Heat Trace of non-local operators

0.16 MB·English

by  Rodrigo Bañuelos

HEAT TRACE OF NON-LOCAL OPERATORS
RODRIGOBAN˜UELOSANDSELMAYILDIRIMYOLCU
ABSTRACT. This paper extends results of M. van den Berg on two-term asymptotics for the trace of
Scho¨dingeroperatorswhentheLaplacianisreplacedbynon-local(integral)operatorscorrespondingto
rotationallysymmetricstableprocessesandothercloselyrelatedLe´vyprocesses.
2
1
0
2

n CONTENTS
a
J 1. Introduction 1

0 2. Stableprocesses, statementofresults 2
1
3. ProofofTheorems2.1and2.2 5

4. AlternativeproofsofTheorems2.1and2.2 9
]
P 5. Extensiontoothernon-local operators 11
S
References 15
.
h
t
a
m
1. INTRODUCTION
[

ThereisanextensiveliteratureoftraceformulæforheatkernelsoftheLaplaciananditsSchro¨dinger
1
v perturbations in spectral and scattering theory. For instance, van den Berg [6] and later Ban˜uelos
1 and Sa´ Barreto [5], explicitly compute several coefficients in the asymptotic expansion of the trace
7 of the heat kernel of the Schro¨dinger operator ∆ + V as t 0 for potentials V (Rd), the
1 − ↓ ∈ S
2 class of rapidly decaying functions at infinity. In particular, in [5] a general formula is obtained for
. these coefficients by using elementary Fourier transform methods. For applications of these results
1
0 to problems in scattering theory, see [5] and references therein. The results in [5] were extended by
2 Donnelly[20]tocertaincompactRiemannianmanifold. Heatasymptoticresultshavealsobeenwidely
1
used in statistical mechanics, for details we refer the reader to the article [26] of E. H. Lieb on the
:
v second virial coefficient of a hard-sphere gas at low temperatures, and to the article [28] of M. D.
i
X Penrose,O.PenroseandG.Stellonthestickyspheresinquantum mechanics. Wealsoreferthereader
r toDatchev and Hezari’s overview article [24]for various other related spectral asymptotic results and
a
applications.
For non-local (integral) operators which arise by replacing the Brownian motion with other Le´vy
processes, many questions concerning spectral asymptotics are wide open at this point. These in-
clude, for example, the Weyltwo-term asymptotics forthe spectral counting function ofthe fractional
Laplacian with Dirichlet boundary conditions and the McKean-Singer [27] result involving the Euler
characteristic of the domain in the third-term asymptotics of the trace of the heat semigroup. On the
other hand, a two-term trace asymptotics involving the volume of the domain and the surface area of
theboundary isproved inBan˜uelos andKulczycki [3](forC1,1 domains) and inBan˜uelos, Kulczycki
and Siudeja [4] (for Lipschitz domains) for the fractional Laplacian associated with symmetric sta-
ble processes. These results are in parallel to the results of van den Berg [7] and Brown [10] for the
R.Ban˜uelosissupportedinpartbyNSFGrant#0603701-DMS.
1
2 RODRIGOBAN˜UELOSANDSELMAYILDIRIMYOLCU
Laplacian. Forfurther recentworkontheDirichletcaseindomainsofEuclidean space,seeFrankand
Geisinger, [22] and [23]. The purpose of this paper is to obtain analogues of the van den Berg result
in [6] for the fractional Laplacian ∆α/2 associated with symmetric α-stable processes, 0 < α < 2,
in Rd and for other closely related non-local operators corresponding to sums of stable processes and
therelativistic Brownianmotion. TheseareallLe´vyprocesses whichareobtained bysubordination of
Brownianmotion.
2. STABLE PROCESSES, STATEMENT OF RESULTS
Before we state our results precisely, we recall the basic definitions and elementary notions related
to stable processes. Let X be the d-dimensional symmetric α–stable process of order α (0,2]
t
∈
in Rd. The process X has stationary independent increments and its transition density p(α)(x,y) =
t t
p(α)(x y), t > 0, x,y Rd,isdeterminedbyitsFouriertransform (characteristic function)
t − ∈
(2.1) e−t|ξ|α =E(eiξ·Xt)= eiξ·yp(α)(y)dy, t > 0, ξ Rd.
