Time changes of local Dirichlet spaces by energy measures of harmonic functions
Naotaka Kajino
∗†‡Graduate School of Informatics, Kyoto University, Kyoto 606-8501, Japan
November 12, 2010
Abstract
Given a (symmetric) recurrent local regular Dirichlet form with state spaceEand an associated symmetric diffusion{Xt}t∈[0,∞) onE, we consider a functionhwhich belongs to the extended Dirichlet space, is harmonic outside F1∪F2 and equal toa onF1and tobonF2, whereF1,F2⊂E are (E-quasi-)closed sets anda,b∈R,a<b.
We prove that the time change of the real-valued process{h(Xt)}t∈[0,∞)by the energy measureµhhiofhis a reflecting Brownian motion on[a,b]. As an application, we also discuss asymptotic analysis of the heat kernel on the harmonic Sierpinski gasket.
Keywords. strong local Dirichlet spaces, time changes, harmonic functions, energy mea- sures.
2010 Mathematics Subject Classification.31C25, 60J55, 31C05, 60J45.
1 Introduction
As presented in the celebrated work of Itˆo and McKean [10], any one-dimensional diffu- sion can be viewed as a suitable reparametrization of one-dimensional Brownian motion.
Such a method of reparametrizations of stochastic processes is known as(random) time changes. The purpose of this paper is to present a natural extension of this fact for a sym- metric diffusion on a general state space subject to certain time changes involving harmonic functions.
We illustrate our main results by treating the scale function of a one-dimensional dif- fusion as a particular example. For simplicity we concentrate on the case with reflecting boundaries. Then the state space has to be a compact interval and therefore without loss of generality we may assume that the state space is[0,1]. Lets:[0,1]→Rbe strictly in- creasing and continuous, and letmbe a finite Borel measure on[0,1]with full support. Set
∗E-mail:kajino.n@acs.i.kyoto-u.ac.jp
†URL:http://www-an.acs.i.kyoto-u.ac.jp/~kajino.n/
‡JSPS Research Fellow PD (20·6088): Supported by the Japan Society for the Promotion of Science.
a:=s(0)andb:=s(1). Following [4, Subsection 2.2.3], we define Fs:=
{
u∈C([0,1])u−u(0) =
∫ (·) 0
du
dsdsfor some du
ds ∈L2([0,1],ds) }
, (1.1) Es(u,v):=1
2
∫ 1 0
du ds
dv
dsds, u,v∈Fs (1.2)
(note that suchdu/ds∈L2([0,1],ds)as in (1.1) is unique for eachu∈Fs). Then(Es,Fs) is an irreducible recurrent strong local regular Dirichlet form onL2([0,1],m)by [4, Propo- sition 2.2.8], and the associatedm-symmetric diffusionXs,m= ({Xts,m}t∈[0,∞),{Px}x∈[0,1]) has the scale functionsand the speed measurem. Clearly,s∈Fsand
Es(s,v) =1
2(v(1)−v(0)), v∈Fs. (1.3) In particular,sis harmonic outside the boundary set{0,1};E(s,v) =0 for anyv∈Fswith v(0) =v(1) =0. Moreover, we see that theEs-energy measure µhsiofs∈Fs is equal to dsand that for anyϕ∈C2(R)and anyu∈Fs,
Es(u,ϕ(s)) =1 2
(u(1)ϕ0(b)−u(0)ϕ0(a))−1 2
∫ 1 0
uϕ00(s)ds. (1.4) Letx∈[0,1]. By the theory of one-dimensional diffusions (see e.g. [19, V.46–47]), under Px we can construct a continuous local martingale M ={Mt}t∈[0,∞) and a one- dimensional Brownian motionB={Bt}t∈[0,∞)on the same sample space as that ofXs,m, so thatM0=B0=0Px-a.s. and
s(Xts,m) =B[a,b]hMi
t=s(x)+BhMit+LahMi
t−LbhMi
t and Mt=BhMit, t∈[0,∞), Px-a.s.; (1.5) hereB[a,b]={B[a,b]t }t∈[0,∞) is the reflecting Brownian motion started at s(x)driven by the Brownian motion B with local times La={Lta}t∈[0,∞) at a and Lb={Lbt}t∈[0,∞) at b, i.e.
(B[a,b],La,Lb)is the pathwisely unique triple ofR-valued continuous processes withB[a,b]
[a,b]-valued and started ats(x),La,Lbnon-decreasing and started at 0 and such that,Px-a.s., Bt[a,b]=s(x) +Bt+Lat −Ltb, t∈[0,∞),
∫ ∞ 0
1(a,b](B[a,b]t )dLta=
∫ ∞ 0
1[a,b)(B[a,b]t )dLtb=0.
