Quasi-Regular Dirichlet Forms: Examples and Counterexamples

Quasi-regular Dirichlet forms: Examples and counterexamples

Michael R¨ockner Byron Schmuland

ABSTRACT We prove some new results on quasi-regular Dirichlet forms. These include results on perturbations of Dirichlet forms, change of speed measure, and tightness. The tightness implies the existence of an associated right continuous strong Markov process. We also discuss applications to a number of examples including cases with possibly degenerate (sub)-elliptic part, diﬀusions on loops spaces, and certain Fleming-Viot processes.

AMS Subject Classiﬁcation: Primary 31C25 Secondary 60J60

Keywords: Dirichlet forms, quasi-regularity, capacities, smooth measures, exceptional sets, tightness, square ﬁeld operator, M processes, degenerate cases.

Short title: Quasi-regular Dirichlet forms

1 0. Introduction. The purpose of this paper is to bring together some new results on quasi-regular Dirichlet forms that were obtained recently. In Section 1 we start with some examples of semi- Dirichlet forms on an open subset of IRd with possibly degenerate (sub)-elliptic part. Our treatment of these forms extends some of the results in [Str 88]. Subsequently, we consider perturbations of Dirichlet forms by smooth measures, along the lines of [AM 91b], and also look at the effect of changing the underlying speed measure (cf. Section 2). In Section 3 we extend our earlier results on tightness to a more general class of Dirichlet forms which consist of a “square field operator”-type form perturbed by a jump and killing term. As a consequence one can construct an associated (special) standard process on the basis of the general theory in [MR 92]. We give several applications in Section 4, i.e., construct diffusions on Banach spaces and loop spaces, and also construct certain Fleming-Viot processes (which are measure-valued). We note that in the Section 1 we look at semi- Dirichlet forms, but afterwards we restrict ourselves to Dirichlet forms (see Definition 0.3 below for the difference). The reason is that we will sometimes use Ancona’s result (see Remark 0.4) and it is not known if this result extends to semi-Dirichlet forms. Many of the results for Dirichlet forms do carry over to semi-Dirichlet forms, we refer the interested reader to [MOR 93]. Until recently, the general theory of Dirichlet forms had been restricted to the case where the underlying space is locally compact. In M. Fukushima’s book [F 80], which is the standard reference in the area, the local compactness is used throughout and is crucial in the construction of the associated Markov processes. Fukushima assumes that E is a locally compact, separable metric space and that m is a positive Radon measure on (E) with full support. He then constructs a Markov process, indeed a Hunt process, B associated to any regular Dirichlet form ( ,D( )) on L2(E; m) (cf. below for definitions) where regularity means E E

D( ) C (E) is 1/2-dense in D( ), and is uniformly dense in C (E). (0.1) E ∩ 0 E1 E 0 Here C0(E) is the space of continuous real-valued functions with compact support. Now the local compactness assumption, of course, eliminates the possibility of using Fukushima’s theory in the study of infinite-dimensional processes. Nevertheless, in the years since the publication of [F 80] several authors (cf. eg. [AH-K 75, 77 a,b], [Ku 82], [AR 89,91], [S 90] and see also the reference list in [MR 92]) have been able to modify Fukushima’s construction in special cases and obtain processes in infinite-dimensional state spaces. Recently a more general framework in which such constructions are possible has been developed. This is the theory of (non-symmetric) quasi-regular Dirichlet forms, which are defined below. The fundamental existence result in this framework is found in [MR 92; Chapter IV, Theorem 6.7] and it says the following:

Theorem 0.1. Let E be a metrizable Lusin space. Then a Dirichlet form ( ,D( )) E E on L2(E; m) is quasi-regular if and only if there exists a pair (M, M) of normal, right continuous, strong Markov processes associated with ( ,D( )). E E c This says that the class of quasi-regular Dirichlet forms is the correct setting for the

2 study of those forms associated with nice Markov processes. Z.M. Ma, L. Overbeck, and M. R¨ockner [MOR 93] have recently proved a one-sided version of the existence result for quasi-regular semi-Dirichlet forms; see Deﬁnition 0.3 below. In this case we do not get a pair of processes but only the process M.

