On unique extension of time changed reflecting brownian motions

www.imstat.org/aihp 2009, Vol. 45, No. 3, 864–875

DOI: 10.1214/08-AIHP301

© Association des Publications de l’Institut Henri Poincaré, 2009

On unique extension of time changed reflecting Brownian motions

Zhen-Qing Chen

and Masatoshi Fukushima

aDepartment of Mathematics, University of Washington, Seattle, WA 98195, USA. E-mail: [email protected] bBranch of Mathematical Science, Osaka University, Toyonaka, Osaka 560-0043, Japan. E-mail: [email protected]

Received 8 May 2008; accepted 26 August 2008

Abstract. LetDbe an unbounded domain inR^dwithd≥3. We show that ifDcontains an unbounded uniform domain, then the symmetric reflecting Brownian motion (RBM) onDis transient. Next assume that RBMXonDis transient and letY be its time change by Revuz measure1_D(x)m(x)dxfor a strictly positive continuous integrable functionmonD. We further show that if there is somer >0 so thatD\B(0, r)is an unbounded uniform domain, thenY admits one and only one symmetric diffusion that genuinely extends it and admits no killings.

Résumé. NotonsDun domaine non borné dansR^d avecd≥3. Nous montrons que siDcontient un domaine uniforme non borné, alors le mouvement brownien réfléchi (RBM) surDest transient. Par ailleurs, supposons que le RBM X surDest transient et notonsY son changement de temps par une mesure de Revuz1_D(x)m(x)dxpour une fonctionmstrictement positive, continue et intégrable surD. Nous démontrons alors que si il existe unr >0 tel queD\B(0, r)soit un domaine uniformément non borné, alorsY admet une unique extension en une diffusion symétrique qui n’est jamais tuée.

MSC:Primary 60J50; secondary 60J60; 31C25

Keywords:Reflecting Brownian motion; Transience; Time change; Uniform domain; Sobolev space; BL function space; Reflected Dirichlet space;

Harmonic function; Diffusion extension

1. Introduction

Consider an unbounded domainD in R^d withd ≥3 possessing continuous boundary. Then(¹₂D, W^1,2(D)) is a regular strongly local Dirichlet form onL²(D;dx), whereD(u, v)=

D∇u(x)· ∇v(x)dx. The symmetric diffusion processXassociated with this form is the symmetric reflecting Brownian motion onDwith infinite lifetime. In this paper, we are concerned with the following two questions:

(i) WhenXis transient?

(ii) IfXis transient, then almost all sample paths ofXapproach to the point at infinity∂ofDas time goes to infinity.

LetY be its time change by Revuz measure1D(x)m(x)dx for a strictly positive continuous integrable function monD. The time-changed processY has finite lifetime with probability one. How many symmetric diffusions are there that genuinely extendsY?

We show that ifD contains an unbounded uniform domain (see Definition 2.1 below), then RBM X on D is transient. We further show that if there is somer >0 so that D\B(0, r)is an unbounded uniform domain, thenY

1Supported in part by NSF Grant DMS-06-000206.

2Supported by Grant-in-Aid for Scientific Research of MEXT No. 19540125.

admits a unique genuine diffusion extension that admits no killings. HereB(x, r)denotes the ball inR^d centered atx with radiusr. The extension diffusion can be obtained through one-point darning ofY at∂.It can also be obtained through the active reflected Dirichlet form ofY. The key of our approach is to identify the reflected Dirichlet space ofY with the space

BL(D):=

u∈L²_loc(D): ∂u

∂x_i ∈L²(D),1≤i≤d

of Beppo Levi functions onDand to show that BL(D)is the linear space spanned byW_e^1,2(D)and constant functions under suitable conditions onD. HereWe^1,2(D)is the extended Dirichlet space of(¹₂D, W^1,2(D)); see next section for its definition.

2. Transience of reflecting Brownian motion

LetEbe a locally compact separable metric space andma positive Radon measure onEwith full support. Numerical functionsf, g onE is said to be m-equivalent ifm(f =g)=0. We write asf =g [m] in this case. Let(E,F) be a Dirichlet form on L²(E;m) and (Fe,E) be its extended Dirichlet space. A functionu is in Fe if and only if |u|<∞ [m] and there exists{u_n} ⊂F called an approximating sequence ofu such that{u_n}isE-Cauchy and limn→∞un=u [m].It holds then that E(u, u)=limn→∞E(un, un).We know thatF=Fe∩L²(E;m)and that every normal contraction operates on(Fe,E).

Denote by{T_t;t >0}be theL²-semigroup generated by the Dirichlet formEand define{S_t, t >0}by the Bochner integralS_tf=_t

0T_sfds, f ∈L²(E;m).The operatorS_t then extends fromL²∩L¹to a bounded linear operator on L¹(E;m)such that 0≤S_tf ≤S_tf for 0≤t < tandf ∈L¹₊(E;m). Hence forf∈L¹₊(E;m),

Gf (x)= lim

N→∞S_Nf (x)(≤ ∞)

defines a functionGf uniquely up to them-equivalence. The Dirichlet form(E,F)is calledtransient(resp.recurrent) if Gf <∞ [m] for some f ∈L¹₊(E;m) withf >0 [m] (resp.m(0< Gf <∞)=0 for everyf ∈L¹₊(E;m)).

