The bottom of the spectrum of time-changed processes and the maximum principle of Schr\"{o}dinger operators

0.19 MB·

by  Masayoshi Takeda

THE BOTTOM OF THE SPECTRUM OF TIME-CHANGED
PROCESSES AND THE MAXIMUM PRINCIPLE OF
SCHRO¨DINGER OPERATORS
7
1
0
MASAYOSHITAKEDA
2

n
a Abstract. We give a necessary and sufficient condition for the maximum
J principle of Schro¨dinger operators interms of the bottom of the spectrum of
time-changedprocesses. Asacorollary,weobtainasufficientconditionforthe
1
LiouvillepropertyofSchro¨dinger operators.
1

]
R
P
. 1. Introduction
h
t
a In[12],wedefinethesubcriticality,criticalityandsuperciriticalityforSchr¨odinger
m forms and characterize these properties in terms of the bottom of the spectrum of
[ time changed processes. In the process, we prove the existence of a harmonic
function (or ground state) of the Schr¨odinger form and study its properties. In
1
particular, we show that it has a bounded, positive, continuous version which is
v
invariant with respect to its Schr¨odinger semigroup. In this paper, we will show,
0
6 as an application of this fact, the maximum principle and Liouville property of
8 Schr¨odinger operators.
2 Let X be a locally compact separable metric space and m a positive Radon
0
measure on X with full topological support. Denote by X := X the one-
. △ ∪{△}
1 pointcompactificationofX. LetM=(P ,X ,ζ)beanm-symmetricHuntprocess
x t
0
withlifetime ζ =inf t>0 X = . WeassumethatMisirreducibleandstrong
t
7 { | △}
Feller. Let µ=µ+ µ− be a signed Radonsmoothmeasure such that the positive
1 −
: (resp. negative)partµ+ (resp. µ−)belongstothe localKatoclass(resp. the Kato
v class). We denote by Aµ+ (resp. Aµ−) the positive continuous additive functional
i t t
X in the Revuz correspondence to µ+ (resp. µ+). Put Aµ = Aµ+ Aµ− and define
r the Feynman-Kac semigroup pµ by t t − t
a { t}t≥0
pµtf(x)=Ex e−Aµtf(Xt) .
(cid:16) (cid:17)
We denote by Mµ+ = (Pµ+,X ,ζ) the subprocess of M by the multiplicative
x t
functionalexp( Aµ+)andby( µ+, ( µ+))theDirichletformgeneratedbyMµ+.
− t E D E
Suppose that the negative part µ− is non-trivial and Green-tight with respect to
1991 Mathematics Subject Classification. 31C25,31C05,60J25.
Keywordsandphrases. Dirichletform,Schro¨dingerform,symmetricHuntprocess,maximum
principle,Liouvilleproperty.
The author was supported in part by Grant-in-Aid for Scientific Research (No.26247008(A))
and Grant-in-Aid for Challenging Exploratory Research (No.25610018), Japan Society for the
PromotionofScience.
1
2 MASAYOSHITAKEDA
Mµ+ (Definition 2.2 (2)). We then define λ(µ) by
(1) λ(µ):=inf (u,u)+ u2dµ+ u ( ), u2dµ− =1 .
E | ∈D E
(cid:26) ZX ZX (cid:27)
λ(µ)isregardedasthebottomofthespectrumofthetime-changedprocessofMµ+
by the continuous additive functional Aµ−. We show in [11, Theorem 2.1]that the
t
minimizerof (1)existsintheextendedDirichletspace ( µ+)anditcanbetaken
e
D E
to be strictly positive on X. The objective of this paper is to prove the maximum
principleofSchr¨odingerformsby usingthe existence ofthe minimizer of(1). More
precisely, let
(2) ba(µ)= h B(X) h is bounded above, pµh h ,
H { ∈ | t ≥ }
where B(X) is the set of Borel functions on X. We here define the maximum
principle by
(MP) If h ba(µ), then h(x) 0 for all x X.
