aa r X i v : . [ m a t h . P R ] M a y REGULAR DIRICHLET SUBSPACES AND MOSCOCONVERGENCE
XIUCUI SONG AND LIPING LI
Abstract.
In this paper, we shall explore the Mosco convergence on regularsubspaces of one-dimensional irreducible and strongly local Dirichlet forms.We find that if the characteristic sets of regular subspaces are convergent, thentheir associated regular subspaces are convergent in sense of Mosco. Finally,we shall show some examples to illustrate that the Mosco convergence doesnot preserve any global properties of the Dirichlet forms. Introduction
What we are concerned in this paper is the theory of Dirichlet forms. TheDirichlet form was first raised by A. Beurling and J. Deny [1] in 1959. Then M.Fukushima proved that the regular Dirichlet forms always possess associated sym-metric Hunt processes in his excellent historical works (e.g. [2] [3]) at the beginningof 1970s. This sets up an exact connection between analysis and probability. Onthe other hand, M. Fukushima and J. Ying introduced a new conception, namedby “regular Dirichlet subpsace”, in 2003, see [4] [5]. Roughly speaking, for a givenDirichlet form, a regular Dirichlet subspace is its closed subspace with Dirichlet andregular properties. In 2005 and 2010, M. Fukushima, J. Ying and their co-authorscharacterized the regular Dirichlet subspaces of 1-dim Brownian motions and 1-dimirreducible diffusions by using a special class of scaling functions, see [6] and [7].Furthermore, the second author of this paper, with J. Ying together, made moredeep descriptions about the regular Dirichlet subspaces, such as [8] [9].To introduce the conception of regular Dirichlet subspace, we need to explainthe basic settings of Dirichlet forms briefly. Let E be a measurable space and m a σ -finite measure on E . Naturally, L ( E, m ) is a real Hilbert space, whose norm andinner product are denoted by k · k m and ( · , · ) m . A Dirichlet form on L ( E, m ) isusually written as ( E , F ). Its definition is standard, see [10] and [11]. In particular,if E is a locally compact separable metric space and m is a fully supported Radonmeasure on E , then we may talk about the regularity of Dirichlet forms. Indeed, aDirichlet form ( E , F ) on L ( E, m ) is said to be regular, if
F ∩ C c ( E ) is dense in F with the norm k · k E and dense in C c ( E ) with the uniform norm, where the norm k · k E corresponds to the inner product E ( u, v ) := E ( u, v ) + ( u, v ) m , and C c ( E ) is the class of continuous functions with compact supports on E . More-over, the Borel measurable structure on E is denoted by B ( E ). Without loss ofgenerality, we use f ∈ B ( E ) to represent that the function f is Borel measurable.Thus B ( E ) is formally the class of all Borel measurable functions on E . Further-more, let B + ( E ) and b B ( E ) be all positive Borel measurable functions and boundedBorel measurable functions on E respectively. On the other hand, C ( E ) is the class Mathematics Subject Classification.
Key words and phrases.
Dirichlet forms, regular suspaces, Mosco convergence, minimaldiffusion. of all continuous functions on E , and C b ( E ) , C ( E ) are its subspaces of boundedfunctions and being 0 at infinity. In particular, C ( R ), C ∞ ( R ) are usual notations.We refer more terminologies of Dirichlet forms and potential theory to [11].Let ( E , F ) and ( E ′ , F ′ ) be two regular Dirichlet forms on L ( E, m ). We say( E ′ , F ′ ) is a regular Dirichlet subspace, or a regular subspace in abbreviation, of( E , F ) provided that F ′ ⊂ F , E ( u, v ) = E ′ ( u, v ) , u, v ∈ F ′ . Furthermore, if F ′ = F , then ( E ′ , F ′ ) is a proper regular subspace of ( E , F ). Inparticular, we use ( E ′ , F ′ ) ≺ ( E , F )to stand for that ( E ′ , F ′ ) is a regular subspace of ( E , F ).Our another focus in this paper is the Mosco convergence of Dirichlet forms. Thiskind of convergence was first introduced by U. Mosco [12] in 1994 and then widelyused by lots of researchers, for instance [13] [14] [15]. In particular, it was alsoemployed in [16] to study the stochastic averaging principle of Halmiton dynamicalsystem.Next, we shall briefly introduce the basic definition and some probabilistic sig-nificances of Mosco convergence. For any Dirichlet form ( E , F ) on L ( E, m ), wealways extend the domain of E to L ( E, m ) by E ( u, u ) = ∞ , u ∈ L ( E, m ) \ F . The following definition is given by U. Mosco [12], in which that u n converges to u weakly in L ( E, m ) means that for any v ∈ L ( E, m ), ( u n , v ) m → ( u, v ) m as n → ∞ , and strong convergence means k u n − u k m → n → ∞ . Definition 1.1.