Rd t ∈
Z
Denoting by Px and Ex the probability and expectation, respectively, of this process starting at x we
havethatanyBorelsubsetB Rd,x Rd,t > 0,
⊂ ∈
Px(X B)= p(α)(x y)dy,
t ∈ t −
ZB
where
1
(2.2) p(α)(x) = e−ix·ξe−t|ξ|αdξ.
t (2π)d Rd
Z
Herewecanalsowrite
(α) ∞ 1 −|x|2 α/2
(2.3) pt (x) = (4πs)d/2e 4s ηt (s)ds,
Z0
α/2
where η (s)isthe density for the α/2-stable subordinator. While explicit formula for the transition
t
density ofsymmetric α-stable processes areonlyavailable forα = 1(the Cauchyprocess) and α = 2
(theBrownianmotion),theseprocessessharemanyofthebasicproperties ofBrownianmotion. Italso
(α)
followstrivially from(2.3)thatp (x)isradial,symmetricanddeceasing inx. Thusexactlyasinthe
t
Brownianmotioncasewehave
(2.4) p(α)(x) t−d/αp(α)(0).
t ≤ 1
In fact, if we denote by ω the surface area of the unit sphere in Rd we can compute p(α)(0) more
d 1
explicitly.
1 ω ∞
p(1α)(0) = (2π)d ZRde−|ξ|αdξ = (2π)ddα Z0 e−ss(αd−1)ds
ω Γ(d/α)
(2.5) = d .
(2π)dα
Throughout thispaperwewilldealwithradialtransition functions andusethenotation
(α) (α)
p (x,y) = p (x y).
t t −
HEATTRACEOFNON-LOCALOPERATORS 3
(α) (α)
In particular, p (x,x) = p (0). Of importance for us in this paper is the scaling property of these
t t
processes. More precisely, by a simple change of variables in (2.2), we see that these processes are
self-similar withscaling
(2.6) p(α)(x,y) = t−d/αp(α)(t−1/αx,t−1/αy).
t 1
(α)
The transition densities p (x) satisfy the following well-known two sided inequality valid for all
t
x Rd andt > 0,
∈
t t
(2.7) C−1 t−d/α p(α)(x) C t−d/α ,
α,d ∧ x d+α ≤ t ≤ α,d ∧ x d+α
(cid:18) | | (cid:19) (cid:18) | | (cid:19)
wherethe constant C only depends onαand d. Hereand throughout thepaper weuse thenotation
α,d
a b = min a,b anda b = max a,b foranya,b R.
∧For rapidl{y dec}aying fu∨nctions f { (}Rd), we have∈the semigroup of the stable processes defined
∈ S
viathetheFourierinversion formulaby
T f(x) = Ex[f(X )] = E0[f(X +x)]
t t t
= f(x+y)p (dy) = p f(x)
t t
Rd ∗
Z
1
= eix·ξe−t|ξ|αf(ξ)dξ.
(2π)d Rd
Z
b
By differentiating this at t = 0 we see that its infinitesimal generator is ∆α/2 in the sense that
\
∆α/2f(ξ) = ξ αf(ξ). Thisisanon-local operator suchthatforsuitabletestfunctions, including all
functions inf−| C| ∞(Rd),wecandefineitastheprinciplevalueintegral
∈ 0
b
f(x+y) f(x)
(2.8) ∆α/2f(x)= lim − dy,
Ad,αǫ→0+Z{|y|>ǫ} |y|d+α
where
Γ d−α
= 2 .