∈H ≤ ∈
We will prove in Theorem 3.1 that under Assumption
(A) Ex e−Aµ∞+;ζ = =0,
∞
(cid:16) (cid:17)
(MP) is equivalentto λ(µ)>1. For the proofof this, it is crucialthat if λ(µ)=1,
then the minimizer h in (1) has a bounded continuous version with pµ-invariance,
t
i.e., E (exp( Aµ)h(X ))=h(x) ([12, Lemma 5.16, Corollary 5.17]).
x − t t
Let us introduce the space b(µ) of bounded pµ-invariant functions:
H t
b(µ)= h B (X) pµh=h .
H { ∈ b | t }
We here define the Liouville property by
(L) If h b(µ), then h(x)=0 for all x X.
∈H ∈
We will show in Corollary 4.1 that under Assumption (A), λ(µ)>1 implies (L).
We remark that Theorem 3.1 and Corollary 4.1 can be applied to non-local
Dirichlet forms. In a remaining part of introduction, we treat these two properties
for stronglylocalDirichlet forms,which areregardedas anextensionof symmetric
ellipticoperatorsofsecondorder. InBerestycki-Nirenberg-Varadhan[2],theydefine
amaximumprincipleforauniformlyellipticoperatorofsecondorder,L=M+c=
a ∂ ∂ +b ∂ +c,onageneralboundeddomainD ofRd. Letu bethe solutionto
i,j i j i i 0
the equation Mu= 1 vanishing on ∂D in a suitable sense: define by the set of
− S
sequences x ∞ D suchthatx convergestoapointofthe boundary∂D and
{ n}n=1 ⊂ n
u (x ) converges to 0. They say that the refined maximum principle holds for L,
0 n
ifa function h boundedabovesatisfies Lh 0 onD andlimsup h(x ) 0 for
≥ n→∞ n ≤
any x ∞ ,thenh 0onD,andprovethatLsatisfiestherefinedmaximum
{ n}n=1 ∈S ≤
principle if and only if the principal eigenvalue λ of L is positive.
0
−
Note that u equals E (τ ), where P is the diffusion process with generator
0 x D x
M and τ is the first exit time from D. We see that if D is bounded (more
D
generally,Green-bounded, i.e., sup E (τ )< ), then is identical to the set
x∈D x D ∞ S
THE MAXIMUM PRINCIPLE OF SCHRO¨DINGER OPERATORS 3
ofsequences x suchthatx ∂DandE (exp( τ )) 1asn (Lemma
{ n} n → xn − D → →∞
3.1, Remark 3.1). Considering this fact, we define
(3) = x ∞ X x and E (e−ζ) 1 as n .
S { n}n=1 ⊂ | n →△ xn → →∞
(cid:8) (cid:9)
Assume that ( , ( )) is strongly local and set
E D E
h ( ) C(X) is bounded above, µ(h,ϕ) 0 for
loc
ba(µ)= h ∈D E ∩ E ≤ ,
H ( | ∀ϕ∈D(E)∩C0+(X) and limsuph(xn)≤0 for ∀{xn}∈S)
n→∞
e
whereC+(X)isthesetofnon-negativecontinuousfunctionswithcompactsupport.
0
Following [2], we here define the refined maximum principle by
(RMP) If h ba(µ), then h(x) 0 for all x X.
∈H ≤ ∈
e
We will show that ba(µ) ba(µ) (Lemma 3.4), and thus see, as a corollary of
H ⊂ H
Theorem of 3.1, that λ(µ) > 1 implies (RMP) (Theorem 3.2). We would like to
emphasize that if Deis bounded and L is symmetric, the principal eigenvalue λ
0
of L is positive if and only if λ(µ) > 1. However, λ(µ) > 1 does not always
−
imply λ >0 for a unbounded domain D, while λ >0 implies λ(µ)>1 in general
0 0
(Lemma 3.5).