(1.6) (1.6) is called theSkorohod equation for the reflecting Brownian motion on[a,b]started at s(x)driven by B. In particular, lettingτt:=inf{u∈[0,∞)| hMiu>t}we see that
s(Xτs,mt ) =B[a,b]t =s(x) +Bt+Lat −Lbt, t∈[0,∞), Px-a.s. (1.7) On the other hand, since the processN:={Nt:=LahMi
t−LbhMi
t}t∈[0,∞)is continuous and of bounded variation, we see that (1.5) actually gives theFukushima decomposition for s:
s(Xts,m)−s(X0s,m) =Mt+Nt for anyt∈[0,∞), Px-a.s. (1.8) It follows that the equality (1.7) is obtained as the time change of the Fukushima decompo- sition (1.8) forsby the right-continuous inverseτ(·)ofhMi(·).
In this paper, we extend these facts to the case of certain harmonic functions on a generalrecurrent strong local regular Dirichlet space. To state our main results, letEbe a locally compact separable metrizable space with one-point compactificationE∆=E∪{∆}, letmbe a Radon measure onEwith full support and letX= ({Xt}t∈[0,∞],{Px}x∈E∆)be an m-symmetric Hunt process onEwhose Dirichlet form(E,F)onL2(E,m)is regular. Let Fedenote the associated extended Dirichlet space and letuedenote anyE-quasi-continuous m-version ofu∈Fe, which is unique up toE-q.e. Leta,b∈R,a<band suppose that F1,F2⊂E are (E-quasi-)closed sets admittingu∈Fesuch thateu=a E-q.e. onF1and e
u=bE-q.e. onF2. Then let h∈Febe a(F1∪F2)-harmonic function satisfyingeh=eu E-q.e. onF1∪F2, which does exist by [9, Theorem 4.6.5]. Let
eh(Xt)−eh(X0) =Mt[h]+Nt[h] (1.9) be the Fukushima decomposition forh, whereM[h]={Mt[h]}t∈[0,∞)is a martingale additive functional andN[h]={Nt[h]}t∈[0,∞)is a continuous additive functional of zero energy. Note that if(E,F)is strong local thenM[h]is continuous by [9, Lemma 5.5.1 (ii)]. LethM[h]i= {hM[h]it}t∈[0,∞)be the quadratic variation ofM[h], which is a positive continuous additive functional with Revuz measure equal to theE-energy measureµhhi ofh. DefineσB:=
inf{t∈(0,∞)|Xt∈B}forB⊂E∆. The following is a summary of the main results of this paper (Theorems 2.12 and 3.6).
Theorem 1.1. Assume that(E,F)is recurrent and strong local.
(1) Letϕ∈C2(R)satisfyϕ0(a) =ϕ0(b) =0. Thenϕ(h)∈Feand E(u,ϕ(h)) =−1
2
∫
Eueϕ00(eh)dµhhi, u∈Fe∩L∞(E,m). (1.10) (2) Suppose additionally thatσF1∨σF2 <∞Pm-a.s., wherePm[(·)]:=∫EPx[(·)]dm(x). Let τt :=inf{s∈[0,∞)| hM[h]is>t}fort∈[0,∞). Then forE-q.e.x∈E, underPx,Bh:=
{Mτ[h]t }t∈[0,∞)is a one-dimensional Brownian motion started at0, and{eh(Xτt)}t∈[0,∞)is the reflecting Brownian motion on[a,b]started ateh(x)driven byBh, with local timesLaataand Lbatbequal respectively to the positive variation and the negative variation of{Nτ[h]t }t∈[0,∞). Remark 1.2. In Theorem 1.1 it is sufficient to assume that(E,F)is recurrent andlocal, since the strong locality of(E,F)easily follows from its recurrence and locality.
Since the positive continuous additive functionalhM[h]ihas the Revuz measure µhhi, {eh(Xτt)}t∈[0,∞), Bh={Mτ[h]t }t∈[0,∞) and{Nτ[h]t }t∈[0,∞) are the time change of the original processeswith respect to theE-energy measureµhhiof the harmonic function h. By (1.10), ϕ(eh)∈Dom(Lµhhi)andLµhhi(
ϕ(eh))=ϕ00(eh)/2 for the generatorLµhhiof thetime change of(E,m,E,F)by µhhi, which is the Dirichlet space associated with{Xτt}t∈[0,∞] and is analytically obtained by replacing the reference measuremof the form(E,F)byµhhi; see [4, Chapter 5] and [9, Section 6.2] for general theory of time changes of Dirichlet spaces.
The original motivation for this research is asymptotic analysis of the heat kernel on a fractal called theharmonic Sierpinski gasket(see Figure 4.2 below), which is the image of an injective harmonic map from the usual Sierpinski gasket (Figure 4.1) intoR2and whose
heat kernel has proved to be subject to the two-sidedGaussianbound by [15, Theorem 6.3].