An m-measurable setA⊂Eis said to be{T_t}-invariantif 1AT_t1A^c =0 [m].The Dirichlet form (E,F)is called m-irreducibleif for any{T_t}-invariant setA, we have eitherm(A)=0 orm(A^c)=0. If the Dirichlet form(E,F)is irreducible, then it is either transient or recurrent. The following criteria are known [6]. The Dirichlet form(E,F)is transient if and only if one of the next conditions is satisfied:

(Fe,E)is a real Hilbert space; (2.1)

u∈Fe, E(u, u)=0 ⇒ u=0[m]. (2.2)

The Dirichlet form(E,F)is recurrent if and only if

1∈Fe and E(1,1)=0. (2.3)

Denote by C_c(E)the space of all continuous functions onE with compact support. For a Dirichlet form (E,F), a subspaceCofF∩C_c(E)is called acoreofEifCisE1-dense inFand uniformly dense inC_c(E).A Dirichlet formE is calledregularif it possesses a core. A Dirichlet form(E,F)is said to bestrongly localifE(u, v)=0 whenever K=Supp[u·m]is compact andvis constant in a neighborhood ofK.Any strongly local regular Dirichlet form on L²(E;m)is known to admit an associatedm-symmetric diffusion process onEthat admits no killing insideE.

In the following, we will take the state spaceEto be a Euclidean domainDinR^d. We denote byL^p(D), p≥1, theL^p-space of functions onDwith respect to the Lebesgue measure dx. We focus our attention on the space

BL(D)=

T: ∂T

∂xi ∈L²(D),1≤i≤d

(2.4)

of Schwartz distributionsT .It is known that any distributionT ∈BL(D)can be identified with a function inL²_loc(D) (cf. Schwartz [13], Deny and Lions [5]) so that

BL(D)=

u∈L²_loc(D): ∂u

∂xi ∈L²(D),1≤i≤d

, (2.5)

where the derivatives are taken in Schwartz distribution sense. Members in BL(D)are called BL (Beppo Levi)func- tionsonD. Foru, v∈BL(D), we put

D(u, v)= d i=1

D

∂u

∂x_i

∂v

∂x_i dx. (2.6)

The space BL(D)is known to enjoy the following properties (cf. [5]):

(BL.1) The quotient spaceBL(D)˙ of BL(D)by the subspace of constant functions is a Hilbert space with inner prod- uctD.AnyD-Cauchy sequenceu_n∈BL(D)admitsu∈BL(D)and constantsc_nsuch thatu_nisD-convergent touandu_n+c_nisL²_loc-convergent tou.

(BL.2) A function uonDis in BL(D)if and only if for eachi (1≤i≤d), there is a versionu⁽ⁱ⁾ofu such that it is absolutely continuous on almost all straight lines parallel toxi-axis and the derivative∂u⁽ⁱ⁾/∂xi in the ordinary sense (which exists a.e. onD) is inL²(D). In this case, the ordinary derivatives coincide with the distribution derivatives ofu.

TheSobolev space of order(1,2)on the domainD⊂R^dis defined by

W^1,2(D)=BL(D)∩L²(D). (2.7)

Then

(E,F)= 1

2D, W^1,2(D)

(2.8) is a Dirichlet form onL²(D).

LetDdenote the class of domains in R^d so that the Dirichlet form(E,F)of (2.8) is regular onL²(D); that is, W^1,2(D)∩Cc(D)is both dense in(W^1,2(D),·1,2)and in(C_∞(D),·∞). HereC_∞(D)is the space of continuous functions onDthat vanishes at infinity. A domainD⊂R^dis inDif one of the following two conditions hold:

(i) Dhas continuous boundary (see Theorem 2 on p. 14 of [12]); that is, for everyz∈∂D, there is somer >0 so thatB(z, r)∩Dis the domain lying above the graph of a continuous function;

(ii) Dis anextendable domainin the sense thatW^1,2(D)=W^1,2(R^d)|D; in other words, every functionu∈W^1,2(D) admits a function u∈W^1,2(R^d) such that u=u a.e. on D. When D is an extendable domain, C_c^∞(D)= C_c^∞(R^d)|_D is a core for(E,F). HereC_c^∞(R^d)is the space of infinitely differentiable functions with compact support onR^d.

Definition 2.1. A domainDis called a locally uniform domain if there areδ∈(0,∞]andC >0such that for every x, y∈D with|x−y|< δ,there is a rectifiable curveγ inD connectingx andy withlength(γ )≤C|x−y|and, moreover,

min

|x−z|,|z−y|

≤Cdist z, D^c

for everyz∈γ .

A domain is said to be a uniform domain if the above property holds withδ= ∞.

The above definition is taken from Väisälä [15], where various equivalent definitions are discussed. Uniform do- main and locally uniform domain are also called(ε,∞)-domain and(ε, δ)-domain, respectively, in [10]. For example, the classical van Koch snowflake domain in the conformal mapping theory is a uniform domain inR². Note that every bounded Lipschitz domain is uniform, and everynontangentially accessible domaindefined by Jerison and Kenig in

[9] is a uniform domain (see (3.4) of [9]), while every Lipschitz domain is an(ε, δ)-domain. LetS^d−1denote the unit sphere{x∈R^d:|x| =1}inR^dand(r, θ )the polar coordinates inR^d. It is easy to check that a truncated infinite cone CA,a:= {(r, θ ): r > aandθ∈A}inR^dfor any connected open setA⊂S^d−1with Lipschitz boundary is a uniform domain. However, the boundary of a uniform domain can be highly nonrectifiable and, in general, no regularity of its boundary can be inferred (besides the easy fact that the Hausdorff dimension of the boundary is strictly less thann).

