff–Young inequality
f2p ˆf2p/(2p−1) (2.3)
wherefˆdenotes the Fourier transform off. We also have that for anyr >0 ˆf2p/(2p2p/(2p−−1)1)=
Rd
(r+ |λ|β)p/(2p−1)
(r+ |λ|β)p/(2p−1)f (λ)ˆ 2p/(2p−1)dλ r+ |λ|β−p/(2p−1)
(2p−1)/(p−1)r+ |λ|βp/(2p−1)f (λ)ˆ 2p/(2p−1)
(2p−1)/p. (2.4) Now r+ |λ|βp/(2p−1)f (λ)ˆ 2p/(2p−1)(2p−1)/p
(2p−1)/p=rf22+Eβ(f, f ) (2.5)
and
cr=:r+ |λ|β−p/(2p−1)(2p−1)/(p−1)
(2p−1)/(p−1)=
Rd
1
(r+ |λ|β)p/(p−1)dλ (2.6)
which is finite ifβ > d(p−1)/p, in which case we also have that limr→∞cr=0. Summarizing, f22pcr(p−1)/p
rf22+Eβ(f, f )
. (2.7)
This gives (2.1) on takingrsufficiently large. This completes the proof of our lemma. 2 SetMψ,p=Mψ,p(1).
Lemma 2. Ifp >1 andβ > d(p−1)/pthen κψ,p=:inf
Cf2pCf12−d(p−1)/pβ
Eψ1/2(f, f )d(p−1)/pβ
, ∀f ∈Fψ
<∞ (2.8)
and
Mψ,p(1)=pβ−d(p−1) d(p−1)
d(p−1)κψ,p2 pβ
pβ/(pβ−d(p−1))
. (2.9)
For anyλ >0
Mψ,p(λ)=λpβ/(pβ−d(p−1))Mψ,p. (2.10)
Proof of Lemma 2. To see that (2.8) is finite, note that if we setf (x)=sd/2g(sx), thenf2= g2,f22p= sd(p−1)/pg22pandEψ(f, f )=sβEψ(g, g)so that from (2.1) we obtain
g22pC
g22+sβEβ(g, g)
s−d(p−1)/p (2.11)
and the fact that (2.8) is finite follows on takingsβ= g22/Eβ(g, g). The same scaling establishes (2.10). Finally, (2.9) follows as in the proof of Lemma 8.2 of [3]. This completes the proof of our lemma. 2
3. Large deviations
In this section we show how to obtain our large deviation result for the intersection local time, Theorem 1, from a large deviation result for an approximate intersection local time together with exponential approximation.
Let X1(t ), . . . , Xp(t ) be independent d-dimensional symmetric stable processes of index β. We assume p(d−β) < d. Recall that the approximate intersection local time is defined by
αp,t,= t 0
· · · t 0
Rd
p j=1
h
Xj(sj)−x dx
ds1· · ·dsp (3.1)
and that
αp,t=lim
→0αp,t,. (3.2)
The following large deviation result forαp,t,will be proven in Section 4.
Theorem 5. Assume thatp(d−β) < d. Then for anyλ >0
tlim→∞
1
t logEexp{λα,t1/p} = sup
g∈Fψ
λ
Rd
(g2∗h)(x)p1/p
−pEψ(g, g)
. (3.3)
The following theorem on exponential approximation will be proven in Section 6.