In the end of this article, we briefly describe how we can determine an on-diagonal short time asymptotic behavior of this heat kernel as an application of Theorem 1.1.
The organization of this paper is as follows. In Section 2, we first recall basics of analytic theory of regular Dirichlet forms, then study fundamental properties of harmonic functions and prove Theorem 1.1 (1) (Theorem 2.12). In Section 3, we present a few general results concerning the sample path properties of additive functionals, then give the precise statement of Theorem 1.1 (2) in Theorem 3.6 and prove it. Section 4 is devoted to an application of the main results to asymptotic analysis of the heat kernel on the harmonic Sierpinski gasket.
Notation. In this paper, we adopt the following notations and conventions.
(1)N={1,2,3, . . .}, i.e. 06∈N.
(2) We set inf /0 :=∞. We writea∨b:=max{a,b},a∧b:=min{a,b},a+:=a∨0 and a−:=−(a∧0)fora,b∈[−∞,∞]. We use the same notations for (equivalence classes of) functions. All functions treated in this paper are assumed to beR-valued or[−∞,∞]-valued.
(3) Let(E,B)be a measurable space. For a positive measureµon(E,B), letBµ denote theµ-completion ofB. Asigned measure on(E,B)is by definition anR-valuedcountably additive set function onB.
(4) LetE be a topological space. The Borelσ-field of E is denoted by B(E). We set C(E):={f |f :E→R, f is continuous}andCc(E):={f ∈C(E)|suppE[f]is compact}, where suppE[f]:={x∈E| f(x)6=0}. Also setkfk∞:=supx∈E|f(x)|for f:E→[−∞,∞].
2 Harmonic functions and their energy measures
In the first half of this section, we briefly recall basic facts from analytic theory of regular Dirichlet forms; see [4, 8, 9, 18] for details. Throughout this section, letE be a locally compact separable metrizable space, mbe a Radon measure onE withfull support, i.e.
such thatm(G)>0 for any non-empty open subsetGofE(recall that aRadon measure on E is by definition a positive Borel measure onE for which every compact set is of finite measure), and let(E,F)be a (symmetric) regular Dirichlet form onL2(E,m).
LetFebe the extended Dirichlet space associated with(E,F);u∈Feif and only ifu is an (m-equivalence class of) Borel measurableR-valued function admitting{un}n∈N⊂F such that limk,`→∞E(uk−u`,uk−u`) =0 and limn→∞un=u m-a.e. We extendE to a non- negative definite symmetric bilinear form onFe by setting E(u,u):=limn→∞E(un,un) withu,unas above, so that limn→∞ku−unkE =0, where we writekukE :=E(u,u)1/2for u∈Fe. We haveF =Fe∩L2(E,m)by [9, Theorem 1.5.2 (iii)]. By [9, Corollary 1.6.3], ϕ(u)∈Fe and E(ϕ(u),ϕ(u))≤E(u,u)for u∈Fe and anormal contraction ϕ, i.e. a functionϕ:R→Rsuch thatϕ(0) =0 and|ϕ(s)−ϕ(t)| ≤ |s−t|for anys,t∈R. We write Fe,b:=Fe∩L∞(E,m), which is an algebra under pointwise sum and multiplication by [9, Corollary 1.6.3].
Definition 2.1. We define the 1-capacityCapE associated with(E,F)by
capE(U):=inf{E1(u,u)|u∈F,u≥1m-a.e. onU}, U⊂Eopen, (2.1) CapE(A):=inf{capE(U)|U⊂Eopen,A⊂U}, A⊂E, (2.2)
whereE1(u,v):=E(u,v) +∫Euvdmforu,v∈F. N⊂E is calledE-polarif and only if CapE(N) =0. Moreover, letA⊂Eand letS(x)be a statement onx∈A. Then we say that S holdsE-q.e. on A, orS(x)forE-q.e. x∈A, if and only if the set{x∈A|S(x)fails} isE-polar. WhenA=Ewe simply say‘S holdsE-q.e.’instead.
Clearly, CapE is an extension of capE andm(A)≤CapE(A)for anyA∈B(E). By [9, Theorem A.1.2], CapE is countably subadditive.
Next we defineE-quasi notions by utilizing CapE, as follows.
Definition 2.2. (1) A non-decreasing sequence{Fk}k∈N of closed sets in E is called an E-nestif and only if limk→∞CapE(K\Fk) =0 for any compact subsetKofE.
(2) A functionu:E\N→[−∞,∞], defined outside anE-polar setN, is calledE-quasi- continuousif and only if there exists anE-nest{Fk}k∈Nsuch that∪k∈NFk⊂E\Nandu|Fk
isR-valued continuous for eachk∈N.