Let { ( E n , F n ) : n ≥ } be a sequence of Dirichlet forms and( E , F ) another Dirichlet form on L ( E, m ). Then ( E n , F n ) is said to be convergentto ( E , F ) in sense of Mosco as n → ∞ , if (a): for any sequence { u n : n ≥ } of functions in L ( E, m ), which is conver-gent to another function u ∈ L ( E, m ) weakly, it holds that(1.1) lim inf n →∞ E n ( u n , u n ) ≥ E ( u, u ); (b): for any function u ∈ L ( E, m ), there always exists a sequence { u n : n ≥ } of functions in L ( E, m ), which is convergent to u strongly as n → ∞ ,such that(1.2) lim sup n →∞ E n ( u n , u n ) ≤ E ( u, u ) . The most important significance of Mosco convergence is that it is equivalentto the convergence of associated semigroups. More precisely, let { T nt : t ≥ } and { G nα : α > } be the semigroup and resolvent of ( E n , F n ), { T t : t ≥ } and { G α : α > } the semigroup and resolvent of ( E , F ). Then ( E n , F n ) is convergentto ( E , F ) in sense of Mosco as n → ∞ , if and only if any one of following assertionsholds: (1): for any t > , f ∈ L ( E, m ), T nt f is convergent to T t f strongly in L ( E, m ) as n → ∞ ; (2): for any α > , f ∈ L ( E, m ), G nα f is convergent to G α f strongly in L ( E, m ) as n → ∞ .Note that the semigoup of Dirichlet form is decided by the probability transitionsemigroup of associated Markov process. Hence the Mosco convergence implies theweak convergence of finite dimensional distributions of associated Markov processes.This fact is one of the reasons why the Mosco convergence is very useful in the theoryof stochastic differential equations. EGULAR DIRICHLET SUBSPACES AND MOSCO CONVERGENCE 3
At the end of this section, let us explain the structure of this paper. In §
2, we shalldescribe the associated Dirichlet forms of irreducible diffusions on 1-dimensionalstate space and characterize their regular subspaces. Particularly, we shall improvethe results of [7] and give another description, say the characteristic sets, of regularsubspaces. In § §
4, we shall provide two conditions on characteristic sets tomake the regular subspaces be Mosco convergent. Finally in §
5, we shall showsome examples to claim that the Mosco convergence cannot maintain the stabilityof global properties of Dirichlet forms. The two convergence methods employed in § The regular subspaces of 1-dim irreducible diffusions and theircharcteristic sets
We always assume that E is R or an open interval of R , which is denoted by I .In other words, E = I = ( a, b ) , where −∞ ≤ a < b ≤ ∞ . The continuous stochastic process X with strong Markovproperty on I is also called a diffusion process. Further assume that(2.1) P x ( σ y < ∞ ) > , ∀ x, y ∈ I, where σ y is the hitting time of { y } . This assumption, usually named by irreducibil-ity, means that any two points of I are connected for X in intuition. Under thiscondition, the diffusion X can be characterized completely by a strictly increasingand continuous function s , which is called the scaling function, and two Radon mea-sures m, k on I . In particular, m is fully supported on I , and X is m -symmetric.Furthermore, k is the so-called killing measure of X , and we may assume that k = 0because of the studies in [9]. Note that for any two constants C, C , if we replace s , m by C · s + C , m/C respectively, then they still describe the same diffusion.We refer more details to [17] and [18]. Definition 2.1.
The boundary point a (resp. b ) of I is called s -approachable,if s ( a +) := lim x ↓ a s ( x ) > −∞ (cid:0) resp. s ( b − ) := lim x ↑ b s ( x ) < ∞ (cid:1) . Furthermore, a (resp. b ) is called an s -regular boundary, if a is s -approachable and there is aconstant c ∈ I such that m (cid:0) ( a, c ) (cid:1) < ∞ (cid:0) resp. b is s -approachable, and there is aconstant c ∈ I such that m (cid:0) ( c, b ) (cid:1) < ∞ (cid:1) .We always assume that the boundary { a, b } of I is the trap of X . That meansif X approaches the boundary, then it dies. Define(2.2) F ( s ,m ) := (cid:26) u ∈ L ( I, m ) : u ≪ s , dud s ∈ L ( I, d s ) (cid:27) , where u ≪ s stands for that u is absolutely continuous with respect to s , or inother words, there is an absolutely continuous function ϕ such that u = ϕ ◦ s . Forany u, v ∈ F ( s ,m ) , set E ( s ,m ) ( u, v ) := 12 Z I dud s dvd s d s . Note that ( E ( s ,m ) , F ( s ,m ) ) is a Dirichlet form on L ( I, m ) but not necessarily aregular one. One may prove that the associated Dirichlet form of diffusion X on L ( I, m ) can be written as(2.3) F ( s ,m )0 := { u ∈ F ( s ,m ) : u ( a ) or u ( b ) = 0 , if a or b is s -regular } , E ( s ,m ) ( u, v ) = 12 Z I dud s dvd s d s , u, v ∈ F ( s ,m )0 . XIUCUI SONG AND LIPING LI
Moreover, ( E ( s ,m ) , F ( s ,m )0 ) is regular and irreducible with a special standard core C ∞ c ◦ s := { ϕ ◦ s : ϕ ∈ C ∞ c ( J ) } , where J := s ( I ) = { s ( x ) : x ∈ I } is also an open interval of R . Here, the irre-ducibility concerns Dirichlet forms (see § E ( s ,m ) , F ( s ,m )0 ) and their global properties. But unfortunately, theydid not assert that all regular subspaces of ( E ( s ,m ) , F ( s ,m )0 ) can be described by thescaling functions, which were provided in [7]. Next, we shall give a brief proof tocover the above shortage. For that, take a fixed point e on I , and set(2.4) S s ( I ) := (cid:26) ˜ s : ˜ s is a strictly increasing and continuous function on I, ˜ s ( e ) = 0 , ˜ s ≪ s , d ˜ s d s = 0 or 1 , d s -a.e. (cid:27) . Note that the choice of e is not essential. Since the scaling functions s and s + C describe the same diffusion process for any constant C , thus we fix the value ofscaling function at a fixed point to avoid the presence of equivalence class. Proposition 2.2.