When ( , ( )) is strongly local, we set
E D E
b(µ)= h ( ) C (X) µ(h,ϕ)=0, ϕ ( ) C (X)
loc b 0
H { ∈D E ∩ |E ∀ ∈D E ∩ }
and definee the property (L) by
e
(L) If h b(µ), then h(x)=0 for all x X.
∈H ∈
e e
WethenseethatifMisconservativeand(A)isfulfilled,thatis,E (exp( Aµ+))=
x − ∞
0, then b(µ) b(µ), and consequently λ(µ) > 1 implies (L) (Corollary 4.2).
H ⊂ H
Grigor’yanand Hansen[7] callsa measureµ+ big if it satisfies(A), and they prove
that for tehe transient Brownian motion M = (P ,B ) on Rd, ifeµ− 0 and µ+ is
x t
≡
big, then (L) holds. Corollary 4.1 tells us that if µ− is small with respect to µ+ in
the sense that λ(µ)>1 then (L) still holds.
Pinsky [e9] treat absolutely continuous potentials dµ=V+dx V−dx and prove
in [9, Theorem 1.1] that if suepx∈RdEx exp( 0∞V−(Bt)dt) <−∞, the Liouville
property (L) is equivalent to
(cid:0) R (cid:1)
∞
e V+(B )dt= , P -a.e. ( (A)).
t x
∞ ⇐⇒
Z0
We will give an example of potential µ that even if supx∈RdEx(exp(Aµ∞−)) = ∞
and E (exp( Aµ+))=0, (L) holds (Example 4.1).
x − ∞
e
4 MASAYOSHITAKEDA
2. Schro¨dinger forms
Let X be a locally compact separable metric space and m a positive Radon
measure on X with full topological support. Let ( , ( )) be a regular Dirichlet
E D E
form on L2(X;m). We denote by u ( ) if for any relatively compact open
loc
∈ D E
set D there exists a function v ( ) such that u = v m-a.e. on D. We denote
∈ D E
by ( ) the family of m-measurable functions u on X such that u < m-
e
D E | | ∞
a.e. and there exists an -Cauchy sequence u of functions in ( ) such that
n
E { } D E
lim u =u m-a.e. We call ( ) the extended Dirichlet space of ( , ( )).
n→∞ n e
Let M=(Ω,F, F , PD E , X ,ζ) be the symmetric HEunDt pErocess
t t≥0 x x∈X t t≥0
generated by ( , ({ )),}whe{re }F { is}the augmented filtration and ζ is the
t t≥0
E D E { }
lifetime ofM. Denote by p and G the semigroupandresolventof M:
t t≥0 α α≥0
{ } { }
∞
p f(x)=E (f(X )), G f(x)= e−αtp f(x)dt.
t x t α t
Z0
We assume that M satisfies next two conditions:
: Irreducibility (I). If a Borel set A is p -invariant, i.e., p (1 f)(x) =
t t A
1 p f(x) m-a.e. for any f L2(X;m) B (X) and t>0, then A satisfies
A t b
either m(A) = 0 or m(X ∈A) = 0. He∩re B (X) is the space of bounded
b
\
Borel functions on X.
: Strong Feller Property (SF). For each t, p (B (X)) C (X), where
t b b
⊂
C (X) is the space of bounded continuous functions on X.
b
We remark that (SF) implies (AC).
: Absolute Continuity Condition (AC). ThetransitionprobabilityofM
is absolutely continuous with respect to m, p(t,x,dy)=p(t,x,y)m(dy) for
each t>0 and x X.
∈
Under (AC), a non-negative,jointly measurable α-resolventkernelG (x,y) ex-
α
ists:
G f(x)= G (x,y)f(y)m(dy), x X, f B (X).
α α b
∈ ∈
ZX
Moreover,G (x,y)isα-excessiveinxandiny ([6,Lemma4.2.4]). Wesimplywrite
α
G(x,y) for G (x,y). For a measure µ, we define the α-potential of µ by
0
G µ(x)= G (x,y)µ(dy).