(3) A subsetE0 of E is called E-quasi-open (resp.E-quasi-closed) if and only if there exists anE-nest{Fk}k∈Nsuch thatE0∩Fkis open (resp. closed) inFkfor eachk∈N, with Fkequipped with the relative topology inherited fromE.
Given anE-nest{Fk}k∈N,E\∪k∈NFkis anE-polar set. If we setE1:=E0∩∪k∈NFk in the situation of (3) above, thenE1∈B(E),E1isE-quasi-open (resp.E-quasi-closed) andE0\E1isE-polar. A[−∞,∞]-valued functionudefinedE-q.e. isE-quasi-continuous if and only ifuisR-valuedE-q.e. andu−1(I)isE-quasi-open for any open subsetIofR. If uisE-quasi-continuous, thenu≥0m-a.e. if and only ifu≥0E-q.e. by [9, Lemma 2.1.4], anduadmits a Borel measurableE-quasi-continuous functionv:E→Rsuch that u=v E-q.e. By [9, Theorem 2.1.7], for anyu∈Fethere exists anE-quasi-continuous function vsuch thatu=v m-a.e., and suchvis called anE-quasi-countinuous m-version of u, which is unique up toE-q.e. Foru∈Fe, letuedenote anyE-quasi-countinuousm-version ofu.
Definition 2.3. Let µ be a positive Borel measure on E charging no E-polar set, i.e.
µ(N) =0 for any E-polarN ∈B(E). (Note that then everyE-polar, E-quasi-open or E-quasi-closed set belongs toB(E)µ and that everyE-quasi-continuous function defined E-q.e. isB(E)µ-measurable.)
(1)µis called anE-smooth measureif and only ifµ(Fk)<∞for anyk∈Nfor someE-nest {Fk}k∈N. The collection of allE-smooth measures is denoted bySE.
(2)Fµ⊂Eis called anE-quasi-support ofµif and only if it isE-quasi-closed,µ(E\Fµ) = 0 andFµ\FisE-polar for anyE-quasi-closed setF⊂Ewithµ(E\F) =0.
Ifµ∈SE and{Fk}k∈Nis anE-nest forµas in Definition 2.3 (1), thenµ(E\∪k∈NFk) = 0 and henceµ isσ-finite. Any Radon measure onE charging noE-polar set belongs to SE; it suffices to setFk:=Gk, where{Gk}k∈N is a non-decreasing sequence of relatively compact open subsets ofEwith∪k∈NGk=E. By [9, Theorem 4.6.3], everyµ∈SE admits anE-quasi-supportFµ∈B(E).
Associated withu∈Fe,bis theE-energy measureµhui; by [9, Theorem 5.2.3] we have
∫ E
f de µhui=2E(u f,u)−E(u2,f), u,f ∈Fe,b, (2.3) where, for eachu∈Fe,b,µhui(∈SE by [9, Lemma 3.2.4]) is defined as the unique positive Borel measure on E satisfying (2.3) for any f ∈F∩Cc(E)with f in place of ef in the
integrand. Foru∈Fe,b, [2, Proposition I.4.1.1] implies thatµhui(E)≤2E(u,u)and that µhϕ(u)i≤µhui for any normal contractionϕ. (2.4) Also foru,v∈Fe,b, we define a Borel signed measureµhu,vi onEbyµhu,vi:= (µhu+vi− µhu−vi)/4. (2.3) yields
∫ E
ef dµhu,vi=E(u f,v) +E(v f,u)−E(uv,f), u,v,f ∈Fe,b, (2.5) and henceFe,b×Fe,b3(u,v)7→µhu,viis bilinear and symmetric. Therefore we easily see that for anyu,v∈Fe,band any bounded Borel measurable f :E→[0,∞),
[∫
E
f dµhu,vi
]2
≤∫
E
f dµhui
∫ E
f dµhvi, (2.6)
[∫
E
f dµhui
]1/2
−[∫
E
f dµhvi
]1/2 2≤∫
E
f dµhu−vi≤2kfk∞ku−vk2E. (2.7) Then by a limiting procedure using (2.4), (2.7) and [9, Corollary 1.6.3], forany u,v∈Fe
we can uniquely define a finiteE-smooth measureµhui andµhu,vi:= (µhu+vi−µhu−vi)/4 so thatµhui(E)≤2E(u,u)andFe×Fe3(u,v)7→µhu,viis bilinear and symmetric. Again we get (2.6) and (2.7) in the same way, and we can also verify (2.4) by using Banach-Saks theorem (see [4, Theorem A.4.1] or [18, Theorem A.2.2]). It is immediate by (2.6) and (2.7) thatµhu1,vi=µhu2,vi foru1,u2,v∈Fe withku1−u2kE =0. Moreover, if(E,F)is strong local, then we have the following chain rule forµh·i, which often plays essential roles in analysis of strong local Dirichlet forms.