Ad,α 2απd/2Γ α
(cid:0) (cid:1)2
(cid:0) (cid:1)
Wewill denote by H the fractional Laplacian operator ∆α/2,α (0,2] and by H its Schro¨dinger
0
∈
perturbation H = ∆α/2 + V, where V L∞(Rd). We let e−tH and e−tH0 be the associated heat
∈
semigroups and let pH and p(α) denote their corresponding transition densities (heat kernels). In fact,
t t
(α)
p isjustasin(2.1)andtheFeynman-Kacformulagives
t
(2.9) pHt (x,y) = p(tα)(x,y)Ext,y e−R0tV(Xs)ds ,
(cid:16) (cid:17)
4 RODRIGOBAN˜UELOSANDSELMAYILDIRIMYOLCU
whereEt istheexpectation withrespecttothestableprocess (bridge)starting atxconditioned tobe
x,y
aty attimet. Themainobjectofstudyinthispaperisthetracedifference
Tr(e−tH e−tH0) = (pH(x,x) p(α)(x,x))dx
− Rd t − t
Z
(2.10) = p(tα)(0) RdExt,x e−R0tV(Xs)ds −1 dx
Z (cid:16) (cid:17)
= t−d/αp(1α)(0) RdExt,x e−R0tV(Xs)ds−1 dx,
Z (cid:16) (cid:17)
(α)
wherep (0)isthedimensional constant givenbytherighthandsideof(2.5).
1
Before we go further let us observe that this quantity is well defined for all t > 0, provided V
L∞(Rd) L1(Rd). Indeed, theelementary inequality ez 1 z e|z| immediately givesthat ∈
∩ | − | ≤| |
t
(cid:12)(cid:12)ZRdExt,x(cid:16)e−R0tV(Xs)ds−1(cid:17)dx(cid:12)(cid:12) ≤ etkVk∞ZRdExt,x(cid:18)Z0 |V(Xs)|ds(cid:19)dx.
However, (cid:12) (cid:12)
(cid:12) (cid:12)
t t
(2.11) Et V(X )ds = Et V(X )ds
x,x | s | x,x| s |
(cid:18)Z0 (cid:19) Z0
t p(α)(x,y)p(α)(y,x)
= s t−s V(y)dyds.
Z0 ZRd p(tα)(x,x) | |
(α) (α)
TheChapman–Kolmogorov equations andthefactthatp (x,x) = p (0,0)givethat
t t
(α) (α)
p (x,y)p (y,x)
s t−s dx = 1
(α)
Rd p (x,x)
Z t
andhence
t
(2.12) Et V(X )ds dx = t V .
ZRd x,x(cid:18)Z0 | s | (cid:19) k k1
Itfollowsthenthat
(2.13) Tr(e−tH e−tH0) t−d/α+1p(α)(0) V etkVk∞,
− ≤ 1 k k1
validforallt > 0andall(cid:12)potentials V L∞(cid:12)(Rd) L1(Rd).
Thepreviousargumen(cid:12)t alsoshowsth∈atfor(cid:12)allpo∩tentials V L∞(Rd) L1(Rd),
∈ ∩
∞ ( 1)k t k
(2.14) Tr e−tH e−tH0 = p(α)(0) − Et V(X )ds dx,
− t k=1 k! ZRd x,x(cid:18)Z0 s (cid:19)
(cid:0) (cid:1) X
wherethesumisabsolutely convergent forallt > 0.
WehavethefollowingtwoTheoremswhichparalleltheresultsinvandenBerg[6]fortheLaplacian.
Note however, that here (ii) in Theorem 2.1 (as well as Theorem 2.2) is more general as we do not
assumethatthepotential isnonnegative.
Theorem2.1. (i)LetV : Rd ( ,0],V L∞(Rd) L1(Rd). Thenforallt > 0
→ −∞ ∈ ∩
1
(2.15) pα(0)t V Tr(e−tH e−tH0) p(α)(0) t V + t2 V V etkVk∞ .