It is known that every locally uniform domainDadmits a bounded linear operator T:

W^1,2(D), · 1,2

→ W^1,2

R^d , · 1,2

(2.9)

(see [10]) and so in particular it is an extendable domain.

WhenD∈D, the Dirichlet form (2.8) is regular and strongly local onL²(D). Its associated diffusion process is called symmetric (or normally) reflected Brownian motion (RBM) onD. WhenD=R^d, then the associated process is thed-dimensional standard Brownian motion and hence (2.8) is transient ifn≥3 and recurrent ifd=1,2.By the following lemma, (2.8) is recurrent for anyD∈Dwheneverd=1,2.

Lemma 2.2. SupposeD∈Dand letX=(X_t,Px)be RBM onD.

(i) RBMX onD is conservative in the sense that Px(ζ = ∞)=1 for q.e.x ∈D,where ζ is the lifetime of X.

Moreover,X(or equivalently,the Dirichlet form(E,F)of(2.8))ism-irreducible.

(ii) SupposeD₁, D₂∈DandD₁⊂D₂.LetX⁽ⁱ⁾be RBM onD_i,i=1,2.IfX⁽¹⁾is transient,then so isX⁽²⁾.IfX⁽²⁾ is recurrent,then so isX⁽¹⁾.

Proof. (i) The conservativeness ofXfollows from [6], Example 5.7.1. The transition function of Xdominates the transition function of the absorbed Brownian motion onDwhich is obtained from then-dimensional standard Brown- ian motion by killing upon leavingDand is known to have a strictly positive transition density onD.ThereforeXis irreducible.

(ii) If we assume the recurrence of (2.8) for D₂, there exist u_n∈W^1,2(D₂), n≥1, which are D-Cauchy and convergent pointwise to 1 onD₂.Thenv_n=u_n|D1, n≥1, satisfy the same properties forD₁, yielding the recurrence of (2.8) forD₁.This implies the first assertion of (ii) because of the irreducibility proven in (i).

Let us denote by(W_e^1,2(D),E)the extended Dirichlet space of (2.8). We may callW_e^1,2(D)theextended Sobolev spaceof order(1,2)onD.We denote byH(D)the space of all harmonic functions onDwith finite Dirichlet integral.

Lemma 2.3. W_e^1,2(D)⊂BL(D)andE(u, u)=¹₂D(u, u)foru∈W_e^1,2(D).If we denote byH^∗(D)the collection of functions inBL(D)that isD-orthogonal to all functions inWe^1,2(D),then

H^∗(D)⊂H(D). (2.10)

Proof. Foru∈W_e^1,2(D), there is a sequence{u_n} ⊂W^1,2(D)which isD-Cauchy and convergent toua.e.D(un, u_n) then converges toE(u, u).By(BL.1), there existv∈BL(D)and constantscn such that{un}isD-convergent tov and the sequence{un+cn}is convergent tovinL²_loc(D).By choosing a subsequence if necessary, we may assume that the latter sequence converges tova.e. Then limn→∞c_n=cexists,u=v−cand consequently,u∈BL(D)and E(u, u)=¹₂D(u, u).

Ifu∈BL(D)isD-orthogonal to all functions inW_e^1,2(D), then, sinceC_c^∞(D)⊂W^1,2(D), we have (u, f )= −D(u, f )=0 for everyf ∈C_c^∞(D),

which implies thatu=0, namely, (a version of)uis harmonic onD.

Informally,H^∗(D)is the space of harmonic functions onDhaving finite Dirichlet energy with zero normal deriv- ative on the boundary∂D.

If (2.8) is transient, then we see from (2.1) that the extended Sobolev space(W_e^1,2(D),¹₂D)is a real Hilbert space and in particular,

u∈W_e^1,2(D)having D(u, u)=0 ⇒ u=0 a.e., (2.11)

which means that the Hilbert space(W_e^1,2(D),¹₂D)is isometrically imbedded into a closed subspace of(BL(D),˙ ¹₂D) by the canonical mapW_e^1,2(D)→ ˙BL(D).Accordingly, we have

Proposition 2.4. Assume that the Dirichlet form(2.8)onL²(D)is transient.Then(W_e^1,2(D),¹₂D)can be regarded as a closed linear subspace of the Hilbert space(BL(D),˙ ¹₂D)by the canonical mapWe^1,2(D)→ ˙BL(D).

BL(D)is a linear space spanned byW_e^1,2(D)andH^∗(D).

The next lemma is well known (cf. [1]).

Lemma 2.5. WhenD=R^d,H^∗(R^d)consists of all constant functions.Ifd≥3,then BL

R^d

is a linear space spanned byW_e^1,2 R^d

and constant functions. (2.12)

See [6], Example 1.5.3 for a proof, where however an incorrect statement that “the spaceWe^1,2(R^d)is obtained from BL(R^d)by removing nonzero constant functions” was made. It should be corrected in the manner of (2.12). See also Remark 2.9 below.