Lemma 2.4([9, Theorem 3.2.2]). Letn∈N,u1, . . . ,un∈Feand letϕ=ϕ(x1, . . . ,xn)∈ C1(Rn)satisfyϕ(0) =0. Suppose either thatu1, . . . ,un∈Fe,bor that∂ϕ/∂xiis bounded onRnfor anyi∈ {1, . . . ,n}. Thenϕ(u1, . . . ,un)∈Fe. Moreover, if in addition(E,F)is strong local, then for anyv∈Fe,
dµhϕ(u1,...,un),vi=
∑
n i=1∂ϕ
∂xi
(eu1, . . . ,uen)dµhui,vi. (2.8) Remark 2.5. [2, Proposition I.4.1.1] and [9, Theorems 3.2.2 and 5.2.3] are stated mainly for functions inF∩L∞(E,m)orF and not necessarily for those inFe,bor Fe, but we easily see that they are valid for functions inFein the following manner:
[2, Proposition I.4.1.1] can be easily extended to functions inFe,bby using Banach- Saks theorem. For the other two theorems, chooseη∈L1(E,m)∩L∞(E,m)so thatη>0 m-a.e., and setFη:=Fe∩L2(E,η·m), where(η·m)(A):=∫AηdmforA∈B(E). Then by [9, Theorem 6.2.1], (E,Fη)is a regular Dirichlet form on L2(E,η·m), and by [9, Theorem 3.1.2, Problems 3.1.1 and 1.4.1] it is strong local if(E,F)is. By [9, Corollary 4.6.1 and the argument before Lemma 6.2.9], the notion ofE-nest and theE-quasi notions with respect to(E,Fη)(onL2(E,η·m)) coincide with those with respect to(E,F)(on L2(E,m)). Moreover,Fη∩L∞(E,η·m) =Fe,b, and for anyu1, . . . ,un∈Fewe can choose ηas above so thatui∈L2(E,η·m)and henceui∈Fηfori∈ {1, . . . ,n}. Now [9, Theorems 3.2.2 and 5.2.3] applied to functions inFη∩L∞(E,η·m)orFηyield the desired assertions for functions inFe,borFe.
We now start our study of harmonic functions and theirE-energy measures. First, we give the definition of harmonic functions.
Definition 2.6. LetF⊂EbeE-quasi-closed and setFFu:={v∈Fe|ev=euE-q.e. onF} foru∈Fe. We callh∈FeF-harmonicif and only if
E(h,h) =inf{E(v,v)|v∈FFh} or equivalently, E(h,v) =0, ∀v∈FF0. (2.9) LetF⊂EbeE-quasi-closed. The equivalence of the two conditions in (2.9) forh∈Fe
is obvious. Letu∈Fe. Then by [9, Theorems 4.1.3, 4.2.1 (ii), 4.6.5 and A.2.6 (i)], there exists anF-harmonic functionh∈FFu. (2.9) implies that, ifh1,h2∈FFuareF-harmonic thenkh1−h2kE =0 and henceµhh1i=µhh2i. Also by (2.9), ifh∈FFuisF-harmonic and ϕis a normal contraction such thatϕ(eu) =ueE-q.e. onF, thenϕ(h)is also anF-harmonic function belonging toFFu.
The following lemma will be used in the proof of Lemma 3.3.
Lemma 2.7. Letu∈Fe. IfF⊂Eis anE-quasi-support ofµhuithenuisF-harmonic.
Proof. Letv∈FF0,`∈Nand setu`:= (−`)∨(u∧`)andv`:= (−`)∨(v∧`). Letn∈N, n≥2. Then (2.5) yields 2E(u`,v`n) =∫Eve`dµhu`,v`n−1i+∫E(ve`)n−1dµhu`,v`i=0, where the latter equality follows byµhui(E\F) =0, (2.4) and (2.6).
Next letϕ∈C2(R)satisfyϕ(0) =ϕ0(0) =0. Thenϕ(v`)∈Fe by Lemma 2.4. By approximatingϕ00uniformly on[−`, `]by polynomials, we see that there exists a sequence of polynomials{ϕn}n∈Nsuch thatϕn(0) =ϕn0(0) =0 and supx∈[−`,`]|ϕn0(x)−ϕ0(x)| →0 asn→∞. The argument in the previous paragraph yieldsE(u`,ϕn(v`)) =0, and letting n→∞results inE(u`,ϕ(v`)) =0 since limn→∞kϕn(v`)−ϕ(v`)kE =0 by [9, (3.2.27)].