For any ˜ s ∈ S s ( I ), it holds that( E (˜ s ,m ) , F (˜ s ,m )0 ) ≺ ( E ( s ,m ) , F ( s ,m )0 ) . On the contrary, if ( E ′ , F ′ ) ≺ (cid:0) E ( s ,m ) , F ( s ,m )0 (cid:1) , then there is a scaling function˜ s ∈ S s ( I ) such that ( E ′ , F ′ ) = ( E (˜ s ,m ) , F (˜ s ,m )0 ) . In particular, the regular subspace ( E (˜ s ,m ) , F (˜ s ,m )0 ) is a proper one, if and only if˜ s = s . Proof.
The first and third assertions are the results of [7], see Theorem 4.1 of[7]. Now let ( E ′ , F ′ ) be a regular subspace of (cid:0) E ( s ,m ) , F ( s ,m )0 (cid:1) . It follows fromTheorem 4.1 of [7] that we only need to prove that ( E ′ , F ′ ) is strongly local andirreducible. In fact, the strongly local property of ( E ′ , F ′ ) is a corollary of Theorem 1of [9]. On the other hand, since (cid:0) E ( s ,m ) , F ( s ,m )0 (cid:1) is irreducible and strongly local,from Theorem 4.6.4 of [11] and the definition of regular subspace, we can easilydeduce that ( E ′ , F ′ ) is also irreducible. (cid:3) In other words, the proposition above claims that the scaling function class S s ( I ) characterizes all regular subspaces of (cid:0) E ( s ,m ) , F ( s ,m )0 (cid:1) . Now we shall turn tointroduce another equivalent description of S s ( I ). Set G s ( I ) := (cid:26) G ⊂ I : Z G ∩ ( c,d ) d s > , ∀ c, d ∈ I, c < d (cid:27) . Obviously, any set G in G s ( I ) is defined in sense of d s -a.e., in other words, it shouldbe regarded as a d s -a.e. equivalence class. The following lemma asserts that G s ( I )has an identical status with S s ( I ) for regular subspaces of (cid:0) E ( s ,m ) , F ( s ,m )0 (cid:1) . Lemma 2.3.
There exists a bijective mapping between the scaling function class S s ( I ) and the class G s ( I ) of sets. EGULAR DIRICHLET SUBSPACES AND MOSCO CONVERGENCE 5
Proof.
For any ˜ s ∈ S s ( I ), define(2.5) G ˜ s := (cid:26) x ∈ I : d ˜ s d s ( x ) = 1 (cid:27) . Clearly G ˜ s is defined in sense of d s -a.e., and for any interval ( c, d ) ⊂ I , it holdsthat Z G ˜ s ∩ ( c,d ) d s = Z dc G ˜ s ( x ) d s ( x ) = Z dc d ˜ s > . That implies G ˜ s ∈ G s ( I ).Now we shall prove that the mapping S s ( I ) → G s ( I ) , ˜ s G ˜ s is a bijective mapping. Firstly, it follows from G ˜ s ∈ G s ( I ) that this mapping isdefined well. Secondly, let us prove that it is an injection. Assume that s , s ∈ S s ( I ) satisfy G s = G s , d s -a.e. Then for any x ∈ I , s ( x ) = Z xe d s = Z xe d s d s d s = Z xe G s ( y ) d s ( y );Similarly, we have s ( x ) = Z xe G s ( y ) d s ( y ) . Hence we can deduce that s = s . Finally we shall explain that the mappingabove is also a surjection. In fact, for any set G ∈ G s , let(2.6) ˜ s ( x ) := Z xe G ( y ) d s ( y ) , x ∈ I. We only need to prove ˜ s ∈ S s ( I ) and G ˜ s ∈ G s ( I ). Indeed, from (2.5) we obtainthat ˜ s is strictly increasing. Furthermore, it follows from (2.6) that ˜ s ( e ) = 0, ˜ s ≪ s and d ˜ s d s = 1 G , d s -a.e.This implies that ˜ s ∈ S s ( I ) and G ˜ s ∈ G s ( I ). That ends the proof. (cid:3) In Proposition 2.2 and Lemma 2.3, we obtain two equivalent characterizations ofall regular subspaces of (cid:0) E ( s ,m ) , F ( s ,m )0 (cid:1) . For each regular subspace (cid:0) E (˜ s ,m ) , F (˜ s ,m )0 (cid:1) ,the set G ˜ s in G s ( I ), which corresponds to the scaling function ˜ s , is called thecharacteristic set of (cid:0) E (˜ s ,m ) , F (˜ s ,m )0 (cid:1) . Therefore, we may write down the followingequivalent descriptions:(2.7) ( E (˜ s ,m ) , F (˜ s ,m )0 ) ⇌ ( E (˜ s ,m ) , F (˜ s ,m ) ) ⇌ ˜ s ⇌ G ˜ s . Note that the two Dirichlet forms in (2.7) are equal if and only if neither a nor b is s -regular.The characteristic set is very important in the research of regular subspaces. Forexample, when I = R , m is the Lebesgue measure on R and s is the natural scalingfunction, (cid:0) E ( s ,m ) , F ( s ,m )0 (cid:1) is exactly the associated Dirichlet form of 1-dimensionalBrownian motion. In another work of the second author and his co-author [8], theyfound that if the characteristic set G is open, such as the complement of generalizedCantor set, then the regular subspace and Brownian motion share the same parton G . That means their difference concentrates on the boundary of G . This factconduces to a study about the traces of Brownian motion and its regular subspace.We refer more details to [8]. In the mean time, we denote a subset of G s ( I ) by(2.8) ◦ G s ( I ) := { G ∈ G s ( I ) : G has an open d s -a.e. version } , which will play an important role in what follows. XIUCUI SONG AND LIPING LI Mosco convergence I
In this section, we shall consider the Mosco convergence on regular subspaces of( E ( s ,m ) , F ( s ,m )0 ). Before presenting the first convergence method, we need to provea very useful lemma. Let G ∈ G s ( I ) be a characteristic set and F := G c . Itsassociated scaling function is denoted by ˜ s . The following lemma provides anotherexpression of Dirichlet form ( E (˜ s ,m ) , F (˜ s ,m ) ) from the viewpoint of characteristicset, a special case of which was already presented for Brownian motion in [8]. Lemma 3.1.