α α
ZX
Definition 2.1. (1) ADirichletspace( , ( ))onL2(X;m)issaidtobetransient
E D E
if there exists a strictly positive,bounded function g in L1(X;m)suchthat for
u ( )
∈D E
ugdm (u,u).
| | ≤ E
ZX
p
(2) A Dirichlet space ( , ( )) on L2(X;m) is said to be recurrent if the constant
E D E
function 1 belongs to ( ) and (1,1) = 0. Namely, there exists a sequence
e
D E E
u ( ) such that lim (u u ,u u )=0 and lim u =
n n,m→∞ n m n m n→∞ n
{ }⊂D E E − −
1 m-a.e.
For other characterizations of transience and recurrence, see [6, Theorem 1.6.2,
Theorem 1.6.3].
THE MAXIMUM PRINCIPLE OF SCHRO¨DINGER OPERATORS 5
We define the (1-)capacity Cap associated with the Dirichlet form ( , ( )) as
E D E
follows: for an open set O X,
⊂
Cap(O)=inf (u,u)+(u,u) u ( ),u 1 m-a.e. on O
m
{E | ∈D E ≥ }
and for a Borel set A X,
⊂
Cap(A)=inf Cap(O) O is open, O A .
{ | ⊃ }
A statement depending on x X is said to hold q.e. on X if there exists a set
∈
N X ofzerocapacitysuchthatthe statementis trueforeveryx X N. “q.e.”
⊂ ∈ \
is an abbreviation of “quasi-everywhere”. A real valued function u defined q.e. on
X is said to be quasi-continuous if for any ǫ > 0 there exists an open set G X
⊂
such that Cap(G) < ǫ and u is finite and continuous. Here, u denotes
X\G X\G
| |
the restriction of u to X G. Each function u in ( ) admits a quasi-continuous
e
\ D E
versionu˜,thatis,u=u˜m-a.e. Inthesequel,wealwaysassumethateveryfunction
u ( ) is represented by its quasi-continuous version.
e
∈D E
We call a positive Borel measure µ on X smooth if it satisfies the following
conditions:
(S1) µ charges no set of zero capacity,
(S2) there exists an increasing sequence F of closed sets that
n
{ }
(4) µ(F )< ,
n
∞
(5) lim Cap(K F )=0 for any compact set K.
n
n→∞ \
We denote by S the set of smooth measures.
A stochastic process A is said to be an additive functional (AF in abbre-
t t≥0
{ }
viation) if the following conditions hold:
(i) A () is F -measurable for all t 0.
t t
(ii) The·re exists a set Λ F =≥σ( F ) such that P (Λ) = 1, for q.e.
∞ t≥0 t x
∈ ∪
x X, θ Λ Λ for all t > 0, and for each ω Λ, A(ω) is a function
t ·
∈ ⊂ ∈
satisfying: A = 0, A (ω) < for t < ζ(ω), A (ω) = A (ω) for t ζ, and
0 t t ζ
∞ ≥
A (ω)=A (ω)+A (θ ω) for s,t 0.
t+s t s t
≥
If an AF A is positive and continuous with respect to t for each ω Λ,
t t≥0
{ } ∈
the AF is called a positive continuous additive functional (PCAF in abbreviation).
The set of all PCAF’s is denoted by A+. The family S and A+ are in one-to-one
c c
correspondence(Revuz correspondence)asfollows: foreachsmoothmeasureµ,
thereexists a unique PCAF A suchthatfor anyf B+(X)andγ-excessive
t t≥0
{ } ∈
function h (γ 0), that is, e−γtp h h,
t
≥ ≤
1 t
(6) lim E f(X )dA = f(x)h(x)µ(dx)
h·m s s
t→0 t (cid:18)Z0 (cid:19) ZX
([6, Theorem 5.1.7]). Here, E ( )= E ( )h(x)m(dx). We denote by Aµ the
h·m · X x · t
PCAF corresponding to µ S. For a signed smooth measure µ = µ+ µ−, we
define Aµ =Aµ+ Aµ−. ∈ R −
t t − t
We introduce some classes of smooth measures.