Finally, choose f ∈C1(R)so that 0≤f ≤1, f(0) =0 and f(x) =1 for|x| ≥1, and set ψn(x):=∫0xf(ny)dy. Thenψn∈C2(R)andψn(0) =ψn0(0) =0. Similarly to [9, Corollary 1.6.3] we have limn→∞kv`−ψn(v`)kE=0 and henceE(u`,v`) =limn→∞E(u`,ψn(v`)) =0.
Now letting`→∞yieldsE(u,v) =0 by [9, Corollary 1.6.3]. ThusuisF-harmonic.
Given anE-quasi-closed setF⊂E andu∈Fe, anF-harmonic functionh∈FFumay notbe unique since(E,F)isnotassumed to be irreducible. Nevertheless we still have a kind of equivalence betweenF-harmonic functions belonging toFFu, as follows.
Lemma 2.8. LetF⊂EbeE-quasi-closed,u∈Feandh1,h2∈FFubeF-harmonic. Then (1)he1=he2µhh1i-a.e.(Recall thatµhh1i=µhh2i.)
(2) Letϕ∈C1(R)satisfyϕ(0) =0. Suppose either thath2∈Fe,bor thatϕ0is bounded on R. Thenϕ(h1),ϕ(h2)∈Feandkϕ(h1)−ϕ(h2)kE =0.
Proof. (1) Let f:=|h1−h2|∧1. Then f ∈FF0andkfkE ≤ kh1−h2kE =0. Let`∈Nand g`:= (−`)∨(h1∧`). (2.3) implies that∫Ef de µhg`i=2E(g`f,g`)−E(g2`,f) =2E(g`f,g`).
[4, Exercise 1.1.10] together withkfkE =0 yieldskg`fkE ≤ kfkL∞(E,m)kg`kE ≤ kh1kE. E(g`f,h1) =0 by (2.9), and then 0≤∫Eef dµhg`i=2E(g`f,g`−h1)≤ kh1kEkg`−h1kE. Since lim`→∞kg`−h1kE =0 by [9, Corollary 1.6.3], letting `→∞ and (2.7) lead to
∫
Ef de µhh1i=0, which yields the assertion since ef=|he1−he2|∧1E-q.e. and henceµhh1i-a.e.
(2) First suppose either thath1,h2∈Fe,bor thatϕ0is bounded. Then for someψ∈C1(R) andc∈(0,∞), cψ is a normal contraction and ϕ(hi) =ψ(hi)m-a.e. fori=1,2. Thus
ϕ(hi)∈Fe andc2µhϕ(hi)i≤µhh1i for i=1,2, in view of (2.4). Let F0 be an E-quasi- support of µhh1i. Then (1) and [9, Theorem 4.6.2] imply thathe1=he2E-q.e. onF0, and henceϕ(he1) =ϕ(he2)E-q.e. onF0. SinceF0isE-quasi-closed andµhϕ(hi)i(E\F0) =0 for i=1,2, Lemma 2.7 yieldsE(ϕ(h1),ϕ(h1)) =E(ϕ(h2),ϕ(h2)) =E(ϕ(h1),ϕ(h2)), from which it is immediate thatkϕ(h1)−ϕ(h2)kE =0.
Next suppose only that h2∈Fe,b. Let `∈Nsatisfy `≥ kh2kL∞(E,m) and let g` be as in (1). Then since |he1|=|eu|=|he2| ≤` E-q.e. onF, g` is also anF-harmonic func- tion belonging to FFu and hence ϕ(g`),ϕ(h2)∈Fe andkϕ(g`)−ϕ(h2)kE =0 by the previous paragraph. Since lim`→∞ϕ(g`) =ϕ(h1) m-a.e. andkϕ(gk)−ϕ(g`)kE =0 for
k, `≥ kh2kL∞(E,m), an argument similar to [4, Proof of Lemma 1.1.12] showsϕ(h1)∈Fe
andkϕ(h1)−ϕ(g`)kE=0 for`≥ kh2kL∞(E,m). Thus we obtainkϕ(h1)−ϕ(h2)kE =0.
In the main results of this paper, we put the following assumption(BC):
(E,F)is recurrent, i.e.1∈FeandE(1,1) =0. F1,F2⊂EareE-quasi-closed and admitu∈Fesuch thateu=0E-q.e. onF1andeu=1E-q.e. onF2.a,b∈R, a<bandh∈Fa,bis(F1∪F2)-harmonic, whereFs,t:=FFs1+(t−s)u1∪F2 fors,t∈R.
(BC)
In the situation of (BC),F1∩F2 is E-polar and there does exist an(F1∪F2)-harmonic functionh∈Fa,b. Suchu∈Feas in(BC)exists ifF1∩F2=/0 and if eitherF1is closed andF2is compact or vice versa, since(E,F)is regular and1∈Fe.