In order to explain what a quasi-regular Dirichlet form is we first need some preparation. For a detailed exposition we refer the reader to [AMR 93a] and, in particular, to the monograph [MR 92]. Let E be a Hausdorff topological space, and (E) be the Borel sets in E. Fix a positive, σ-finite measure m on (E). B B Definition 0.2. A pair ( ,D( )) is called a coercive closed form on (real) L2(E; m) if D( ) is a dense linear subspaceE E of L2(E; m) and if : D( ) D( ) IR is a bilinear form suchE that the following conditions hold: E E × E → (i) (u, u) 0 for all u D( ). E ≥ ∈ E (ii) D( ) is a Hilbert space when equipped with the inner product E ˜ (u, v) := (1/2) (u, v) + (v, u) + (u, v) 2 . E1 {E E } L (E;m)

(iii) ( 1,D( )) satisﬁes the sector condition, i.e., there exists a constant K > 0 such that E (u, vE) K (u, u)1/2 (v, v)1/2, for all u, v D( ). |E1 | ≤ E1 E1 ∈ E

Here and henceforth, α(u, v) := (u, v) + α(u, v)L2(E;m) for α 0, and (u) := (u, u). For the one-to-one correspondenceE E between coercive closed forms,≥ their generators,E E resol- vents, and semigroups we refer to [MR 92; Chapter I].

Definition 0.3. A coercive closed form ( ,D( )) on L2(E; m) is called a semi-Dirichlet E E form (cf. [CaMe 75], [MOR 93]) if it has the following (unit) contraction property: for all u D( ), we have u+ 1 D( ) and ∈ E ∧ ∈ E (u + u+ 1, u u+ 1) 0. (0.2) E ∧ − ∧ ≥ If, in addition, (u u+ 1, u + u+ 1) 0, then ( ,D( )) is called a Dirichlet form. E − ∧ ∧ ≥ E E Remark 0.4. If ψ : IR IR satisfies ψ(0) = 0 and ψ(t) ψ(s) t s for all t, s IR, → | − | ≤ | − | ∈ then ψ is called a normal contraction. Ancona [An 76] has shown that if ( ,D( )) is a Dirichlet form and ψ is a normal contraction, then the mapping u ψ(u)E is stronglyE ˜ → continuous on the Hilbert space (D( ), 1). It follows easily that this conclusion also holds if ψ is a function with a bounded firstE derivativeE and ψ(0) = 0.

Deﬁnition 0.5. Let ( ,D( )) be a semi-Dirichlet form on L2(E; m). E E (i) For a closed subset F E we deﬁne ⊆ D( ) := u D( ) u = 0 m-a.e. on E F . (0.3) E F { ∈ E | \ } Note that D( ) is a closed subspace of D( ). E F E 3 (ii) An increasing sequence (Fk)k IN of closed subsets of E is called an -nest if ˜1/2 ∈ E k 1D( )F is -dense in D( ). ∪ ≥ E k E1 E c (iii) A subset N E is called -exceptional if N k 1Fk for some -nest (Fk)k IN .A ⊂ E ⊆ ∩ ≥ E ∈ property of points in E holds -quasi-everywhere (abbreviated -q.e.), if the property holds outside some -exceptionalE set. It can be seen that everyE -exceptional set has m-measure zero. E E

(iv) An -q.e. deﬁned function f : E IR is called -quasi-continuous if there exists E → E an -nest (Fk)k IN so that f F is continuous for each k IN. E ∈ | k ∈

(v) Let f, fn, n IN, be -q.e. deﬁned functions on E. We say that (fn)n IN converges - ∈ E ∈ E quasi-uniformly to f if there exists an -nest (Fk)k IN such that fn f uniformly E ∈ → on each Fk.

We shall use the following result throughout this paper (cf. [MR 92; Chapter III, Propo- sition 3.5] and [MOR; Proposition 2.18]).

Lemma 0.6. Let ( ,D( )) be a semi-Dirichlet form on L2(E; m). Let u D( ), E E n ∈ E which have -quasi-continuous m-versions u˜n, n IN, such that un u D( ) with 1E/2 ∈ → ∈ E respect to ˜ . Then there exists a subsequence (un )k IN and an -quasi-continuous E1 k ∈ E m-version u˜ of u so that (˜un )k IN converges -quasi-uniformly to u˜. k ∈ E We are now able to deﬁne a quasi-regular semi-Dirichlet form.