Definition 2.2. Suppose that µ S is a positive Radon measure.
∈
6 MASAYOSHITAKEDA
(1) A measure µ is said to be in the Kato class of M ( in abbreviation) if
K
lim G µ =0.
α ∞
α→∞k k
A measure µ is said to be in the local Kato class ( in abbreviation) if for
loc
K
any compact set K, 1 µ belongs to .
K
· K
(2) Suppose that M is transient. A measure µ is said to be in the class if for
∞
K
any ǫ>0, there exists a compact set K =K(ǫ)
sup G(x,y)µ(dy)<ǫ.
x∈XZKc
A measure µ in is called Green-tight.
∞
K
We note that every measure treated in this paper is supposed to be Radon. We
denote the Green-tight class by (G) if we would like to emphasize the depen-
∞
K
denceofthe Greenkernel. Chen[3]definethe Green-tightclassinslightlydifferent
way; however the two definitions are equivalent under (SF) ([8, Lemma 4.1]).
Let µ=µ+ µ− . We define the Schr¨odinger form by
loc
− ∈K −K
µ(u,u)= (u,u)+ u2dµ
(7) E E ZX
 ( µ)= ( ) L2(X;µ+).
D E D E ∩
Denotingby µ = µtheself-adjointoperatorgeneratedbytheclosedsymmetric
L L−
form ( µ, ( µ)), ( µu,v) = µ(u,v), we see that the associated semigroup
m
exp(t Eµ) iDs eExpresse−dLas exp(t µ)fE(x)=E (exp( Aµ)f(X )) (cf. [1]).
L L x − t t
Let Mµ+ = (Pµ+,X ,ζ) the subprocess of M by the multiplicative functional
x t
exp( Aµ+) and suppose that Mµ+ is also strong Feller (For this conditions, refer
− t
[4]).
3. Maximum Principle
In this section we consider the maximum principle for Schr¨odinger forms. For
h B(X) we denote by h+ and h− the positive and negative part of h.
∈
Theorem 3.1. Assume (A). Then
λ(µ)>1 (MP).
⇐⇒
Proof. For h ba(µ)
∈H
h(x)≤Ex e−Aµth(Xt) =Eµx+ eAµt−h(Xt) ≤Eµx+ eAµζ−h+(Xt)
(cid:18) (cid:19) (cid:18) (cid:19)
(cid:16) (cid:17)
≤kh+k∞·Eµx+ eAµζ−;t<ζ .
(cid:18) (cid:19)
If λ(µ) > 1, then sup Eµ+(exp(Aµ−)) < by [3, Theorem 5.1]. Hence the
x∈X x ζ ∞
right-hand side tends to 0 as t because
→∞
lim Pµ+(t<ζ)=E e−Aµ∞+1 =0
x x {ζ=∞}
t→∞
(cid:18) (cid:19)
by Assumption (A).
THE MAXIMUM PRINCIPLE OF SCHRO¨DINGER OPERATORS 7
Suppose λ(µ) 1. By the definition of λ(µ)
≤
(8) inf µ+(u,u) λ(µ) u2dµ− =1 =1.
E |
(cid:26) ZX (cid:27)
It follows from [12, Lemma 5.16, Corollary 5.17] that the minimizer h in (8) is a
bounded positive continuous with pµ+−λ(µ)µ−-invariance, h(x) = pµ+−λ(µ)µ−h(x).
t t
Hence
h(x)=pµ+−λ(µ)µ−h(x) pµ+−µ−h(x)=pµh(x),
t ≤ t t
and (MP) does not hold. (cid:3)
In the sequel of this section, we deal with a strongly local Dirichlet form and
extend a result of [2]. We set
= x ∞ X x and lim E e−ζ =1. ,
S { n}n=1 ⊂ | n →△ n→∞ xn
=n x ∞ X x and lim P ((cid:0)ζ >(cid:1)ǫ) 0ofor any ǫ>0 .