We shall be concerned with a finite measurem(dx)=m(x)dxonDwith a density functionm(x)satisfying (A.1) m(x) >0 for everyx∈Dandm∈C_b(D)∩L¹(D).

We then consider the form defined by E^∗,F^∗

= 1

2D,BL(D)∩L²(D;m)

, (2.13)

which is obtained just by replacingL²(D)withL²(D;m)in (2.7) and (2.8).

Proposition 2.6. The symmetric form(E^∗,F^∗)of(2.13)is a recurrent Dirichlet form on L²(D;m). Its extended Dirichlet space(E^∗,Fe^∗)coincides with the space(¹₂D,BL(D)).

Proof. Since the convergence inL²(D;m)implies the convergence inL²_loc(D), (2.13) can be readily seen to be a closed symmetric form onL²(D;m).Its Markovian property is an immediate consequence of(BL.2). It satisfies the recurrence condition (2.3) becausemis a finite measure onD.We further have

u∈Fe^∗having E^∗(u, u)=0 ⇒ uis constant a.e. (2.14)

To see this, supposeu∈F_e^∗withE^∗(u, u)=0 and putu =ϕ ◦uwith the normal contraction ϕ (t ):=

(− )∨t

∧ , ∈N. (2.15)

Thenu ∈L²(E;m)∩F_e^∗=F^∗and henceE^∗(u , u )=¹₂D(u, u )=0.Thereforeu is a constant and we get (2.14) by letting → ∞.

Denote byF˙_e^∗the quotient space ofF_e^∗by the subspace of constant functions. Just as in the proof of the preceding proposition but using (2.14) in place of (2.11), we conclude that the space(F˙e^∗,E^∗)is isometrically embedded into the space(BL(D),˙ ¹₂D).

Take anyu∈BL(D)and putu =ϕ ◦uas above. By(BL.2),u ∈BL(D)and D(u−u , u−u )=

{x:|u(x)|> }|∇u|²dx→0, → ∞. (2.16)

Sinceu ∈F^∗andu converges toupointwise,umust be an element ofF_e^∗.Hence the above isometric embedding

is an onto map and soFe^∗=BL(D).

When the Lebesgue measure of the domainDis finite, then we can take the functionm≡1 in (2.13) in reducing F^∗toW^1,2(D).Hence

Corollary 2.7. If the domainDis of finite Lebesgue measure,then(¹₂D, W^1,2(D))is a recurrent Dirichlet form on L²(D)and

W_e^1,2(D)=BL(D). (2.17)

We will show in Theorem 3.1 below that BL(D)is the reflected Dirichlet space of(¹₂D, W^1,2(D))for anyD∈D in any dimensiond≥1.

In what follows, we are concerned with the following condition on the domainD⊂R^d: (A.2) D∈Dand the Dirichlet form (2.8) onL²(D)is transient.

This condition forces dimensiond to be greater than or equal to 3 andD to have infinite Lebesgue measure in view of Lemma 2.2 and Corollary 2.7. We shall see from Theorem 2.10 below that, ifd≥3 andD∈Dcontains an unbounded uniform domain, thenDsatisfies the condition (A.2); in other words, RBM onDis transient. This result is almost sharp since RBMX on an infinite cylinderD= {x=(x₁, x₂,·, x_d)∈R^d: _d

i=2x_k²<1}whered≥3 is recurrent becauseXis direct product of the one-dimensional Brownian motion and the reflecting Brownian motion on the closed unit ball. Note that an infinite cylinder isnota uniform domain.

We first prepare a proposition.

Proposition 2.8. LetDbe an unbounded uniform domain inR^d withd≥3andf ∈W_e^1,2(R^d).Iff =ca.e.onD for some constantc,thenc=0.

Proof. Without loss generality, we assume that the origin 0∈D. SinceDis unbounded, there is a sequence{x_k, k≥0}

so that|xk| =2Ck, whereC≥1 is the constant in the Definition 2.1 for the uniform domainD. By definition of uniform domain, for everyk≥1, there is a rectifiable curveγ_k connecting 0 andx_k with 2Ck= |x_k| ≤length(γk)≤ C|x_k| =2C²kand

min

|z|,|z−x_k|

≤Cdist z, D^c

for everyz∈γ_k.

Choosey_k∈γ_ksuch that|y_k| = |y_k−x_k|.Then clearly|y_k| ≥ |x_k|/2=Ck,2|y_k| ≤length(γk)≤2C²kand further dist

y_k, D^c

≥(1/C)(Ck)=k.

AccordinglyDcontains a sequence of open balls{B(yk, k), k≥1}withCk≤ |yk| ≤C²k.

LetX=(X_t,Px)be the Brownian motion inR^d, which is transient asd≥3, ande₁:=(1,0, . . . ,0)∈R^d. Observe that by the Brownian scaling and the rotation invariance of the Brownian motionX,

P0(Xt∈Dfor somet≥Ck)

≥P0

X_t∈B(y_k, k)for somet≥Ck

≥P0

X_t∈B(y_k, k)for somet≥ |y_k|

=P0

X_t∈B

e₁, k/|y_k|

for somet≥1/|y_k|

≥P0

X_t∈B

e1, C⁻²

for somet≥C⁻¹

=:p0>0.