It holds that(3.1) F (˜ s ,m ) = (cid:26) u ∈ F ( s ,m ) : dud s = 0 , d s -a.e. on F (cid:27) . Proof.
Note that F (˜ s ,m ) has the expression (2.3). For any u ∈ F (˜ s ,m ) , since u ≪ ˜ s ,it follows that u ≪ s and dud s = dud ˜ s · d ˜ s d s = dud ˜ s · G , d s -a.e.Thus we have du/d s = 0, d s -a.e. on F . On the contrary, assume that u is afunction in the class of right side of (3.1), we only need to prove u ≪ ˜ s and du/d ˜ s ∈ L ( I, d ˜ s ). In fact, for any x, y ∈ I , u ( x ) − u ( y ) = Z xy dud s d s = Z xy dud s · G d s = Z xy dud s d ˜ s . Hence u ≪ ˜ s and du/d ˜ s = du/d s , d ˜ s -a.e. It follows from du/d s ∈ L ( I, d s ) that du/d s ∈ L ( I, d ˜ s ). That implies u ∈ F (˜ s ,m ) , which completes the proof. (cid:3) Now, we assume that { G n : n ≥ } is a sequence of sets in G s ( I ). For each n , G n corresponds to the scaling function s n . Set ( E n , F n ) := ( E ( s n ,m ) , F ( s n ,m ) ).Take another set G ∈ G s ( I ), its associated scaling function is ˜ s , and set ( E , F ) :=( E (˜ s ,m ) , F (˜ s ,m ) ). The following theorem asserts that if a sequence of characteristicsets is decreasing to another characteristic set, then the sequence of their associatedDirichlet forms is convergent in sense of Mosco. Theorem 3.2. If G n ↓ G , d s -a.e., then ( E n , F n ) is convergent to ( E , F ) in senseof Mosco. Proof.
Firstly, we claim that(3.2)
F ⊂ · · · F n +1 ⊂ F n ⊂ · · · F ⊂ F ( s ,m ) . Indeed, for n and n + 1, from G n ⊂ G n +1 , we can deduce that F n +1 := G cn +1 ⊂ G cn := F n . It follows from Lemma 3.1 that F n +1 ⊂ F n . Similarly, from G ⊂ G n ,we have F ⊂ F n . Clearly, F ⊂ F ( s ,m ) .Secondly, we shall prove (b) of Definition 1.1. If u / ∈ F , then E ( u, u ) = ∞ .Clearly, (1.2) is right. For any u ∈ F , let u n := u ∈ F ⊂ F n . Obviously, u n isstrongly convergent to u . Note that ( E n , F n ) and ( E , F ) are both regular subspacesof ( E ( s ,m ) , F ( s ,m ) ). We have E n ( u n , u n ) = E ( s ,m ) ( u n , u n ) = E ( u n , u n ) . Particularly, E n ( u n , u n ) = E ( u, u ). Hence lim sup n →∞ E n ( u n , u n ) = E ( u, u ), whichimplies that (b) is proved.Finally, we turn to prove (a) of Definition 1.1. Assume u n is weakly convergentto u in L ( E, m ). Without loss of generality, we may assume that u n belongs to F n .Or, E n ( u n , u n ) = ∞ , which implies that u n is useless in the left side of (1.1). Fix aninteger N , for any n > N , it follows from u n ∈ F n ⊂ F N that { u n : n ≥ N } ⊂ F N . EGULAR DIRICHLET SUBSPACES AND MOSCO CONVERGENCE 7
In particular, E n ( u n , u n ) = E N ( u n , u n ). Note that a sequence of the same Dirichletform is convergent to itself in sense of Mosco. That implies that(3.3)lim inf n →∞ E n ( u n , u n ) = lim inf n ≥ N,n →∞ E n ( u n , u n ) = lim inf n ≥ N,n →∞ E N ( u n , u n ) ≥ E N ( u, u ) . If for some integer N , u / ∈ F N , then lim inf n →∞ E n ( u n , u n ) ≥ E N ( u, u ) = ∞ .Naturally, lim inf n →∞ E n ( u n , u n ) ≥ E ( u, u ) . Now, assume that for any N , u ∈ F N . From Lemma 3.1, we know that u ∈ F ( s ,m ) ,and du/d s = 0, d s -a.e. on F N . It follows from G N ↓ G that ∪ N ≥ F N = F , where F := G c . Hence we can obtain that du/d s = 0, d s -a.e. on F . By using Lemma 3.1again, we can deduce that u ∈ F . Particularly, since ( E , F ) and ( E N , F N ) in (3.3)are both regular subspaces of ( E ( s ,m ) , F ( s ,m ) ), it follows that E N ( u, u ) = E ( u, u ).From (3.3), we obtain that lim inf n →∞ E n ( u n , u n ) ≥ E ( u, u ) , which completes the proof. (cid:3) Although ( E n , F n ) and ( E , F ) in Theorem 3.2 have the relation (2.7) with thecorresponding regular subspaces, they are not exactly the regular subspaces. Next,we shall discuss some examples of Mosco convergence of real regular subspaces.Particularly, if there is a constant c ∈ I such that m (cid:0) ( a, c ) (cid:1) = m (cid:0) ( c, b ) (cid:1) = ∞ (weuse m ( a +) = m ( b − ) = ∞ to stand for this property), then any irreducible diffusionon I with speed measure m would not have a regular boundary. Hence, the Dirichletspaces in (2.2) and (2.7) are the same. On the other hand, the speed measure isnot essential for the structure of regular subspaces. In [9], we found that after atime change with full quasi support, the structure of regular subspaces maintains. Corollary 3.3.