S { n}n=1 ⊂ | n →△ n→∞ xn →
n o
Lemmea 3.1. It holds that
= .
S S
Proof. For x ∞
{ n}n=1 ∈S e
E (e−ζ) e−ǫP (ζ >ǫ)+P (ζ ǫ)=1 (1 e−ǫ)P (ζ >ǫ),
xn ≤ xn xn ≤ − − xn
and thus
1 E (e−ζ)
lim P (ζ >ǫ) lim − xn =0.
n→∞ xn ≤n→∞ 1 e−ǫ
−
For x ∞
{ n}n=1 ∈S
E (e−ζ)=E (e−ζ;ζ >ǫ)+E (e−ζ;ζ ǫ) e−ǫP (ζ ǫ),
xn e xn xn ≤ ≥ xn ≤
and thus lim E (e−ζ) e−ǫ and lim E (e−ζ)=1. (cid:3)
n→∞ xn ≥ n→∞ xn
A Dirichlet form ( , ( )) is said to be strongly local, if (u,v) = 0 for any
E D E E
u,v ( ) such that u is constant on a neighborhood of supp[v]. In the sequel of
∈D E
this section, we assume that ( , ( )) is strongly local. We introduce
E D E
h ( ) C(X) is bounded above, µ(h,ϕ) 0 for any
loc
ba(µ)= h ∈D E ∩ E ≤ .
H ( | ϕ∈D(E)∩C0+(X), nl→im∞h(xn)≤0 for any {xn}∞n=1 ∈S. )
e
Lemma 3.2. Let τ ∞ be a sequence of stopping times such that τ < ζ and
{ n}n=1 n
τ ζ, as n , P -a.s. Then there exists a subsequence σ ∞ of τ ∞
n ↑ → ∞ x { n}n=1 { n}n=1
such that
(9) P ( X )=1.
x { σn}∈S
Proof. First note
ζ(θ )>ǫ, τ <ζ = τ +ζ(θ )>τ +ǫ, τ <ζ
{ τn n } { n τn n n }
= ζ >τ +ǫ, τ <ζ = ζ >τ +ǫ .
n n n
{ } { }
8 MASAYOSHITAKEDA
We then have by the strong Markov property
E (P (ζ >ǫ))=E (P (ζ >ǫ);τ <ζ)
x Xτn x Xτn n
=E (P (ζ(θ )>ǫ, τ <ζ F ))
x x τn n | τn
=P (ζ >τ +ǫ) 0 as n .
x n
−→ →∞
Hence there exists a subsequence τ(1) ∞ of τ ∞ such that
{ n }n=1 { n}n=1
P (ζ >1) 0 as n , P -a.s..
Xτn(1) −→ →∞ x
By the same argument
E (P (ζ >1/2);τ(1) <ζ) 0
x Xτn(1) n −→
and there exists a subsequence τ(2) ∞ of τ(1) ∞ such that
{ n }n=1 { n }n=1
P (ζ >1/2) 0 as n , P -a.s.
Xτn(2) −→ →∞ x
By continuing this procedure we can take a subsequence τ(k) ∞ of τ(k−1) ∞
{ n }n=1 { n }n=1
such that
P (ζ >1/k) 0 as n , P -a.s.
Xτn(k) −→ →∞ x
The sequence σ :=τ(n) ∞ is a desired one. (cid:3)
{ n n }n=1
Lemma 3.3. Suppose ( , ( )) is strongly local. Let D ∞ be a sequence of
relatively compact open sEetsDsuEch that D X. Define S{ =n}inn=f1t > 0 Aµ− > n
n ↑ n { | t }
and T =S τ . Then for h ba(µ)
n n∧ Dn ∈H
Ex e−AµTn∧the(XTn∧t) ≥h(x) q.e. x.