Consequently, P0

n≥1

k≥n

{Xt∈Dfor somet≥Ck}

≥p0. (2.18)

Let∂be the one-point compactification ofR^d. As is well known, Px

tlim→∞X_t=∂

=1, x∈R^d. (2.19)

Anyf ∈W_e^1,2(R^d)admits a versionfwhich is quasi-continuous in the restricted sense with respect to the transient extended Dirichlet space(W_e^1,2(R^d),¹₂D), namely, there exists a decreasing sequence of open subsets{G_n}of R^d such that Cap₍₀₎(Gn)→0, n→ ∞, and the restriction offto each set(R^d\Gn)∪ {∂}is continuous there if we set f (∂) =0. By (2.9) of [3], it then holds that Px(σ_Gn = ∞for somen≥1)=1, q.e. x ∈ R^d, which combined with (2.19) yields thatϕ(x)=1 for q.e.x∈R^d, whereϕ(x)=Px(limt→∞f (X _t)=0).Hence, by the Markov prop- erty,

P0

tlim→∞f (X _t)=0

=(2π)^−d/2

R^de^−|x|2/2ϕ(x)dx=1. (2.20)

Supposef ∈W_e^1,2(R^d) equals a constantca.e. on D, then f=c q.e. onD, and we can deduce from (2.18)

and (2.20) thatc=0.

Remark 2.9. We can make the statement(2.12)more specific as follows:Whend≥3,W_e^1,2(R^d)is obtained from the quotient spaceBL(˙ R^d)by choosing from each equivalence class a special function whose quasi-continuous version has zero limit at∂in the sense of(2.20).

Theorem 2.10. Assume thatd≥3andDis a domain inDthat contains an unbounded uniform domain.Then RBM onDis transient;in other words,Dsatisfies condition(A.2).

Proof. In view of the comparison result Lemma 2.2(ii), we may assume, without loss of generality, thatD is an unbounded uniform domain.

We first show thatW_e^1,2(D)=W_e^1,2(R^d)|D. By [8], Theorem 4.13, there is a linear bounded extension operator S: BL(D)→BL(R^d)in the following sense: for everyu∈BL(D),Su∈BL(R^d)withSu=ua.e. onD, moreover,

∇(Su)

L²(R^d)≤M∇u_L²_(D) for everyu∈BL(D). (2.21)

Foru∈W^1,2(D),Su∈BL(R^d)and so by Lemma 2.5, Su=v₀+c for somev₀∈W_e^1,2(R^d)and constantc. On the other hand, by (2.9), there isT u∈W^1,2(R^d)so thatT u=ua.e. onD. It follows thatT u−v0∈W_e^1,2(R^d)and T u−v0=conD. Proposition 2.8 now implies thatc=0. This proves that the extension operatorSmapsW^1,2(D) intoWe^1,2(R^d). Now foru∈We^1,2(D), there is aD-Cauchy sequence{uk, k≥1} ⊂W^1,2(D)so thatuk→ua.e. onD.

By (2.21) and what we just established,{Su_k, k≥1}is aD-Cauchy sequence inW_e^1,2(R^d). Since(W_e^1,2(R^d),D)is a Hilbert space,SukisD-convergent to somef ∈W_e^1,2(R^d). Taking a subsequence if necessary,Sukconverges tofa.e.

onR^d. It follows thenu=f a.e. onD. This proves thatW_e^1,2(D)=W_e^1,2(R^d)|D, as clearlyW_e^1,2(R^d)|D⊂W_e^1,2(D).

(The above also implies thatSmapsW_e^1,2(D)intoW_e^1,2(R^d). This is because foru∈W_e^1,2(D),Su∈BL(R^d)and so Su=v₀+cfor somev₀∈W_e^1,2(R^d)and constantc. On the other hand, there is somef∈W_e^1,2(R^d)so thatf =u a.e. onD. Thus we havef −v₀=conD. Sincef −v₀∈W_e^1,2(R^d), we have by Proposition 2.8 thatc=0. This proves thatSu=v0∈W_e^1,2(R^d).)

To show that RBM on D is transient, by (2.2) and Lemma 2.3, it suffices to show that if u∈W_e^1,2(D) has D(u, u)=0, thenu=0 onD. Supposeu∈W_e^1,2(D)andD(u, u)=0. Clearly,u=ca.e. onDfor some constantc.

We know from the above that there isu0∈We^1,2(R^d)so thatu0=u=ca.e. onD. It follows from Proposition 2.8

thatc=0 and sou=0.

3. Unique symmetric extension and the space of BL-functions

We start this section with a domainD∈D, together with a measurem(dx)=m(x)dx onDwhose densitym(x)is strictly positive for everyx∈Dand satisfiesm∈Cb(D). Consider the following symmetric form

E⁽⁰⁾,F⁽⁰⁾

= 1

2D, W_e^1,2(D)∩L²(D;m)

, (3.1)

obtained by just replacing BL(D)withW_e^1,2(D)in (2.13). As for (2.13), the above form can be readily checked to be a Dirichlet form on L²(D;m).Furthermore, it is a regular and strongly local Dirichlet form on L²(D;m).The associated diffusion processY =(Y_t,Px)onDis obtained from the reflecting Brownian motionXby a time change with respect to the additive functionalA_t=_t

0m(X_s)dsin view of [6], Section 6.2.