t k k1 ≤ − ≤ t k k1 2 k k1k k∞
(cid:18) (cid:19)
HEATTRACEOFNON-LOCALOPERATORS 5
Inparticular
Tr(e−tH e−tH0) = p(α)(0) t V + (t2)
− t k k1 O
(2.16) = t−d/αp(1(cid:0)α)(0) tkVk1+O(cid:1)(t2) ,
ast 0. (cid:0) (cid:1)
(ii↓)IfweonlyassumethatV L∞(Rd) L1(Rd),thenforallt > 0,
∈ ∩
(2.17) Tr(e−tH e−tH0)+p(α)(0)t V(x)dx p(α)(0)Ct2 V V etkVk∞,
− t Rd ≤ t k k1k k∞
(cid:12) Z (cid:12)
forsomeunive(cid:12)rsal constantC. Fromthisweconclude that(cid:12)
(cid:12) (cid:12)
(2.18) Tr(e−tH e−tH0) = p(α)(0) t V(x)dx+ (t2) ,
− t (cid:18)− ZRd O (cid:19)
ast 0.
↓
Theorem 2.2. Suppose V L∞(Rd) L1(Rd) and that it is also uniformly Ho¨lder continuous of
orderγ (thereexistsaconsta∈ntM (0,∩ )suchthat V(x) V(y) M x y γ,forallx,y Rd)
∈ ∞ | − | ≤ | − | ∈
with 0 < γ < α 1, whenever 0 < α 1, and with 0 < γ 1, whenever 1 < α < 2. Then for all
∧ ≤ ≤
t > 0,
1
Tr(e−tH e−tH0) +p(α)(0)t V(x)dx p(α)(0) t2 V(x)2dx
(cid:12) − t ZRd − t 2 ZRd| | (cid:12)
(2.19) (cid:12)(cid:12)(cid:0)C V p(α)(0)(cid:1) V 2 etkVk∞t3+tγ/α+2 , (cid:12)(cid:12)
(cid:12)≤ α,γ,dk k1 t | k∞ (cid:12)
(cid:16) (cid:17)
wheretheconstant C depends onlyonα,γ andd. Inparticular,
α,γ,d
1
(2.20) Tr(e−tH e−tH0)= p(α)(0) t V(x)dx+ t2 V(x)2dx+ (tγ/α+2) ,
− t (cid:18)− ZRd 2 ZRd| | O (cid:19)
ast 0.
↓
Theorems2.1and2.2areprovedin 2. In 3,weprovideextensions toothernon-local operators.
§ §
3. PROOF OF THEOREMS 2.1 AND 2.2
ProofofTheorem2.1. Setting
t
a = V(X )ds, and b = t V ,
s ∞
k k
Z0
weobservethat b a 0. Wethenusethefollowingelementary inequality from[6]
− ≤ ≤
1
(3.1) a e−a 1 a 1+ beb
− ≤ − ≤ − 2
(cid:18) (cid:19)
toconclude that
tV(Xs)ds e−R0tV(Xs)ds 1 tV(Xs)ds 1+ 1t V ∞etkVk∞ .
− ≤ − ≤ − 2 k k
Z0 (cid:16) (cid:17) (cid:20) Z0 (cid:21)(cid:18) (cid:19)
Taking expectations of both sides of this inequality with respect to Et and then integrating on Rd
x,x
with respect to x, it follows exactly asin the derivation of the general bound (2.13), that (2.15)holds.
Thisconcludes theproofof(i)inTheorem2.1.
6 RODRIGOBAN˜UELOSANDSELMAYILDIRIMYOLCU
For(ii),weobservethatby(2.14)wehave(againusing(2.12))
(3.2) Tr(e−tH e−tH0)+p(α)(0)t V(x)dx
− t Rd
(cid:12) Z (cid:12)
(cid:12) ∞ 1 t k (cid:12)
(cid:12) p(α)(0) Et V(X )ds d(cid:12)x
≤ t Xk=2 k!ZRd x,x(cid:12)Z0 s (cid:12)
∞ tk−1 V k−1(cid:12)(cid:12) (cid:12)(cid:12)t
p(α)(0) k k∞ E V(X )ds dx
≤ t k=2 k! ZRd x,x(cid:18)Z0 | s | (cid:19)
X
∞ tk−1 V k−1
= p(α)(0)t V k k∞ Cp(α)(0)t2 V V etkVk∞,
t k k1 k! ≤ t k k1k k∞
k=2
X
forsomeabsolute constant C. Thisproves(ii)andconcludes theproofofthetheorem. (cid:3)
Remark 3.1. Weobserve that the above holds for any operator which arises from subordination of a
Brownianmotion. Forsuchoperatorsthetransitionprobabilities p aresymmetricradialanddecreas-
t
ing,hencep (x,x) = p (0). Insection5belowwegiveexamplesofothernonlocaloperatorsforwhich
t t
Theorem2.2alsoholds.