The following proposition is due to Fitzsimmons [7].
Proposition 2.9([7, (2.7)]). Assume(BC). Then there exists a unique Borel signed mea- sureλ onEcharging noE-polar set, such that
E(h,v) =−(b−a)
∫
Evde λ, v∈Fe,b, (2.10) andλ is independent of particular choices ofa,bandh. Moreover, letλ1(A):=λ(A\F2) andλ2(A):=−λ(A\F1)for A∈B(E). Thenλ1,λ2∈SE, λ =λ1−λ2,λ1(E\F1) = λ2(E\F2) =0andλ1(F1) =λ2(F2) = (b−a)−2E(h,h).
Note that, ifλ is a Borel signed measure onEcharging noE-polar set, then so is its total variation|λ|and hence∫Eevdλ forv∈Fe,bandλ(A\Fi)forA∈B(E),i=1,2 are defined.
The proof of Proposition 2.9 given by Fitzsimmons [7, (2.7)] is based on its probabilistic counterpart shown in [3, Proof of Theorem 3.2]. We give an alternative analytic proof here.
Proof. Leth0,1∈F0,1be(F1∪F2)-harmonic. Thena1+ (b−a)h0,1(∈Fa,b)is also(F1∪ F2)-harmonic and hencek(b−a)h0,1−hkE =0 by (2.9). ThusE(h,v) = (b−a)E(h0,1,v) for v∈Fe, and therefore it suffices to show the assertions for h0,1 instead of h. Since kh0,1−(0∨h0,1)∧1kE =0 by (2.9), we may assume 0≤h0,1≤1m-a.e. Leth1,0:=1−h0,1 andv∈Fe,b. Choose(F1∪F2)-harmonic functionsu0,1∈FFvh1∪F0,12 andu1,0∈FFvh1∪F1,02so that
|u0,1| ∨ |u1,0| ≤ kvkL∞(E,m)m-a.e. (2.9) yieldsE(h0,1,u1,0h0,1) =E(h1,0,u0,1h1,0) =0 and
therefore by (2.3) and (2.9),
E(h0,1,v) =E(h0,1,vh1,0) +E(1−h1,0,vh0,1) =E(h0,1,u1,0)−E(h1,0,u0,1)
=−(
2E(h0,1,u1,0h0,1)−E((h0,1)2,u1,0)) +(
2E(h1,0,u0,1h1,0)−E((h1,0)2,u0,1))
=−
∫
Egu1,0dµhh0,1i+
∫
Egu0,1dµhh1,0i. (2.11)
It follows from (2.11) that|E(h0,1,v)| ≤4kvkL∞(E,m)E(h0,1,h0,1)for anyv∈Fe,b. Then [8, Theorem 4.2] and a time change argument as in Remark 2.5 imply the existence of a Borel signed measureλ onEcharging noE-polar set and satisfying (2.10).
LetG⊂EbeE-quasi-open. By [9, Lemma 4.6.1] we can chooseuG∈FE\G0 so that f
uG>0E-q.e. onG. Letuk:= (0∨kuG)∧1 fork∈N. Thenuk∈FE\G0 and limk→∞uek=1G E-q.e. Now (2.10) yieldsλ(G) =limk→∞∫
Euekdλ =−limk→∞E(h0,1,uk). Therefore the values ofλ forE-quasi-open sets are uniquely determined by the property (2.10), and the Dynkin class theorem [12, Theorem 2.1.3] implies the uniqueness ofλ.
Next we prove thatλ1∈SE. Let λ1=λ1+−λ1−be the Hahn decomposition of λ1. It suffices to show thatλ1is a positive measure, i.e. λ1−=0. Sinceλ1|B(F2)=0 we can chooseL∈B(E\F2)so that λ1+(L) =λ1−(E\L) =0. Let K⊂L be a closed subset of E. SinceE\F2 andE\(F2∪K)are E-quasi-open, by the previous paragraph there exist{uk}k∈N,{vk}k∈N⊂FF02 such that|uk| ∨ |vk| ≤1m-a.e. fork∈N, limk→∞uek=1E\F2 E-q.e. and limk→∞vek=1E\(F2∪K) E-q.e. Then we easily see from (2.10) and (2.11) that
∫
E(uek−vek)+dλ=−E(h0,1,(uk−vk)+)≥0, and lettingk→∞yields 0≤λ(K) =λ1(K) =
−λ1−(K), i.e.λ1−(K) =0. Nowλ1−(L) =sup{λ1−(K)|K⊂L,Kis closed inE}=0 by [5, Theorem 7.1.3] and hence λ1−=0. In exactly the same way we haveλ2∈SE, and in particular λ|B(E\(F1∪F2))=0. Therefore λ =λ1−λ2 andλ1(E\F1) =λ2(E\F2) =0.