For a general regular Dirichlet form(E,F), the notions of its reflected Dirichlet space(F^ref,E^ref)and its active reflected Dirichlet form(E^ref,F_a^ref)were originally introduced by Silverstein (see [14]) and have been further studied in [2] when(E,F)is transient. These notions are well defined for the present Dirichlet form (3.1).

Assume thatm∈C_b(D)withm(x) >0 for everyx∈Dand let(E⁽⁰⁾,F⁽⁰⁾)be the Dirichlet form onL²(D;m) defined by (3.1). Following [2], let

F^ref:=

f: f_k:=ϕ^k◦f ∈F_loc⁽⁰⁾and sup

k≥1

D(fk, f_k) <∞

,

F_a^ref:=F^ref∩L²(D;m), E^ref(f, f ):=1

2

D

∇f (x)²dx=1

2D(f, f ) forf∈F^ref.

We call(E^ref,F^ref)and(E^ref,Fa^ref)the reflected Dirichlet form and the active reflected Dirichlet form, respectively, of(E⁽⁰⁾,F⁽⁰⁾)onL²(D;m). In [2], the above notions are defined for transient(E⁽⁰⁾,F⁽⁰⁾)and it is proved there that (E^ref,Fa^ref)is a Dirichlet form onL²(D;m). But these notions can also be defined when(E⁽⁰⁾,F⁽⁰⁾)is recurrent.

Theorem 3.1. LetD∈Dbe a domain inR^d withd≥1.Assume thatm∈C_b(D)withm(x) >0 for everyx∈D.

Then the Dirichlet form(E^∗,F^∗)defined by(2.13)withm(dx)=m(x)dxis the active reflected Dirichlet form of the Dirichlet form(E⁽⁰⁾,F⁽⁰⁾)given by(3.1)andBL(D)is the reflected Dirichlet space of(E⁽⁰⁾,F⁽⁰⁾).

Proof. Forf ∈F^ref∩L^∞(D),f ∈F_loc⁽⁰⁾ and sof ∈BL(D). For generalf ∈F^refandk≥1, definef_k=ϕ^k◦f for k ≥1, where ϕ^k is defined in (2.15). From above we have fk ∈BL(D) with E^ref(fk, fk)= ¹₂D(fk, fk).

Since sup_k≥1D(fk, fk) <∞, there is a Cesàro mean {gk, k≥1} of {fk, k≥1} that is D-Cauchy. Observe that g_k∈F^ref∩L^∞(D)⊂BL(D). By (BL.1), there isu∈BL(D)and a sequence of constants{c_k, k≥1}so thatg_k is D-convergent touandg_k+c_kconverges toulocally inL²(D)ask→ ∞. But limk→∞g_k=f onD. This implies that limk→∞c_k=candf +c=u. Thereforef∈BL(D)and soF^ref⊂BL(D).

Now forf ∈BL(D), clearlyf_k=ϕ^k◦f∈BL(D)∩L²_loc(D)⊂F_loc⁽⁰⁾for everyk≥1. SinceD(fk, f_k)≤D(f, f ), we conclude that f ∈Fa^ref. This completes the proof that BL(D)=F^ref. It then follows immediately that F^∗=

BL(D)∩L²(D;m)=F^ref∩L²(D;m)=Fa^ref.

Taking m(x)≡1 on D in Theorem 3.1 yields in particular that BL(D) is the reflected Dirichlet space for (¹₂D, W^1,2(D))for everyD∈Din any dimension.

We shall next consider a domainD∈Dsatisfying the transience condition (A.2) and the functionmsatisfying condition (A.1). Thend≥3 andDis of infinite Lebesgue measure as is noted in Section 2. Abusing the notation a bit, we also use∂to denote the point at infinity ofD, namely, we letD^∗=D∪ {∂}be the one point compactification ofD.Under condition (A.2), the reflecting Brownian motionX=(X_t,Px)onDsatisfies

Px

tlim→∞X_t=∂

=1 for q.e.x∈D (3.2)

on account of [3], Theorem 2.4. In this case, the Dirichlet form(E⁽⁰⁾,F⁽⁰⁾)of (3.1) is transient and its associated diffusion process, time-changed Brownian motionY =(Y_t,Px)onD has lifetimeζ satisfyingPx(ζ <∞)=1 for q.e.x∈D. This is because, sincem∈L¹(D,dx)and the RBMXonDis transient, by Lemma 1.5.1 of [6],E^Y_x[ζ] = Ex[_∞

0 m(Xs)ds]<∞for q.e.x∈D. Moreover,Yt approaches to∂ast→ζ in view of (3.2).

Let us further make the following assumption on the domainD:

(A.3) H^∗(D) consists of constant functions onD.

HereH^∗(D)is the space of harmonic functions onDappearing in Lemma 2.3. Thus under the transience condition (A.2), (A.3) amounts to assuming that

BL(D)is a linear space spanned byW_e^1,2(D)and constant functions. (3.3) We will show in Theorem 3.5 below that conditions (A.2) and (A.3) hold for every unbounded uniform domain.

Moreover Proposition 3.6 will imply that the condition (A.3) is satisfied by any domainD∈Dthat is the complement of a compact set.