We make the same assumptions as Theorem 3.2, i.e. G n ↓ G , d s -a.e. If any one of following conditions is satisfied: (1): m ( a +) = m ( b − ) = ∞ ; (2): if there is a constant c ∈ I such that d s (cid:0) G ∩ ( a, c ) (cid:1) < ∞ (cid:0) resp. d s (cid:0) G ∩ ( c, b ) (cid:1) < ∞ (cid:1) , then there exists an integer N such that d s (cid:0) G N ∩ ( a, c ) (cid:1) < ∞ (cid:0) resp. d s (cid:0) G N ∩ ( c, b ) (cid:1) < ∞ (cid:1) ;then ( E ( s n ,m ) , F ( s n ,m )0 ) is convergent to ( E (˜ s ,m ) , F (˜ s ,m )0 ) in sense of Mosco. Proof.
The sufficiency of first condition is clear. We only prove the sufficiency ofsecond one. In fact, it suffices to prove(3.4) F (˜ s ,m )0 ⊂ · · · F ( s n ,m )0 ⊂ · · · F ( s ,m )0 ⊂ F ( s ,m )0 and(3.5) F (˜ s ,m )0 = (cid:26) u ∈ F ( s N ,m )0 : dud s = 0 , d s -a.e. (cid:27) . Indeed, from G n +1 ⊂ G n , we have s n +1 ≪ s n , and d s n +1 /d s n = 1 or 0, d s n -a.e.Then it follows from Proposition 2.2 that F ( s n +1 ,m )0 ⊂ F ( s n ,m )0 . Similarly, we candeduce that (3.4) is right. On the other hand, the second condition of Corollary 3.3means that, a or b is ˜ s -regular, if and only if it is s N -regular. That is because, if a is s N -regular, then from˜ s ≪ s N , d ˜ s d s N = 1 or 0 , d s N -a.e. , we can deduce that a is also ˜ s -regular; on the contrary, if a is ˜ s -regular, whichimplies that m ( a +) < ∞ , and there is a constant c ∈ I such that d s (cid:0) G ∩ ( a, c ) (cid:1) < ∞ , XIUCUI SONG AND LIPING LI then from the second condition, we may obtain that a is s N -regular. Therefore wecan complete the proof of (3.5), which is similar to that of Lemma 3.1. (cid:3) In particular, if a and b are both s -regular boundaries, then a and b are also˜ s -regular and s n -regular. Thus the second condition in Corollary 3.3 is naturallysatisfied. At the end of this section, we shall give another example to show that iftwo conditions above are not satisfied, then G n ↓ G , d s -a.e. may not imply that( E ( s n ,m ) , F ( s n ,m )0 ) is convergent to ( E (˜ s ,m ) , F (˜ s ,m )0 ) in sense of Mosco. Example 3.4.