ProofofTheorem2.2. Webeginbyobservingthat(exactlyasintheproofofinequality (3.2))wehave
e−R0tV(Xs)ds 1+ tV(Xs)ds 1 tV(Xs)ds 2
(cid:12)(cid:12) − Z0 − 2 (cid:20)Z0 (cid:21) (cid:12)(cid:12)
(cid:12) t (cid:12)
(3.3) (cid:12)(cid:12)C(t V ∞)2etkVk∞ V(Xs)ds, (cid:12)(cid:12)
≤ k k | |
Z0
for some constant C. By taking expectation of both sides of (3.3) with respect to Et and then inte-
x,x
gratingwithrespecttox,weobtain
ZRdExt,x (cid:12)(cid:12)e−R0tV(Xs)ds −1+Z0tV(Xs)ds− 12 (cid:20)Z0tV(Xs)ds(cid:21)2(cid:12)(cid:12)!dx
(cid:12) t (cid:12)
≤ C(tkVk∞)(cid:12)(cid:12)2etkVk∞ZRdExt,x(cid:18)Z0 |V(Xs)|ds(cid:19)dx (cid:12)(cid:12)
= C(t V )2etkVk∞t V ,
∞ 1
k k k k
whereweuseagain(2.12).
Returningtothedefinition ofthetracedifferences in(2.10)weseethatthisleadsto
1 1 t 2
(Tr(e−tH e−tH0))+t V(x)dx Et V(X )ds dx
(cid:12)(cid:12)p(tα)(0) − ZRd − 2 ZRd x,x (cid:20)Z0 s (cid:21) ! (cid:12)(cid:12)
(3.4) (cid:12)(cid:12) C(t V ∞)2etkVk∞t V 1. (cid:12)(cid:12)
(cid:12) ≤ k k k k (cid:12)
It remains to estimate the term Et ([]2). Since V is uniformly Ho¨lder with exponent γ and constant
x,x ·
M,wehave
(3.5) V(X +x) V(x) M X γ.
s s
| − | ≤ | |
HEATTRACEOFNON-LOCALOPERATORS 7
Hence,
t 2 t 2 t 2
Et V(X )ds t2V2(x) = Et V(X )ds V(x)ds
(cid:12)(cid:12) x,x(cid:20)Z0 s (cid:21) − (cid:12)(cid:12) (cid:12)(cid:12) x,x(cid:20)Z0 s (cid:21) −(cid:20)Z0 (cid:21) (cid:12)(cid:12)
(cid:12)(cid:12) (cid:12)(cid:12) (cid:12)(cid:12) t 2 t (cid:12)(cid:12)2
(cid:12) (cid:12) = (cid:12)Et V(X )ds V(x)ds(cid:12)
(cid:12)(cid:12) x,x (cid:20)Z0 s (cid:21) −(cid:20)Z0 (cid:21) !(cid:12)(cid:12)
(cid:12) t (cid:12)
(3.6) = (cid:12)(cid:12)E0t,0( (V(Xs+x)−V(x))ds · (cid:12)(cid:12)
(cid:20)Z0 (cid:21)
t
V(X +x)+V(x))ds ).
s
(cid:20)Z0 (cid:21)
Byemploying (3.5),weobtain
t 2 t t
Et V(X )ds t2V2(x) MEt X γds (V(X +x) + V(x))ds .