(cid:16) (cid:17)
Proof. This lemma can be derived by the argument similar to that in [12, Lemma
4.7]. In fact, put = ( ) C (X). Then is a Stone vector lattice, i.e., if
0
L D E ∩ L
f,g , then f g , f 1 . For h ba(µ) define the functional I by
∈L ∨ ∈L ∧ ∈L ∈H
(10) I(ϕ)= µ(h,ϕ), ϕ .
−E e ∈L
Then I(ϕ) is a pre-integral, that is, I(ϕ ) 0 whenever ϕ and ϕ (x) 0
n n n
↓ ∈ L ↓
for all x X. Indeed, let ψ ( ) C (X) such that ψ = 1 on supp[ϕ ]. Then
0 1
∈ ∈ D E ∩
ϕ ϕ ψ and
n n ∞
≤k k
I(ϕ ) ϕ I(ψ) 0, n .
n n ∞
≤k k · ↓ →∞
Notice that by the regularity of ( , ( )) the smallest σ-field generated by is
E D E L
identical with the Borel σ-field. We then see from [5, Theorem 4.5.2] that there
exists a positive Borel measure ν such that
(11) I(ϕ)= ϕdν.
ZX
Bythedefinitionofν weseethatν isaRadonmeasureandsatisfies(S2)forany
increasing sequence F of compact sets with F X. Let K be a compact set of
n n
{ } ↑
zero capacity. Then for a relatively compact open set D such that K D, there
existsasequence ϕ ( ) C+(D)suchthatϕ 1onK and (ϕ⊂,ϕ ) 0
{ n}⊂D E ∩ 0 n ≥ E1 n n →
as n ([6, Lemma 2.2.7]). For ψ ( ) C (X) with ψ =1 on D,
0
→∞ ∈D E ∩
I(ϕ )= µ(h,ϕ )= µ(hψ,ϕ ) |µ|(hψ,hψ)1/2 |µ|(ϕ ,ϕ )1/2,
n n n n n
−E −E ≤E ·E
THE MAXIMUM PRINCIPLE OF SCHRO¨DINGER OPERATORS 9
where µ =µ++µ−. Note that 1 µ and G (1 µ) < . We then see
D 1 D ∞
| | | |∈ K k | | k ∞
from the Stollmann-Voigt inequality ([10]) that
ϕ2dµ = ϕ21 dµ G (1 µ) (ϕ ,ϕ ) 0, n
n | | n D | |≤k 1 D| | k∞·E1 n n −→ →∞
ZX ZX
and |µ|(ϕ ,ϕ ) 0 as n . Since
n n
E → →∞
ν(K) ϕ dν =I(ϕ ) 0, n
n n
≤ → →∞
ZX
ν satisfies (S1), consequently the measure ν is smooth.
The equations (10), (11) lead us to
(h,ϕ)= ϕhdµ ϕdν = ϕ(hdµ+dν).
E − − −
ZX ZX ZX
On account of [6, Theorem 5.5.5], we have
t
h(X )=h(X )+M[h]+ h(X )dAµ+Aν P -a.s., q.e. x.
t 0 t s s t x
Z0
Hence, by Itoˆ’s formula
t t
e−Aµth(Xt)=h(X0)+ e−AµsdMs[h]+ e−Aµsh(Xs)dAµs
Z0 Z0
t t
+ e−AµsdAνs − e−Aµsh(Xs)dAµs
Z0 Z0
t t
=h(X0)+ e−AµsdMs[h]+ e−AµsdAνs Px-a.s., q.e. x.
Z0 Z0
Since 0Tn∧te−AµsdMs[h] is a martingale and 0te−AµsdAνs ≥0,
R Ex e−AµTn∧th(XTn∧t)R≥h(x) q.e. x.
(cid:16) (cid:17)
(cid:3)
Lemma 3.4. It holds that
ba(µ) ba(µ).
H ⊂H
Proof. Let h be a function in Heba(µ) and {Tn}∞n=1 a sequence of stopping times
defined in Lemma 3.3. We fix a point x X such that
∈
e
Ex e−AµTn∧th(XTn∧t) ≥h(x).