Under the stated three conditions, we shall compare the form (E^∗,F^∗) defined by (2.13) and the (E⁽⁰⁾,F⁽⁰⁾) defined by (3.1). They are Dirichlet forms onL²(D;m)and related to each other by

F^∗=

u=u₀+c: u₀∈F⁽⁰⁾, cis constant ,

E^∗(u, u)=E⁽⁰⁾(u0, u0)=¹₂D(u0, u0) (3.4)

in view of Proposition 2.4.

Letm^∗be the extension ofmfromDtoD^∗obtained by setting m^∗

∂D∪ {∂}

=0.

By identifyingL²(D;m)withL²(D^∗;m^∗), we can regard(E^∗,F^∗)as a Dirichlet form onL²(D^∗;m^∗).

Theorem 3.2. Assume that conditions(A.1), (A.2)and(A.3)hold.

(i) (E^∗,F^∗)is a recurrent,strongly local and regular Dirichlet form onL²(D^∗, m^∗).

(ii) The associated diffusion processY^∗=(Y_t^∗,P^∗_x)onD^∗is a conservative extension of the time changed transient reflected Brownian motionY onDand satisfies

P^∗_x(σ∂<∞)=1 for q.e.x∈D^∗. (3.5)

Proof. (i) The recurrence condition (2.3) is trivially satisfied by(E^∗,F^∗).If we letC= {u+c: u∈C_c^∞(D), cis constant}, thenC⊂C(D^∗)andCis readily seen to be a core of the Dirichlet form(E^∗,F^∗).The strong locality of (E^∗,F^∗)can be proved in the same way as in the proof of Theorem 3.2 of [7], where a Dirichlet form quite similar to (3.4) was studied in a rather general context.

(ii) SinceY is associated with(E⁰,F⁰),Y^∗is an extension ofY fromDtoD^∗=D∪ {∂}. The point∂is notm^∗- polar forX^∗becauseP^∗_m∗(σ_∂<∞)dominates the quantityPm(ζ <∞) >0 forY.On the other hand, the irreducibility of(E^∗,F^∗)can be verified in a similar manner to [7]. Hence (3.5) holds in view of [6], Theorem 4.6.6.

Remark 3.3. The extended diffusionY^∗ofYin the above theorem can be also constructed stochastically by a darning ofY at∂as in[7]and[3],namely,by piecing together the excursions ofY around∂according to an excursion-valued Poisson point process{pt}.The characteristic measurenof{pt}is described as[7], (4.4)in terms of the transition function{qt}ofY and the{qt}-entrance law{μt}onEuniquely determined by the equation

m= _∞

0

μtdt.

Using Theorem 3.1, we can further show thatY^∗is the uniquem-symmetric continuous genuine extension ofY in the following sense.

Theorem 3.4. Suppose that there is a Luzin spaceEso thatDis embedded continuously inEas a dense open subset and there is an m-symmetric diffusionZ on E that admits no killings inside and thatY is the proper subprocess ofZ killed upon leaving D.Here the measure mis extended to E by settingm(E\D)=0.Then the symmetric Dirichlet form ofZonL²(E;m)coincides with that ofX^∗onL²(D;m);in other words,ZunderPmhas the same finite-dimensional distributions as that ofY^∗underP^∗_m.

Note that althoughZandX^∗live on different state spaces, sincem(E\D)=0 andm({∂})=0, Pm(Z_t∈E\D)=P^∗_m

Y_t^∗=∂

=0 for everyt≥0

and so we can compare the finite-dimensional distributions ofZunderPmwith those ofY^∗underP^∗_m.

Proof of Theorem 3.4. Let(E^Z,F^Z)denote the symmetric Dirichlet form ofZonL²(E;m). It is known (cf. [11]) that(E^Z,F^Z)is quasi-regular onEand so, by [4], it is quasi-homeomorphic to a regular Dirichlet form on a locally compact metric space. Consequently, the results established under the regular Dirichlet form framework in [6] and [14] apply toZand(E^Z,F^Z).

LetLandL⁰be theL²-generators of the Dirichlet forms(E^Z,F^Z)onL²(E;m)and(E⁽⁰⁾,F⁽⁰⁾)onL²(D;m), respectively. SinceY is the part process ofZ killed upon leavingD, for everyf ∈Dom(L), there is a uniqueE^Z- orthogonal decompositionf =f0+h, wheref0∈F⁽⁰⁾ andh∈F^Z so thatE^Z(g, h)=0 for everyg∈F⁽⁰⁾. Note thathis harmonic with respect to the diffusionY (see [6]). Sincef ∈Dom(L), we have in particular

(−Lf, g)_L2(D;m)=E^Z(f, g)=E⁽⁰⁾(f0, g) for everyg∈F⁽⁰⁾⊂F^Z.

The above implies thatf₀∈Dom(L⁰)andL⁽⁰⁾f₀=Lf. In the terminology of Silverstein ([14], Definition 15.1 on p. 152),L is contained in thelocal generator of (E⁽⁰⁾,F⁽⁰⁾). So by [14], Theorem 15.2, and its proof as well as Theorem 3.1 above, we have

F^Z⊂F^∗ and E^Z(f, f )≥E^∗(f, f ) forf∈F^Z.