(cid:12)(cid:12) x,x (cid:20)Z0 s (cid:21) !− (cid:12)(cid:12) ≤ 0,0(cid:18)(cid:20)Z0 | s| (cid:21)(cid:20)Z0 | s | | | (cid:21)(cid:19)
(cid:12) (cid:12)
I(cid:12)ntegrating bothsidesofthisinequality(cid:12) withrespect toxandusingFubini’s theorem, wehavethatthe
(cid:12) (cid:12)
valueofthesecondintegralbecomes2t V . Thuswearriveat
1
k k
t 2
(3.7) Et V(X )ds dx t2 V(x)2dx
(cid:12)(cid:12)ZRd x,x (cid:20)Z0 s (cid:21) ! − ZRd| | (cid:12)(cid:12)
(cid:12) t (cid:12)
≤ (cid:12)(cid:12)2tMkVk1E0t,0 |Xs|γds . (cid:12)(cid:12)
(cid:18)Z0 (cid:19)
Now,itremainstoestimatetheexpectation ontherightsideof(3.7). Asin(2.11)wehave
t t
(3.8) Et X γds = Et (X γ)ds
0,0 | s| 0,0 | s|
(cid:18)Z0 (cid:19) Z0
t p(α)(0,y)p(α)(y,0)
= s t−s y γdyds.
Z0 ZRd p(tα)(0,0) | |
Toestimatetherighthandsidewerecallthefollowing“5P-inequality”
(α) (α)
p (x,z)p (z,y)
(3.9) s t C p(α)(x,z)+p(α)(z,y) ,
p(α)(x,y) ≤ α,d s t
s+t (cid:16) (cid:17)
valid for all x,y,z Rd, s,t > 0 and 0 < α < 2, where C is a constant depending only on α
α,d
∈
and d. This inequality is proved by Bogdan and Jakubowski in [9, p.182]. It is derived form the the
“3P-inequality”
(3.10) p(α)(x,z) p(α)(z,y) C p(α)(x,y).
s ∧ t ≤ α,d s+t
(α)
Inequality(3.10)itselfisprovedbyasimplecomputationreplacingp withthequantityfrom(2.7).
s
From (3.10) and the fact that for a,b 0 wehave ab = (a b)(a b) and (a b) (a+b)), (3.9)
≥ ∧ ∨ ∨ ≤
followsimmediately.
8 RODRIGOBAN˜UELOSANDSELMAYILDIRIMYOLCU
Returningto(3.8),wehave,
t t
Et X γds C (p(α)(0,y)+p(α)(y,0))y γdyds
0,0(cid:18)Z0 | s| (cid:19) ≤ α,dZ0 ZRd s t−s | |
t
= 2C p(α)(0,y)y γdyds
α,dZ0 ZRd s | |
t
= 2C E0(X γ)ds
α,d s
| |
Z0
t
= 2C sγ/αE0(X γ)ds
α,d 1
| |
Z0
tγ/α+1
= 2C E0(X γ) ,
α,d 1
| | γ/α+1
where we used the scaling property of the stable process coming from (2.6). We now recall that
E0(X γ) is finite under our assumption that γ < α. (This fact follows trivially from (2.7).) Thus
1
| |
weseethat
t
(3.11) Et X γds C tγ/α+1,
0,0 | s| ≤ α,γ,d
(cid:18)Z0 (cid:19)
wheretheconstantC depends onlyonα,γ andd. Weconclude that
α,γ,d
t 2 t
Et V(X )ds dx t2 V(x)2dx 2tM V Et X γds
(cid:12)(cid:12)ZRd x,x (cid:20)Z0 s (cid:21) ! − ZRd| | (cid:12)(cid:12) ≤ k k1 0,0(cid:18)Z0 | s| (cid:19)
(3.12(cid:12)(cid:12)) (cid:12)(cid:12) M V 1Cα,γ,dtγ/α+2.