Let I = R , s ( x ) = x , and assume that m ( R ) < ∞ . Further assumethat G is the set, which is given by Example 5.2 of [7]. Clearly, G ∈ G s ( R ) and | G | < ∞ , where | · | represents the Lebesgue measure on R . Define(3.6) G n := G [ (cid:18) ∪ k ∈ Z ( k − n , k + 1 n ) (cid:19) . Let ˜ s and s n denote the associated scaling functions of G and G n respectively.Particularly, ˜ s ( −∞ ) > −∞ , ˜ s ( ∞ ) < ∞ . Note that m ( R ) < ∞ , which implies that −∞ and ∞ are both ˜ s -regular boundaries. Hence we can obtain F (˜ s ,m ) = F (˜ s ,m )0 . On the other hand, one may easily check that G n ∈ G s ( R ) and G n ↓ G, a.e.For each n , it follows that s n ( −∞ ) = −∞ , s n ( ∞ ) = ∞ . Thus −∞ and ∞ are not s n -regular boundaries, and F ( s n ,m ) = F ( s n ,m )0 . Therefore, it follows from Theorem 3.2 that ( E ( s n ,m ) , F ( s n ,m )0 ) is convergent to( E (˜ s ,m ) , F (˜ s ,m ) ) in sense of Mosco. However, F (˜ s ,m ) = F (˜ s ,m )0 . Then by the unique-ness of Mosco convergence, we know that ( E ( s n ,m ) , F ( s n ,m )0 ) cannot converge to( E (˜ s ,m ) , F (˜ s ,m )0 ) in sense of Mosco.4. Mosco convergence II In §
3, we considered the Mosco convergence for decreasing characteristic sets. Inthis section, we shall discuss the increasing case.We first assert that Mosco convergence is invariant under spatial transforms ofDirichlet forms. More precisely, let { ( E n , F n ) : n ≥ } be a sequence of Dirichlet forms on L ( E, m ), ( E , F ) another Dirichlet form on L ( E, m ).Moreover, ( E n , F n ) is convergent to ( E , F ) in sense of Mosco. Assumethat ˆ E is another measurable space and j : E → ˆ E, x ˆ x is a measurable mapping. Let ˆ m := m ◦ j − be the image measure of m with respectto j . Then j ∗ : L ( ˆ E, ˆ m ) → L ( E, m ) , ˆ f ˆ f ◦ j is an isometric mapping, and the image space of j ∗ is a closed subspace of L ( E, m ).Set further ˆ F := (cid:8) ˆ f ∈ L ( ˆ E, ˆ m ) : j ∗ ˆ f ∈ F (cid:9) , ˆ E ( ˆ f , ˆ g ) := E ( j ∗ ˆ f , j ∗ ˆ g ) , ˆ f , ˆ g ∈ ˆ F . If j ∗ maps L ( ˆ E, ˆ m ) onto L ( E, m ), then ( ˆ E , ˆ F ) is a Dirichlet form on L ( ˆ E, ˆ m ).Similarly, we can define the image Dirichlet form ( ˆ E n , ˆ F n ) of ( E n , F n ) under j ∗ . EGULAR DIRICHLET SUBSPACES AND MOSCO CONVERGENCE 9
Lemma 4.1.
Assume that j ∗ is a surjection. On L ( ˆ E, ˆ m ), the Dirichlet form( ˆ E n , ˆ F n ) is convergent to ( ˆ E , ˆ F ) in sense of Mosco as n → ∞ . Proof.
Denote the semigroups of ( E , F ), ( E n , F n ), ( ˆ E , ˆ F ) and ( ˆ E n , ˆ F n ) by ( T t ) t ≥ ,( T nt ) t ≥ , ( ˆ T t ) t ≥ and ( ˆ T nt ) t ≥ respectively. We only need to prove that for anyˆ f ∈ L ( ˆ E, ˆ m ) and t ≥
0, ˆ T nt ˆ f converges to ˆ T t ˆ f strongly. In fact, since j ∗ issurjective, one may easily check thatˆ T t ˆ f = T t ( j ∗ ˆ f ) ◦ j − , ˆ T nt ˆ f = T nt ( j ∗ ˆ f ) ◦ j − . Since ( E n , F n ) is Mosco convergent to ( E , F ), it follows that || ˆ T nt ˆ f || m = Z ˆ E (cid:18) T nt (cid:0) j ∗ ˆ f (cid:1)(cid:0) j − (ˆ x ) (cid:1)(cid:19) ˆ m ( d ˆ x ) = || T nt ( j ∗ ˆ f ) || m → || ˆ T t ˆ f || m , which completes the proof. (cid:3) If in addition, j is a homeomorphism, then the Mosco convergences of Dirichletforms and their transforms under j are exactly equivalent. Note that the irreduciblediffusion X on I with scaling function s will be transformed to another irreduciblediffusion with natural scaling function after spatial transform s . Thus without lossof generality, we shall always assume that s is the natural scaling function on I inthis section. Let ( E , F ) := ( E ( s ,m ) , F ( s ,m )0 ) . Take a sequence of characteristic sets { G n ∈ ◦ G s ( I ) : n ≥ } . Furthermore, assumethat all of them are open. For each n , denote the associated scaling function of G n by s n , and set ( E n , F n ) := ( E ( s n ,m ) , F ( s n ,m )0 ) . The following theorem is our main result of this section.
Theorem 4.2. If G n ↑ I , then the Dirichlet form ( E n , F n ) is convergent to ( E , F )on L ( I, m ) in sense of Mosco as n → ∞ . Proof.