Finally, lettingv:=1andv:=h0,1in (2.10) yieldsλ1(F1) =λ2(F2) =E(h0,1,h0,1).
Remark 2.10. The boundary valueeh={aonF1
bonF2 E-q.e. is essential in Proposition 2.9. In fact, for generalu∈Feand anE-quasi-closed setF⊂E, there may not exist such a Borel signed measureλ onEas in (2.10)even if h∈FFuis F-harmonic.
A simple application of Lemma 2.4 and Proposition 2.9 yields the following fact due to Fitzsimmons [7], which is used in Section 4. See [7, Proposition 2.9] for a proof. Note that, ifu∈Fethen byµhui∈SE we can regardeuas a measurable mapue:(
E,B(E)µhui,µhui)
→ (R,B(R)), and therefore the image measureµhui◦ue−1on(R,B(R))is defined.
Corollary 2.11([7, Proposition 2.9]). Assume(BC)and that(E,F)is strong local. Let dydenote the Lebesgue measure on(R,B(R)). Then
µhhi◦eh−1=2E(h,h)
b−a 1[a,b]dy. (2.12)
Now we can state and prove the main theorem of this section, which is in fact an easy consequence of the strong locality, Lemmas 2.4, 2.8 and Proposition 2.9. Note that µhu,vi(E) =2E(u,v)foru,v∈Fe if(E,F)is recurrent, which follows by (2.3). Recall thatϕ(h)∈Fefor anyϕ∈C1(R)in the situation of(BC)by Lemma 2.8 and1∈Fe.
Theorem 2.12. Assume(BC)and that(E,F)is strong local.
(1) Setκ:=2(b−a)−1E(h,h). Then for anyϕ,ψ∈C1(R), E(ϕ(h),ψ(h)) =κ
2
∫ b
a ϕ0(y)ψ0(y)dy. (2.13) (2) Letϕ∈C2(R)andu∈Fe,b. Then
E(u,ϕ(h)) = (b−a)
(ϕ0(b)∫
F2eudλ2−ϕ0(a)∫
F1ude λ1
)−1 2
∫
Eueϕ00(eh)dµhhi. (2.14) Proof. (1) Since we may assume thath∈Fe,bby Lemma 2.8, it follows from Lemma 2.4 and (2.12) that forϕ,ψ∈C1(R),
2E(ϕ(h),ψ(h)) =µhϕ(h),ψ(h)i(E) =
∫
Eϕ0(eh)ψ0(eh)dµhhi=κ∫ b
a ϕ0(y)ψ0(y)dy.
(2) Using (2.5), Lemma 2.4 and Proposition 2.9, we have 2E(u,ϕ(h)) =µhu,ϕ(h)i(E) =
∫
Eϕ0(eh)dµhu,hi
=E(uϕ0(h),h) +E(hϕ0(h),u)−E(uh,ϕ0(h))
=2E(ϕ0(h)u,h)−(
E(hu,ϕ0(h)) +E(ϕ0(h)u,h)−E(hϕ0(h),u))
=−2(b−a)
∫
Eϕ0(eh)ude λ−∫
Eude µhh,ϕ0(h)i
=2(b−a) (ϕ0(b)∫
F2
e
udλ2−ϕ0(a)∫
F1
e udλ1
)−∫
Eueϕ00(eh)dµhhi,
proving (2.14).
3 Reflecting Brownian motion arising from time change by µµµ
hhhhiThroughout this section, we follow the notations introduced in the previous section. The main purpose of this section is to give the precise statement of Theorem 1.1 (2) in Theorem 3.6 and to prove it. In the first part of this section, we recall basics on them-symmetric Hunt process corresponding to(E,F)and its additive functionals. See [4, 9] for details.
LetE∆:=E∪{∆}denote the one-point compactification ofE. In what follows, the measure mis extended toB(E∆)by settingm({∆}):=0, and a[−∞,∞]-valued function f defined (E-q.e.) onEis always set to be 0 at∆when needed; f(∆):=0.
We fix anm-symmetric Hunt processX=(
Ω,M,{Xt}t∈[0,∞],{Px}x∈E∆)
onEwith life timeζ and shift operators{θt}t∈[0,∞]whose Dirichlet form on L2(E,m)is(E,F). Such X does exist by [9, Theorem 7.2.1]. Let F∗={Ft}t∈[0,∞] be the minimum completed admissible filtration as in [9, p.311], which is right-continuous by [9, Theorem A.2.1].
For eachσ-finite positive Borel measure µ on E∆ andA∈F∞, the functionE∆3x7→
Px[A] is B(E∆)µ-measurable, and associated with µ is a measurePµ on (Ω,F∞) given