(cid:12) (cid:12) ≤ k k
Returningto(3.4)wefindthat
1 1
Tr(e−tH e−tH0) +t V(x)dx t2 V(x)2dx
(cid:12)pαt(0) − ZRd − 2 ZRd| | (cid:12)
(3.13) (cid:12)(cid:12)(cid:12)≤ Ct3k(cid:0)Vk2∞etkVk∞kVk1+(cid:1)MkVk1Cα,γ,dtγ/α+2 (cid:12)(cid:12)(cid:12)
C V V 2 etkVk∞t3+tγ/α+2 .
≤ α,γ,dk k1 | k∞
Rewritingthisintheformstatedin(cid:16)Theorem2.2wearriveat(cid:17)theannounced bound
1
Tr(e−tH e−tH0) +pα(0)t V(x)dx p(α)(0)t2 V(x)2dx
(cid:12) − t ZRd − t 2ZRd| (cid:12)
(3.14) (cid:12)(cid:12)(cid:0)C V p(α)(0)(cid:1) V 2 etkVk∞t3+tγ/α+2 , (cid:12)(cid:12)
≤(cid:12) α,γ,dk k1 t | k∞ (cid:12)
validforallt > 0. (cid:16) (cid:17) (cid:3)
Remark3.2. WeremarkthatinthecaseofBrownianmotion,α = 2,onemayusethefact(asdonein
[6])that
(2) (2)
p (0,y)p (y,0)
(3.15) s t−s = p(2) (0,y)
(2) s(t−s)
p (0,0) t
t
to explicitly compute the right hand side of (3.8). While the inequality (3.9) does not hold for α =
2, we may estimate the right hand side of (3.8) in the case of Brownian motion without the explicit
estimatethatcomesfrom(3.15). Indeed,byobservingthatinlawtheBrownianbridgestartedat0and
HEATTRACEOFNON-LOCALOPERATORS 9
conditioned toreturnto0intimetisthesameasB sB ,whereB isstandardBrownianmotion,a
s− t t s
simplecomputation leadstothesameconclusion as(3.11)forα = 2. Indeed,
t t s
Et B γds = E0 B B γds
0,0 | s| | s− t t|
(cid:18)Z0 (cid:19) Z0
t s γ
(3.16) C E0 B γ + B γ ds
γ s t
≤ | | t | |
Z0 (cid:16) (cid:16) (cid:17) (cid:17)
t s γ
= C E0 B γ sγ/2+ tγ/2 ds
γ 1
| | t
Z0 (cid:16) (cid:16) (cid:17) (cid:17)
= C tγ/2+1,
γ,d
whereC dependsonlyonγ andd.
γ,d
Unfortunately, the simple path construction X sX only leads to the stable bridge when α = 2
s − t t
and we are not able to repeat (3.16) in the case 0 < α < 2, hence our use of (3.9). However, it may
be that the construction by Chaumont in [12] (or perhaps some of the estimates of Fitzsimmons and
Getoorin[21])canbeusedtobypasstheestimate(3.9).
4. ALTERNATIVE PROOFS OF THEOREMS 2.1 AND 2.2
Recallingformula(2.14)wehave
∞ ( 1)k t k
(4.1) Tr e−tH e−tH0 = p(α)(0) − Et V(X )ds dx,
− t k=1 k! ZRd x,x(cid:18)Z0 s (cid:19)
(cid:0) (cid:1) X
where as mentioned there the sum is absolutely convergent for all potentials V L∞(Rd) L1(Rd)
∈ ∩
andallt > 0. Wenotealsothatinfact,foreveryN 1,
≥
N ( 1)k t k
(4.2) Tr e−tH e−tH0 = p(α)(0) − Et V(X )ds dx+0(tN+1) ,
− t k=1 k! ZRd x,x(cid:18)Z0 s (cid:19) !
(cid:0) (cid:1) X
ast 0. Thisfollowsfromthefactthat
↓
∞ ( 1)k t k ∞ 1
(4.3) − Et V(X )ds dx tk V k−1 V
(cid:12)k=XN+1 k! ZRd x,x(cid:18)Z0 s (cid:19) (cid:12) ≤ k=XN+1k! k k∞ k k1
(cid:12) (cid:12)
(cid:12) (cid:12) tN+1kVkN∞kVk1etkVk∞
≤ (N +1)!