Similarly to Theorem 3.2, we can obtain that F ⊂ · · · ⊂ F n ⊂ · · · ⊂ F . Now we shall prove (a) of Definition 1.1. For any sequence { u n : n ≥ } in L ( E, m ), which is weakly convergent to u , we may always assume that u n ∈ F n .Or, E n ( u n , u n ) = ∞ . Then u n is useless in the left side of (1.1). It follows that u n ∈ F n ⊂ F , and E n ( u n , u n ) = E ( u n , u n ) . Because of the same reason as that of the proof of Theorem 3.2, we have(4.1) lim inf n →∞ E ( u n , u n ) ≥ E ( u, u ) . Thus lim inf n →∞ E n ( u n , u n ) ≥ E ( u, u ), i.e. (a) is proved.Finally, we turn to prove (b) of Definition 1.1. We assert that ∪ n ≥ F n is densein F with the norm || · || E . Note that C ∞ c ( G n ) ⊂ F n , and C ∞ c ( I ) is dense in F .For any function u ∈ C ∞ c ( I ), since the support of u is compact, andsupp[ u ] ⊂ I = ∪ n ≥ G n , it follows that there is an integer N such that supp[ u ] ⊂ G N , which implies that u ∈ C ∞ c ( G N ) ⊂ F N . Hence C ∞ c ( I ) ⊂ ∪ n ≥ F n . Clearly, ∪ n ≥ F n is dense in F . For any function u ∈ L ( E, m ), if u / ∈ F , then (1.2) is naturally satisfied. Nowassume that u ∈ F . From the above assertion, we may find a sequence of functions { u n : n ≥ } such that u n ∈ F n and || u n − u || E → n → ∞ . In particular, u n ∈ F and E n ( u n , u n ) = E ( u n , u n ). Therefore,lim sup n →∞ E n ( u n , u n ) = lim n →∞ E ( u n , u n ) = E ( u, u ) , which implies (1.2). That completes the proof. (cid:3) The instability of global properties under Mosco convergence
The global properties of a Dirichlet form stand for its recurrence, transience,irreducibility, conservativeness and etc. We refer their standard definitions to § (1): a sequence of recurrent Dirichlet forms is convergent to a transient Dirich-let form in sense of Mosco; (2): a sequence of transient Dirichlet forms is convergent to a recurrent Dirich-let form in sense of Mosco; (3): a sequence of conservative Dirichlet forms converges to a non-conservativeDirichlet form in sense of Mosco; (4): a sequence of non-conservative Dirichlet forms is convergent to a conser-vative Dirichlet form in sense of Mosco.Before that, we need to point out some facts. The Dirichlet form ( E ( s ,m ) , F ( s ,m )0 ),which is given by (2.3), is transient if and only if a or b is s -approachable. Thefollowing example is about the instability of recurrence/transience. Example 5.1.
Let I = R , m the Lebesgue measure on R and s ( x ) = x . In otherwords, ( E ( s ,m ) , F ( s ,m )0 ) corresponds to 1-dimensional Brownian motion on R . Moreprecisely, ( E ( s ,m ) , F ( s ,m )0 ) = (cid:0) D , H ( R ) (cid:1) , where H ( R ) is the 1-dim Sobolev space, and D ( f, g ) := R R f ′ ( x ) g ′ ( x ) dx, f, g ∈ H ( R ). Clearly, this Dirichlet form is recurrent.Similarly to Example 3.4, let G be the set given by Example 5.2 of [7], and G n the characteristic set defined by (3.6). Denote the associated scaling functions of G and G n by ˜ s and s n . From Example 3.4, we know that˜ s ( −∞ ) > −∞ , ˜ s ( ∞ ) < ∞ , whereas for each n , s n ( −∞ ) = −∞ , s n ( ∞ ) = ∞ . That implies that ( E (˜ s ,m ) , F (˜ s ,m )0 ) is transient, but ( E ( s n ,m ) , F ( s n ,m )0 ) is recurrent.Note that m satisfies the first condition of Corollary 3.3. It follows that as n → ∞ ,a sequence of recurrent Dirichlet forms { ( E ( s n ,m ) , F ( s n ,m )0 ) : n ≥ } is convergentto a transient Dirichlet form ( E (˜ s ,m ) , F (˜ s ,m )0 ) in sense of Mosco.Now we still take G above. Note that G is open. For any integer n , define(5.1) U n := G ∪ ( − n, n ) . EGULAR DIRICHLET SUBSPACES AND MOSCO CONVERGENCE 11
One may easily check that U n ∈ ◦ G s ( R ), { U n : n ≥ } is an increasing sequenceof open sets, and ∪ n ≥ U n = R . Denote the associated regular subspace of U n by( E n , F n ). It follows from Theorem 4.2 that ( E n , F n ) is convergent to (cid:0) D , H ( R ) (cid:1) in sense of Mosco. Finally, we assert that for each n , ( E n , F n ) is transient. Infact, since | G | < ∞ , it follows that | U n | < ∞ . Furthermore, its associated scalingfunction s n satisfies s n ( −∞ ) > −∞ , s n ( ∞ ) < ∞ , which implies that ( E n , F n ) istransient.We refer the definition of approachable boundary in finite time of 1-dimensionaldiffusion to Example 3.5.7 of [10]. In particular, a (resp. b ) is approachable in finitetime, if and only if for some constant c ∈ ( a, b ), Z ca m (cid:0) ( x, c ) (cid:1) d s ( x ) < ∞ , (resp. Z bc m (cid:0) ( c, x ) (cid:1) d s ( x ) < ∞ ) . Apparently, regular boundary is always approachable in finite time. On the otherhand, a minimal diffusion is conservative, if and only if neither a nor b is approach-able in finite time. At the end of this paper, we shall present an example for theinstability of conservativeness under Mosco convergence. Example 5.2.