(cid:16) (cid:17)
Since T <ζ and T ζ, we can take a subsequence σ of T satisfying (9) in
n n n n
↑ { } { }
Lemma 3.2. Since h+ is bounded continuous and
lim h+(X )=0, P -a.s. on t ζ
n→∞ σn∧t x { ≥ }
10 MASAYOSHITAKEDA
by (9), we have
nl→im∞Ex e−Aµσn∧th+(Xσn∧t)
≤nl→im∞Ex (cid:16)e−Aµσn∧th+(Xσn∧t);(cid:17)t<ζ +nl→im∞Ex e−Aµσn∧th+(Xσn∧t);t≥ζ
≤Ex( e−A(cid:16)µth+(Xt);t<ζ +Ex nl→im(cid:17)∞e−Aµσn∧th+(cid:16)(Xσn∧t);t≥ζ (cid:17)
=Ex (cid:16)e−Aµth+(Xt) . (cid:17) (cid:16) (cid:17)
(cid:16) (cid:17)
Here, the second inequality above follows from the inverse Fatou’s lemma because
e−Aµσn∧th+(Xσn∧t)≤eAµσn−∧th+(Xσn∧t)≤kh+k∞·eAµt− ∈L1(Px)
by µ− .
∈K
Besides, we have
lim Ex e−Aµσn∧th−(Xσn∧t)
n→∞
(cid:16) (cid:17)
≥ lim Ex e−Aµσn∧th−(Xσn∧t);t<ζ + lim Ex e−Aµσn∧th−(Xσn∧t);t≥ζ
n→∞ n→∞
(cid:16) (cid:17) (cid:16) (cid:17)
≥Ex lim e−Aµσn∧th−(Xσn∧t);t<ζ =Ex e−Aµth−(Xt) .
(cid:18)n→∞ (cid:19) (cid:16) (cid:17)
Hence
h(x)≤ lim Ex e−Aµσn∧th(Xσn∧t)
n→∞
(cid:16) (cid:17)
≤nl→im∞Ex e−Aµσn∧th+(Xσn∧t) −nl→im∞Ex e−Aµσn∧th−(Xσn∧t)
(cid:16) (cid:17) (cid:16) (cid:17)
Ex e−Aµth+(Xt) Ex e−Aµth−(Xt) =Ex e−Aµth(Xt) ,
≤ −
and h(x) pµh(x(cid:16)) for q.e. x. S(cid:17)ince pµ(cid:16)is strong Fell(cid:17)er, pµ(h(cid:16) ( n))(x)(cid:17) h(x) for
all x X≤andt pµh(x) h(x) for all xt X by letting n tot ∨. − ≥ (cid:3)
∈ t ≥ ∈ ∞
Following [2], we define the refined maximum principle:
(RMP) If h ba(µ), then h(x) 0 for all x X.
∈H ≤ ∈
Combining Lemmea 3.4 with Theorem 3.1, we have the next theorem.
Theorem 3.2. Suppose ( , ( )) is strongly local. Then under Assumption (A)
E D E
λ(µ)>1= (RMP).
⇒
Remark 3.1. Suppose D is a bounded domain in Rd and consider the absorbing
Brownian motion (P ,B ,τ ) on D, where τ is the first exit time from D. If
x t D D
D is Green-bounded, i.e., sup E (τ ) < , then is identical to the set of
x∈D x D ∞ S
sequences x such that x ∂D and E (τ ) 0 as n . Indeed, take
{ n} n → xn D → → ∞
δ > 0 so that sup E (δτ ) < 1. Then since sup E (exp(δτ )) < by
x∈D x D x∈D x D ∞
Has’minskii’s lemma, we see sup E (τ2) < . Hence if P (τ > ǫ) 0 as
x∈D x D ∞ xn D →
n , then
→∞
E (τ ) E (τ2) 1/2 (P (τ >ǫ))1/2+ǫP (τ ǫ) ǫ.
xn D ≤ xn D · xn D xn D ≤ →
(cid:0) (cid:1)