Theorem 6. Assume thatp(d−β) < d. Then for anyλ >0, lim sup
→0
lim sup
t→∞
1
t logEexp
λ|αp,t,−αp,t|1/p
=0. (3.4)
Proof of Theorem 1. By Hölder’s inequality, Eexp{λa−1αp,t,1/p }
Eexp{λαp,t1/p}1/a Eexp
ba−1λ|αp,t,−αp,t|1/p1/b
(3.5) wherea, b >1 anda−1+b−1=1. Hence by Theorem 5
sup
g∈Fβ
λa−1
Rd
(g2∗h)(x)p1/p
−pEβ(g, g)
lim inf
t→∞
1
atlogEexp{λα1/pp,t} +lim sup
t→∞
1
btlogEexp
ba−1λ|αp,t,−αp,t|1/p
. (3.6)
Letting→0 and using Theorem 6 and (2.10) lim inf
t→∞
1
t logEexp{λα1/pp,t}a sup
g∈Fψ
λa−1
Rd
g(x)2pdx 1/p
−pEψ(g, g)
=ap−pβd(p−1)−d(p−1)(λa−1)pβ−pβd(p−1)Mψ,p. (3.7)
Lettinga→1, lim inf
t→∞
1
t logEexp{λα1/pp,t}p−pβd(p−1)−d(p−1)λ
pβ
pβ−d(p−1)Mψ,p. (3.8)
On the other hand, Eexp{λα1/pp,t}
Eexp{aλαp,t,1/p }1/a Eexp
bλ|αp,t,−αp,t|1/p1/b
. (3.9)
Therefore, using Theorem 5 lim sup
t→∞
1
t logEexp{λαp,t1/p} a−1 sup
g∈Fψ
aλ
Rd
(g2∗h)(x)pdx 1/p
−pEψ(g, g)
+lim sup
t→∞
1
bt logEexp
bλ|αp,t,−αt|1/p a−1 sup
g∈Fψ
aλ
Rd
g(x)2pdx 1/p
−pEψ(g, g)
+lim sup
t→∞
1
btlogEexp
bλ|αp,t,−αp,t|1/p . (3.10) Letting→0 and using Theorem 6 and (2.10)
lim sup
t→∞
1
t logEexp{λαp,t1/p}a−1 sup
g∈Fψ
aλ
Rd
g(x)2pdx 1/p
−pEψ(g, g)
=a−1p−
d(p−1) pβ−d(p−1)(λa)
pβ
pβ−d(p−1)Mψ,p. (3.11)
Lettinga→1, lim sup
t→∞
1
t logEexp{λαp,t1/p}p−
d(p−1) pβ−d(p−1)λ
pβ
pβ−d(p−1)Mψ,p. (3.12)
Combining what we have,
tlim→∞
1
t logEexp{λαp,t1/p} =p−pβd(p−1)−d(p−1)λ
pβ
pβ−d(p−1)Mψ,p, λ >0. (3.13)
Finally, by the Gärtner–Ellis theorem, Theorem 2.3.6 of [6],
tlim→∞
1
t logP{αp,t1/pht} = −sup
λ>0
λh−p−
d(p−1) pβ−d(p−1)λ
pβ
pβ−d(p−1)Mψ,p
= −hpβ/d(p−1)d(p−1) β
βp−d(p−1) βpMψ,p
βp−d(p−1)d(p−1)
. 2 (3.14)
4. Exponential asymptotics for the approximate intersection local time In this section, we fix >0 and write
L(t, x, )= t 0
h
X(s)−x
ds x∈Rd, t0. (4.1)
For each 1jp, letLj(t, x, )be the analogues ofL(t, x, )withX(t )being replaced byXj(t ).
For anyθ >0, using∗to denote convolution, write Mψ,p,(θ )= sup
f∈Fψ
θ
Rd
f2∗h(x)p
dx 1/p
−Eψ(f, f )
, (4.2)
Nψ,p,(θ )= sup
f∈Fψ
θ
Rd
f2∗h(x)p
dx 1/p
−pEψ(f, f )
. (4.3)
By the fact thatf2=1
Rd
f2∗h(x)p
dx sup
x∈Rd
f2∗h(x)p−1
h∞
d p−1
so that the functionsMψ,p,(·)andNψ,p,(·)are continuous for any fixed >0.
Theorem 7. For anyθ >0 and integersd1,p2,
tlim→∞
1
t logEexp
θ
Rd
Lp(t, x, )dx 1/p
=Mψ,p,(θ ), (4.4)
and
tlim→∞
1
t logEexp
θ
Rd
p j=1
Lj(t, x, )dx 1/p
=Nψ,p,(θ ). (4.5)
Proof of Theorem 7. We start with the following result based on the Feynman–Kac formula:
tlim→∞
1
t logEexp t
0
f X(s)
ds
sup
g∈Fψ
Rd
f (x)g2(x)dx−Eψ(g, g)
(4.6) wheref can be any bounded, measurable functionf onRd. This can be proven in a manner similar to our proof of (4.2) in [5], which deals with the one-dimensional case. (Alternatively, (4.6) can be derived by the methods used in [7], which also deals with the one-dimensional case. Using those methods one can show that we have equality in (4.6), although we will not need that.)