SinceZ is a genuine extension ofY,F^Z must be the same asF^∗ in view of (3.4). For anyf =f0+c1∈F^∗with f0∈F⁽⁰⁾, since(E^Z,F^Z)is local and admits no killing, we have

E^Z(f, f )=E^Z(f0, f0)=E⁽⁰⁾(f0, f0)=E^∗(f, f ).

This proves that(E^Z,F^Z)=(E^∗,F^∗).

Finally we give sufficient conditions for (A.3) to hold.

Theorem 3.5. LetD⊂R^dbe a uniform domain.ThenH^∗(D)consists of all constant functions onD.In particular, ifd≥3andD⊂R^d is an unbounded uniform domain,thenDsatisfies conditions(A.2)and(A.3).

Proof. Again we use [8], Theorem 4.13, which states that there is a linear bounded extension operatorS: BL(D)→ BL(R^d)in the following sense: for everyu∈BL(D),Su∈BL(R^d)withSu=ua.e. onDand with bound (2.21).

Foru∈H^∗(D)⊂BL(D),Su∈BL(R^d). Therefore, by Lemma 2.5,Su=f0+cfor somef0∈W_e^1,2(R^d)and some constantc. Note thatf₀|D∈W_e^1,2(D). Sincef₀|D=u−c∈H^∗(D), we obtain

D(f0, f0)=0, (3.6)

which implies thatf0=c1for some constant. Henceu∈H^∗(D)equals a constantc+c1onD.

The second statement of the theorem follows from the above and Theorem 2.10.

We can readily deduce the next proposition from the above theorem. In particular, any unbounded domainD⊂ R^d, d≥3, inDwith compact boundary satisfies (A.3).

Proposition 3.6. IfD⊂R^d,d≥3is an unbounded domain inDsuch thatD\B(0, r)is a uniform domain for some r >0,then condition(A.3)holds.

Proof. By normal contraction property, it suffices to show that for every bounded nonnegativeu∈BL(D),u=u₀+c for someu0∈W_e^1,2(D)and a constantc.

We putU=D\B(0, r)and takeφ∈C_c^∞(R^d)such thatφ=1 onB(0, r+1). Asu∈BL(D)⊂BL(U )andUis an unbounded uniform domain, we see by the above theorem that there is a constantcwithu0:=u−c∈W_e^1,2(U ). We claim that(1−φ)u0∈We^1,2(U ). This is because there is a sequencefn∈W^1,2(U )such that sup_n≥1

U|∇fn|²dx=0 andf_n→u₀, n→ ∞, a.e. on U. Since u₀ is bounded, by the normal contraction property, we may assume that {f_n, n≥1}are uniformly bounded. Clearly(1−φ)f_n∈W^1,2(D),(1−φ)f_nconverges a.e. onDto(1−φ)u₀and

sup

n≥1

D

∇

(1−φ)f_n²dx

≤2 sup

n≥1

fn²_L∞(U )

R^d|∇φ|²dx+2(1−φ)²

∞sup

n≥1

U

|∇fn|²dx <∞.

We conclude that(1−φ)u0∈W_e^1,2(D). On the other hand, sinceφu0∈W^1,2(D), we haveu−c=u0=φu0+(1−

φ)u0∈We^1,2(D). This proves the theorem.

Remark 3.7. The second statement of Theorem3.5is “sharp” for(A.3)in view of the following example.

IfDis an unbounded domain inR^dwith two or more infinite branches,then condition(A.3)may fail.Consider D=B(0,1)∪

x=(x₁, x₂, . . . , x_d): x_d²>

d−1

k=1

x_k²

inR^d ford≥3.HereB(x, r)denote the ball centered atx with radiusr >0.ClearlyDis a Lipschitz domain but it is not a uniform domain as it has a bottleneckB1.HoweverDcontains an unbounded uniform domain

C₁:=B(0,1)^c∩ x_d>

x₁²+ · · · +x_d²₋₁ . So by Theorem2.10,condition(A.2)holds forD.

We claim that condition(A.3)does not hold forD.Define C2=B(0,1)^c∩

xd<−

x₁²+ · · · +x²_d−1 .

Letf∈C_b²(D)be such thatf=1onC1andf=2onC2.Clearlyf ∈BL(D)and so we can write it asf=f0+h withf0∈W_e^1,2(D)andh∈H^∗(D).LetXbe RBM onD,which is transient as is noted in the above.By(3.2)

Px

tlim→∞f (Xt)=1 ·Px

tlim→∞f (Xt)=2

>0 for q.e.x∈D.

On the other hand,f0is quasi-continuous in the restricted sense with respect to the transient extended Dirichlet space(W_e^1,2(D),¹₂D):there exists a decreasing sequence of open subsets{Gn}ofD such thatCap₍₀₎(Gn)→0as n→ ∞, and the restriction of f₀ to each set (D\G_n)∪ {∂}is continuous there if we set f₀(∂)=0. By(2.9) of [3], it then holds that Px(σ_Gn = ∞for somen≥1)=1, q.e.x ∈D, which combined with (3.2) leads us to limt→∞f0(X_t)=0 Px-a.s.for q.e.x ∈D.Thus we havelimt→∞h(X_t)=limt→∞f (X_t)Px-a.s.for q.e.x∈D.

This yields thathcannot be a constant.

Remark 3.8. In the case where the reflecting Brownian motion onDis recurrent,we have the identification

W_e^1,2(D),1 2D

=

BL(D),1 2D

. (3.7)