Deﬁnition 0.7. A semi-Dirichlet form ( ,D( )) on L2(E; m) is called quasi-regular if: E E (QR1) There exists an -nest (Fk)k IN consisting of compact sets. E ∈ ˜1/2 (QR2) There exists an 1 -dense subset of D( ) whose elements have -quasi-continuous m-versions. E E E

(QR3) There exist u D( ), n IN, having -quasi-continuous m-versionsu ˜ , n IN, n ∈ E ∈ E n ∈ and an -exceptional set N E such that u˜n n IN separates the points of E N. E ⊂ { | ∈ } \ Remark 0.8. Let ( ,D( )) be a quasi-regular semi-Dirichlet form. E E (i) By [MOR 93; Proposition 3.6] (cf. [MR 92; Chapter IV, Remark 3.2 (iii)]) the compact sets Fk in (QR1) can always be chosen to be metrizable.

(ii) (QR2) implies that every u D( ) has an -quasi-continuous m-versionu ˜. Hence- forth, for any subset D of ∈D( )E we will useE D˜ to denote the set of all -quasi- continuous m-versions of elementsE of D. That is, D˜ = u˜ u D . E { | ∈ } (iii) By [MOR 93; Proposition 2.18(ii)] (cf. [MR 92; Chapter III, Proposition 3.6]) there exists an -nest (Fk)k IN such that m(Fk) < for each k. Using this and [Bou 74; Chapter IX,E Sect. 6, Def.∈ 9, Théorème6 and Proposition∞ 10] we can prove that m is inner regular on (E), i.e., m(B) = sup m(K) K B and K is compact for all B { | ⊆ } 4 B (E). ∈ B (iv) We define the symmetric part of ( ,D( )) by setting ˜(u, v) := 1/2 (u, v) + (v, u) for u, v D( ). If ( ,D( E)) isE a Dirichlet form,E then ( ˜,D({E)) is also Ea Dirichlet} form.∈ We noticeE thatE theE definitions of -nest, -quasi-continuityE E and quasi-regularity only depend on through its symmetricE partE ˜. E E (v) The property (QR1) is equivalent (see [MR92; Chapter III, Theorem 2.11] and [MOR 93; Theorem 2.14]) to the tightness of an associated capacity and is absolutely vital in the construction of an associated Markov process (see [LR 92], [RS 92], [MOR 93]). In this paper we will not use the notion of capacity, instead we will stick with the equivalent “nest” formulation.

The new concept of a quasi-regular semi-Dirichlet form includes the classical concept of a regular semi-Dirichlet form, this follows from the next proposition (cf. [MR 92; Chapter IV, Example 4a]). We repeat the proof here for the convenience of the reader.

Proposition 0.9. Assume E is a locally compact, separable, metric space and m is a positive Radon measure on (E). If ( ,D( )) is a regular semi-Dirichlet form on B E E L2(E; m) (see (0.1)), then ( ,D( )) is quasi-regular. E E Proof. We only show (QR1), as (QR2) and (QR3) are easy exercises. By the topological assumptions on E, we may write E = k∞ Fk, where (Fk)k IN is an increasing sequence ∪ =1 ∈ of compact sets in E so that Fk is contained in the interior of Fk+1 for all k 1. It is then easy to see that ≥ C (E) D( ) ∞ D( ) , (0.4) 0 ∩ E ⊆ ∪k=1 E Fk which concludes the proof.

Remark 0.10. The first example in Section 5 is a Dirichlet form that satisfies (0.1), but is not quasi-regular. The space E is this example is a separable, compact, non-metrizable Hausdorff space.

5 1. Degenerate semi-Dirichlet forms in ﬁnite dimensions.