We begin by proving the lower bounds for (4.4) and (4.5). Notice that for anyr >0, and any measurable function f onRdwith|f|q=1 andf (x)=0 for|x|> r
{|x|r}
Lp(t, x, )dx 1/p
Rd
f (x)L(t, x, )dx= t
0
f ∗h
X(s)
ds. (4.7)
By (4.6) we have lim inf
t→∞
1
t logEexp
θ
{|x|r}
Lp(t, x, )dx 1/p
sup
g∈Fψ
θ
Rd
f∗h(x)g2(x)dx−Eψ(g, g)
= sup
g∈Fψ
θ
{|x|r}
f (x)g2∗h(x)dx−Eψ(g, g)
. (4.8)
Taking the supremum overf on the right-hand side, lim inf
t→∞
1
t logEexp
θ
{|x|r}
Lp(t, x, )dx 1/p
sup
g∈Fψ
θ
{|x|r}
g2∗h(x)pdx 1/p
−Eψ(g, g)
. (4.9)
In particular, lettingr= ∞gives the lower bound for (4.4).
To prove the lower bound for (4.5), we letr >0 be finite in (4.9). For any functionf (x), letRrf (x)be the restriction off (x)toBr, the closed ball of radiusrcentered at the origin. It follows from the definition (4.1) that L(t,·, )∞h∞tand|L(t, x, )−L(t, y, )|∇h∞t|x−y|for allx, y. Hence if we set
Ar,=
f∈C(Br) f (x)h∞, ∀x∈Br andf (x)−f (y)∇h∞|x−y|, ∀x, y∈Br
(4.10) we have that
1
tRrL(t,·, )∈Ar,. (4.11)
Note that by Ascoli’s LemmaAr, is a precompact subset ofC(Br)in the uniform norm, and a fortioriAr, is a precompact subset ofLp(Br). We useKr,to denote the closure ofAr,inLp(Br).
Consider the continuous, non-negative functionalΨ defined on(Lp(Br))p: Ψ (f1, . . . , fp)=1
p p j=1 {|x|r}
fj(x)pdx 1/p
−
{|x|r}
p j=1
fj(x)dx 1/p
. (4.12)
Clearly,Ψ ≡0 on the diagonal (f1, . . . , fp); f1= · · · =fp
.
Hence, for givenδ >0 and anyg∈Lp(Br)there exists ab=b(g, δ) >0 such that
Ψ (f1, . . . , fp)δ iffj∈B(g, b)for∀1jp (4.13)
whereB(g, b)stands for the open ball inLp(Br)with the centergand the radiusb. Therefore, Eexp
θ
{|x|r}
p j=1
Lj(t, x, )dx 1/p
e−δtE
exp
θ p
p j=1 {|x|r}
Lpj(t, x, )dx 1/p
; 1
tRrLj(t,·, )∈B(g, b), ∀1j p
=e−δt
E
exp θ
p {|x|r}
Lp(t, x, )dx 1/p
; 1
tRrL(t,·, )∈B(g, b) p
. (4.14)
Let{B(g1, b1), . . . , B(gN, bN)}be a finite sub-family of the open sets B
g, b(g, δ)
; g∈Kr, which coversKr,. Then by (4.11)
E
exp θ
p {|x|r}
Lp(t, x, )dx 1/p
N
i=1
E
exp θ
p
Br
Lp(t, x, )dx 1/p
; 1
tRrL(t,·, )∈B(gi, bi)
. (4.15)
Therefore, lim inf
t→∞
1
t log max
1iN
E
exp θ
p {|x|r}
Lp(t, x, )dx 1/p
; 1
tRrL(t,·, )∈B(gi, bi)