Nextobservethat
t k
V(X )ds = k!J
s k
(cid:18)Z0 (cid:19)
where
J = V(X )ds V(X )ds V(X )ds .
k s1 1 s2 2··· sk k
Z0≤s1≤s2≤···≤sk≤t
Recallingthatthefinitedimensional distributions ofthestablebridgearegivenby
k
1
Pt X dy ,X dy ,...,X dy = p(α) (y y ),
x,x{ s1 ∈ 1 s2 ∈ 2 sk ∈ k} p(α)(x,x) sj+1−sj j+1− j
t j=0
Y
wherey = y = x,s = 0,s = t,wecanwrite
0 k+1 0 k+1
10 RODRIGOBAN˜UELOSANDSELMAYILDIRIMYOLCU
∞
(4.4) Tr e−tH e−tH0 = ( 1)k I˜ (s ,s ,...,s )ds ds ds ,
k 1 2 k 1 2 k
− − ···
k=1 Z0≤s1≤s2≤···≤sk≤t
(cid:0) (cid:1) X
where
k k
I˜ (s ,s ,...,s ) = p(α) (y y ) V(y )dy dy . dy dx.
k 1 2 k Z(Rd)k+1j=0 sj+1−sj j+1− j j=1 j 1 2 ··· k
Y Y
(α) (α)
Here(asseveraltimesbefore)weusedthefactthatp (x,x) = p (0).ByFubini’stheoremwemay
t t
integratewithrespecttoxfirstandusethefactthat
p(α)(y x)p(α) (y x)dx = p(α) (y y )
Rd s1 1− t−sk k − t−(sk−s1) k − 1
Z
toarriveat
∞
(4.5) Tr e−tH e−tH0 = ( 1)k I (s ,s ,...,s )ds ds ds ,
k 1 2 k 1 2 k
− − ···
k=1 Z0≤s1≤s2≤···≤sk≤t
(cid:0) (cid:1) X
where
(α)
I (s ,s ,...,s ) = p (y y )V(y )
k 1 2 k Z(Rd)k (cid:16) t−(sk−s1) k − 1 1 (cid:17)·
k−1
(α)
p (y y )V(y ) dy dy dy .
 sj+1−sj j+1− j j+1  1 2··· k
j=1
Y
 
Wecanthenalsore-writetherighthandsideof(4.2)as
N ( 1)k
(4.6) p(α)(0) − I (s ,s ,...,s )ds ds ds +0(tN+1) .
t k=1 p(tα)(0) Z0≤s1≤s2≤···≤sk≤t k 1 2 k 1 2··· k !
X
From these formulas we can also compute coefficients in the asymptotic expansion of the trace as
t 0fairlydirectlyandinparticular givealternative proofsofTheorems2.1and2.2.
↓
Ifk = 1,weseethat
(α) (α)
p (y y ) = p (0)
t−(sk−s1) k − 1 t
and
(α)
I (s ) = p (0) V(y)dy,
1 1 t
Rd
Z
whichimmediatelyleadsto(2.18)inTheorem2.1.
Ifk = 2,wewrite
(α) (α)
(4.7) I (s ,s )= p (y y )p (y y )V(y )V(y )dy dy
2 1 2 Rd Rd t−(s2−s1) 2− 1 (s2−s1) 2− 1 1 2 1 2
Z Z
(α) (α)
= p (y y )p (y y )(V(y ) V(y )+V(y ))V(y )dy dy
Rd Rd t−(s2−s1) 2− 1 (s2−s1) 2− 1 1 − 2 2 2 1 2
Z Z
= p(α) (y y )p(α) (y y )V2(y )dy dy
Rd Rd t−(s2−s1) 2− 1 (s2−s1) 2− 1 2 1 2
Z Z
(α) (α)
+ p (y y )p (y y )(V(y ) V(y ))V(y )dy dy .
Rd Rd t−(s2−s1) 2− 1 (s2−s1) 2− 1 1 − 2 2 1 2
Z Z