The purpose of this section is to generalize the standard class of examples of semi- Dirichlet forms on an open (not necessarily bounded) set U IRd, d 3 (cf. [MR 92; Chapter II, Subsection 2d)]). In particular, we want to allow sub-elliptic,⊂ ≥ possibly degenerate diffusion parts. We need some preparations. We adopt the terminology of Chapters I and II from [MR 92]. Let σ, ρ L1 (U; dx), σ, ρ > 0 dx a.e. where dx denotes Lebesgue measure. The ∈ loc − following symmetric form will serve as a “reference form". Set for u, v C0∞(U) (:= all infinitely differentiable functions with compact support in U) ∈

d ∂u ∂v (u, v) = ρ dx. (1.1) Eρ ∂x ∂x i,j i j X=1 Z Assume that 2 ( ,C∞(U)) is closable on L (U; σdx). (1.2) Eρ 0

Remark 1.1. A sufficient condition for (1.2) to hold is that ρ, σ satisfy Hamza’s condition (see [MR 92; Chapter II, Subsection 2a)]). We recall that a (U)-measurable function B f : U [0, ) satisfies Hamza’s condition if for dx-a.e. x U, f(x) > 0 implies that for some →¯ > 0 ∞ ∈ 1 (f(y))− dy < , (1.3) y: y x ¯ ∞ Z{ k − k≤ } 1 d where we set 0 := + and denotes Euclidean distance in IR . In particular, σ, ρ may have zeros, and (1.2)∞ holds if,k · fork example, σ, ρ are lower semi-continuous. However, there is also a generalized version, a kind of “Hamza condition on rays", which, if it is fulfilled for σ, ρ, also implies (1.2) (cf. [AR 90; (5.7)] and [AR 91; Theorem 2.4]). In particular, if σ, ρ are weakly differentiable then (1.2) holds. 1 Now let a , b , d , c L (U; dx), 1 i, j d, and define for u, v C∞(U) ij i i ∈ loc ≤ ≤ ∈ 0

d ∂u ∂v d ∂u (u, v) = aij dx + vbi dx E ∂xi ∂xj ∂xi i,j=1 i=1 X Z X Z (1.4) d ∂v + u d dx + uvc dx. ∂x i i i X=1 Z Z 2 1 Then ( ,C0∞(U)) is a densely deﬁned bilinear form on L (U; σdx). Seta ˜ij := 2 (aij + aji), aˇ := E1 (a a ), b := (b , . . . , b ), and d := (d , . . . , d ). We deﬁne F to be the set of ij 2 ij − ji 1 d 1 d all functions g L1 (U; dx) such that the distributional derivatives ∂g , 1 i d, are loc ∂xi 1 ∈ 1/2 p p+1 p/q 1/2≤ ≤d in Lloc(U; dx) such that g (gσ)− L∞(U; dx) or g (g σ )− L (U; dx) for some p, q [1, ] withk∇1 k+ 1 = 1, p∈ < . We say thatk∇ ak (U)-measurable∈ function f ∈ ∞ p q ∞ B has property (Aρ,σ) if one of the following conditions holds:

6 1/2 (i) f(ρσ)− L∞(U; dx) ∈ p p+1 p/q 1/2 d 1 1 (ii) f (ρ σ )− L (U; dx) for some p, q [1, ] with p + q = 1, p < , and ρ F . ∈ ∈ ∞ ∞ ∈ Theorem 1.2. Suppose that