lim inf
t→∞
1 t logE
exp
θ p
{|x|r}
Lp(t, x, )dx 1/p
. (4.16)
Combining this with (4.9), (withθbeing replaced byp−1θ), and with (4.14) we have
lim inf
t→∞
1
t logEexp
θ
{|x|r}
p j=1
Lj(t, x, )dx 1/p
−δ+p sup
g∈Fψ
θ p {|x|r}
g2∗h(x)pdx 1/p
−Eψ(g, g)
. (4.17)
Lettingδ→0+andr→ ∞we obtain the lower bound for (4.5):
lim inf
t→∞
1
t logEexp
θ
Rd
p j=1
Lj(t, x, )dx 1/p
Nψ,p,(θ ). (4.18)
We now prove the upper bound for (4.4). We may lett→ ∞along the integers. Letm >0 be fixed and letGm be the discrete subgroup ofRdconsisting of vectors whose coordinates are integer multiples ofm. LetTmd be the quotient ofRdmoduloGmand letι:Rd→Tmdbe the canonical map. Then theTmd-valued processX∗(t )=ι
X(t ) is Markov process, the symmetric stable process on the torusTmd. Notice thatTmdbecomes a compact group under the induced distanced(x∗, y∗)= |x−y|, wherexandysatisfyι(x)=x∗,ι(y)=y∗andx, y∈ [0, m)d. Letλ(dx∗) be the Lebesgue (Haar) measure onTmdand write
L∗(t, x∗, )=
k∈Zd
L(t, x+mk, )= t
0
h∗
X∗(s)−x∗
ds, t0, x∗∈Tmd (4.19)
whereh∗is a function onTmddefined by h∗(x∗)=
k∈Zd
h(x+mk). (4.20)
Notice that
Rd
Lp(t, x, )dx=
k∈Zd
[0,m]d
Lp(t, x+mk, )dx
[0,m]d k∈Zd
L(t, x+mk, ) p
dx=
Tdm
L∗(t, x∗, )p
λ(dx∗). (4.21)
Using the methods we used in the proof of Lemma 6 of [5], which deals with the one-dimensional case, we can show that for any bounded, measurable functionf onTmd
tlim→∞
1
t logEexp t
0
f X∗(s)
ds
= sup
g∈Fψ,T d
m Tmd
f (x)g2(x)λ(dx)−Eψ,Tmd(g, g)
. (4.22)
Here
Eψ,Tmd(g, g)=:
λ∈(2πm)Zd
ψ (λ)g(λ)ˆ 2 1
md (4.23)
whereg(λ)ˆ denotes the Fourier transform ofg∈L2(Tmd), and Fψ,Tmd =
g∈L2(Tmd)| g2,Tmd =1, Eψ,Tmd(g, g) <∞
. (4.24)
We will use the notationf∗mg(x)= Td
mf (x−y)g(y)λ(dy)for convolution of functions onTmd. By (4.22),
tlim→∞
1
t logEexp
Tmd
f (x∗)L∗(t, x∗, )λ(dx∗)
= lim
t→∞
1
t logEexp t
0
f ∗mh,∗ X∗(s)
ds
= sup
g∈Fψ,T d
m Tmd
f∗mh,∗(x)g2(x)λ(dx)−Eψ,Tmd(g, g)
= sup
g∈Fψ,T dm Tmd
f (x)g2∗mh,∗(x)λ(dx)−Eψ,Tmd(g, g)
. (4.25)
From (4.20) and the fact thath∈S(Rd)we see thath∗∞and∇h∗∞are both finite. It follows from the definitions (4.19) thatL∗(t,·, )∞h∗∞t and|L∗(t, x, )−L∗(t, y, )|∇h∗∞t|x−y| for allx, y.