We first set I = R and assume that m ( R ) < ∞ . The scalingfunction s is the natural scaling function. Let { U n : n ≥ } be the sequence(5.1) of sets in Example 5.1. Since ( E ( s ,m ) , F ( s ,m )0 ) is recurrent, it is also conser-vative. We assert that the associated regular subspace of U n is not conservative.In fact, since m ( R ) < ∞ and the scaling function s n , which corresponds to U n ,satisfies s n ( −∞ ) > −∞ , s n ( ∞ ) < ∞ , it follows that a and b are both the ap-proachable boundaries in finite time of ( E n , F n ). In particular, ( E n , F n ) is notconservative. Therefore, from Theorem 4.2, we obtain that as n → ∞ , the non-conservative Dirichlet form ( E n , F n ) is convergent to a conservative Dirichlet form( E ( s ,m ) , F ( s ,m )0 ) in sense of Mosco.Finally, we still set I = R , s is the natural scaling function and m will be decidedlater. Assume that G and G n are the characteristic sets in Example 5.1, ˜ s and s n are their associated scaling functions. Set (˜ a, ˜ b ) := ˜ s ( R ), then ˜ a > −∞ , ˜ b < ∞ .Take a strictly increasing and integral function F on (˜ a, ˜ b ) such that F (˜ a +) = −∞ , F (˜ b − ) = ∞ . The existence of F is clear. For example, take a constant 0 < α <
1, define F ( x ) := 1 / | b − x | α near ˜ b and F ( x ) := − / | x − a | α near ˜ a . Since ˜ s is strictlyincreasing, it follows that F ◦ ˜ s is a strictly increasing function on R . Without lossof generality, assume that F ◦ ˜ s (0) = 0. Furthermore, let m be the Lebesgue-Stieltjesmeasure with respect to F ◦ ˜ s . In particular, we have m (cid:0) ( −∞ , (cid:1) = m (cid:0) (0 , ∞ ) (cid:1) = ∞ . That implies that the first condition of Corollary 3.3 is satisfied. Thus the Dirichletform ( E ( s n ,m ) , F ( s n ,m )0 ) is convergent to ( E (˜ s ,m ) , F (˜ s ,m )0 ) in sense of Mosco. Note that( E ( s n ,m ) , F ( s n ,m )0 ) is recurrent, hence they are all conservative (see Lemma 1.6.5of [11]). In the end, we assert that ( E (˜ s ,m ) , F (˜ s ,m )0 ) is not conservative. It sufficesto prove that −∞ or ∞ is an approachable boundary of ( E (˜ s ,m ) , F (˜ s ,m )0 ) in finitetime. Indeed, since F is integral, we can obtain that Z −∞ m (cid:0) ( x, (cid:1) d ˜ s ( x ) = − Z −∞ F ◦ ˜ s ( x ) d ˜ s ( x ) = Z ˜ s (0)˜ a | F ( y ) | dy < ∞ . That implies that, as n → ∞ , the conservative Dirichlet form ( E ( s n ,m ) , F ( s n ,m )0 ) isconvergent to a non-conservative Dirichlet form ( E (˜ s ,m ) , F (˜ s ,m )0 ) in sense of Mosco. References [1] Beurling A, Deny J. Dirichlet spaces [J]. Proc Nat Acad Sci USA., 1959, 45:208–215.[2] Fukushima M. Dirichlet spaces and strong Markov processes [J]. Trans Amer MathSoc., 1971, 162:185–224.[3] Fukushima M. Regular representations of Dirichlet spaces [J]. Trans Amer Math Soc.,1971, 155(2):455–473.[4] Fukushima M, Ying J. A note on regular Dirichlet subspaces [J]. Proc Amer MathSoc., 2003, 131(5):1607–1610.[5] Fukushima M, Ying J. Erratum to: “A note on regular Dirichlet subspaces”[Proc. Amer. Math. Soc. 2003, 131(5):1607–1610] [J]. Proc Amer Math Soc., 2004,132(5):1559–1560.[6] Fang X, Fukushima M, Ying J. On regular Dirichlet subspaces of H ( I ) and associatedlinear diffusions [J]. Osaka J Math., 2005, 42(1):27–41.[7] Fang X, He P, Ying J. Dirichlet forms associated with linear diffusions [J]. Chin AnnMath Ser B., 2010, 31(4):507–518.[8] Li L, Ying J. On structure of regular subspaces of one-dimensional Brownian motion[J]. arXiv: 1412.1896, 2014.[9] Li L, Ying J. Regular subspaces of Dirichlet forms [J]. In: Festschrift MasatoshiFukushima. In Honor of Fukushima’s Sanju: World Scientific, 2015,397–420.[10] Chen Z-Q, Fukushima M. Symmetric Markov Processes, Time Change, and BoundaryTheory [M]. Princeton NJ: Princeton University Press, 2012.[11] Fukushima M, Oshima Y, Takeda M. Dirichlet Forms and Symmetric Markov Pro-cesses [M]. extended. Berlin: Walter de Gruyter & Co., 2011.[12] Mosco U. Composite media and asymptotic Dirichlet forms [J]. J Funct Anal., 1994,123(2):368–421.[13] Barlow M T, Bass R F, Chen Z-Q, Kassmann M. Non-local Dirichlet forms andsymmetric jump processes [J]. Trans Amer Math Soc., 2009, 361(4):1963–1999.[14] Kolesnikov A V. Mosco convergence of Dirichlet forms in infinite dimensions withchanging reference measures [J], J Funct Anal., 2006, 230(2):382–418.[15] Suzuki K, Uemura T. On instability of global path properties of symmetric Dirichletforms under Mosco-convergence [J], arXiv: 1412.0725, 2014.[16] Barret F, Renesse von M. Averaging principle for diffusion processes via Dirichletforms [J]. Potential Anal., 2014, 41(4):1033–1063.[17] Itˆo K, McKean HP Jr. Diffusion Processes and Their Sample Paths [M]. Berlin-NewYork: Springer-Verlag, 1974.[18] Rogers L C G, Williams D. Diffusions, Markov Processes, and Martingales [M]. Vol.2. 2nd ed. Cambridge: Cambridge University Press, 2000. School of Mathematical Sciences, Fudan University, Shanghai 200433, China.
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