Hence if we set A∗,r,=
f ∈C(Tmd) f (x)h∗∞,∀x∈Tmdandf (x)−f (y)∇h∗∞|x−y| ,
∀x, y∈Tmd (4.26)
we have that 1
tL∗(t,·, )∈A∗,r,. (4.27)
Note as before that by Ascoli’s LemmaA∗,r,is a precompact subset ofC(Tmd)in the uniform norm, and a fortiori A∗,r,is a precompact subset ofLp(Tmd). We useK∗,r, to denote the closure ofA∗,r, inLp(Tmd). Letq >1 be the conjugate ofpand letδ >0 be fixed. By the Hahn–Banach Theorem and compactness, there are finitely many bounded functionsf1, . . . , fNin the unit sphere ofLq(TM)such that
Tmd
h(x)pλ(dx) 1/p
< max
1iN
Tmd
fi(x)h(x)λ(dx)+δ ∀h∈K∗,r,. (4.28)
In particular, using (4.27) E
exp
θ
Tmd
L∗(t, x, )p
λ(dx) 1/p
eδt N
i=1
Eexp
θ
Tmd
fi(x)L∗(t, x, )λ(dx)
. (4.29)
Hence by (4.25), lim sup
t→∞
1
t logEexp
θ
Tmd
L∗(t, x, )p
λ(dx) 1/p
δ+ max
1iN sup
g∈Fψ,T d
m
θ
Tmd
fi(x)g2∗mh,∗(x)λ(dx)−Eψ,Tmd(g, g)
δ+ sup
g∈Fψ,T d
m
θ
Tmd
g2∗mh,∗(x)p
dx 1/p
−Eψ,Tmd(g, g)
. (4.30)
In view of the relation (4.21) and Lemma 3 below, lettingδ→0 and thenm→ ∞ we obtain the upper bound for (4.4):
lim sup
t→∞
1
t logEexp
θ
Rd
Lp(t, x, )dx 1/p
Mψ,p,(θ ). (4.31)
By the inequality
Rd
p j=1
Lj(t, x, )dx 1/p
1
p p j=1 Rd
Lpj(t, x, )dx 1/p
(4.32)
we have Eexp
θ
Rd
p j=1
Lj(t, x, )dx 1/p
Eexp θ
p
Rd
Lp(t, x, )dx
1/pp
. (4.33)
From (4.31), (withθbeing replaced byp−1θ), we have the upper bound for (4.5):
lim sup
t→∞
1
t logEexp
θ
Rd
p j=1
Lj(t, x, )dx 1/p
Nψ,p,(θ ). 2 (4.34)
5. Localization
In this section we assumeβ <2. The case of Brownian motion was developed in [3]. By the Lévy–Khintchine formula
ψ (λ)=2
Rd
1−cos(λ·y)
Jd+β(y) dy (5.1)
whereJd+β(y)0 is a symmetric positively homogeneous function of degreed+βand we may assume that for some 0< c < C <∞
c|y|d+βJd+β(y)C|y|d+β. (5.2)
Using Parseval’s formula we find that Eψ(f, f )=
Rd
Rd
|f (y)−f (x)|2
Jd+β(y−x) dydx. (5.3)
Recall that for a functionh(x)onRd we seth∗(x)=
k∈Zdh(x+mk).
Lemma 3. Letp >1 and lethbe any non-negative measurable function satisfying h(x) C
1+ |x|d+ζ (5.4)
for someζ >0. Then for anyθ >0
lim sup
m→∞ sup
¯ g∈Fψ,T d
m
θ
Tmd
g¯2∗mh∗(x)p
dx 1/p
−Eψ,Tmd(g,¯ g)¯
sup
g∈Fψ
θ
Rd
g2∗h(x)p
dx 1/p
−Eψ(g, g)
. (5.5)
Proof of Lemma 3. Letg¯∈Fψ,Tmd be fixed. We may considerg¯to be extended toRd by periodicity. Then
[0,m]d
¯
g2(x)dx=1. (5.6)
and
¯
g2∗mh∗(x)=
[0,m]d
¯
g2(x−y)
k∈Zd
h(y+mk)dy= ¯g2∗h(x). (5.7)
Hence sup
x∈[0,m]dg¯2∗h(x)= sup
x∈[0,m]d
[0,m]d
¯
g2(y)
k∈Zd
h(x−y−mk)dyc
[0,m]d
¯
g2(y)dyc (5.8)
using (5.4) and (5.6).
We also have Eψ,Tmd(g,¯ g)¯ =
[0,m]d
Rd
| ¯g(x+y)− ¯g(x)|2
Jd+β(y) dydx (5.9)
where the last equality follows as in the proof of (5.3).
Note that by (5.6) we have
[0,m]d
¯
g2∗h(x)dx=
Rd
h(x)dx <∞. (5.10)
Throughout,cwill denote a constant which may depend onh. Write E=
d i=1
{0xi2√
m} ∪ {m−2√
mxim} .
By Lemma 3.4 in Donsker–Varadhan [7], there is ana∈Rdsuch that
E
¯
g2∗h(x+a)dx c
√m. We may assumea=0, i.e.,
E
¯
g2∗h(x)dx c
√m (5.11)