(1.5) ξ 2 := d a˜ ξ ξ ρ ξ 2 dx-a.e. for all ξ = (ξ , . . . , ξ ) IRd. k ka˜ i,j=1 ij i j ≥ k k 1 d ∈ 1 (1.6) aˇ ρ− PL∞(U; dx). ij ∈ 1/2 (1.7) For all K U, K compact, 1K b + d and 1K c have property (Aρ,σ), and ⊂ d ∂di k k (c + α0σ)dx is a positive measure on (U) for some α0 (0, ). − i=1 ∂xi B ∈ ∞ (1.8) b d has propertyP (A ). k − k ρ,σ 1 d ∂γi (1.9) b = β + γ such that β , γ L (U; dx), (c + α0σ)dx is a positive k k k k ∈ loc − i=1 ∂xi measure on (U) and β has property (Aρ,σ). B k k P Then: 2 (i) There exists α (0, ) such that ( ,C∞(U)) is closable on L (U; σdx) and ∈ ∞ Eα 0 its closure ( ,D( )) is a regular semi-Dirichlet form. In particular, the Eα Eα corresponding semigroup (Tt)t>0 is sub-Markovian and there exists a diffusion process M properly associated with ( ,D( )) (cf. [MR 92; Chapter IV]). Eα Eα (ii) If β 0 in (1.9) then α can be taken to be α as given in (1.7), and ( ,D( )) ≡ 0 Eα Eα is a regular Dirichlet form. In particular, both corresponding semigroups (Tt)t>0, (Tt)t>0 are sub-Markovian and there exists a pair (M, M) of diffusion processes properly associated with ( ,D( )) (cf. [MR 92; Chapter IV]). Eα Eα b c Remark 1.3. (i) Theorem 1.2 extends a result obtained by different techniques by D.W. Stroock (cf. [Str 88, Theorem II 3.8]) in the strictly elliptic case (ρ const) with σ 1, a ≡ ≡ ij ∈ L∞(U; dx),a ˇij 0 for 1 i, j d, γ, d 0, and β L∞(U; dx). We emphasize, however, that Stroock’s≡ result≤ in≤ this particular≡ casek k is ∈ stronger than ours since he even proves the corresponding semigroup to be strongly Feller and to have a density with respect to Lebesgue measure.

(ii) The analytic part of the proof of Theorem 1.2 is quite elementary. One of the main ingredients is the classical Sobolev Lemma (cf. [Da 89; Theorem 1.7.1]), i.e., if 2(d 1) − λ := (d 2)d1/2 , then −

u λ u for all u C∞(U), (1.10) k kq ≤ k∇ k 2 ∈ 0 where 1 + 1 = 1 and for p 1, denotes the usual norm in Lp(U; dx). A part q d 2 ≥ k kp of the proof of Theorem 1.2 is close to the classical one in [St 65] where ρ σ 1. However even in this case our proof of the Dirichlet property is quite diﬀerent≡ ≡

7 and shorter (cf. [Bl 71, (10.7)] for the classical proof), and we need less restrictive d assumptions on the coeﬃcients (namely, e.g. merely b , d Lloc(U; dx), c d/2 k k k k ∈ ∈ Lloc (U; dx) instead of the global integrability conditions in [St 65]). This is mainly due to our more reﬁned techniques to prove closability which also permit us to take so general σ and ρ.

(iii) We stress that in the situation of Theorem 1.2 we can replace U by a Riemannian manifold M as long as condition (1.10) or more generally the following inequality holds for some α > 0

1/2 u 2d const. ( ∆ + α) u 2 for all u C∞(M), (1.11) d 2 0 k k − ≤ k − k ∈ where ∆ is the Laplacian on M. We refer e.g. to [VSC 92] for examples.

Before we prove Theorem 1.2 we discuss some examples for the function ρ in (1.1), (1.5).

Examples 1.4. (i) In the case σ 1 it is easy to check that our conditions allow ρ to have zeros of order x α, α 2.≡ k k ≥ d d (ii) Suppose that U = IR and that for every K IR , K compact, there exists cK (0, ) such that ⊂ ∈ ∞ d a˜ ξ ξ c ξ 2 dx-a.e. (1.12) ij i j ≥ K k k i,j X=1 for all ξ = (ξ , . . . , ξ ) IRd (i.e., we have local strict ellipticity). Then there exists 1 d ∈ a strictly positive C1-function ρ satisfying (1.5).

(iii) Suppose ∂U is smooth and let g be a smooth distance function in the sense of [LM 72]. Deﬁne ρ := gα, α 2. (1.13) ≥ 1/2 Then it is easy to check that ρ ρ− L∞(U; dx). k∇ k ∈ For the proof of Theorem 1.2 we need two lemmas.

Lemma 1.5. Let f be a (U)-measurable function having property (A ). Then B ρ,σ there exist δ, η (0, ), with δ arbitrarily small, such that for all u C∞(U), ∈ ∞ ∈ 0 2 1 2 2 2 f ρ− u dx δ u ρ dx + η u σ dx. (1.14) ≤ k∇ k Z Z Z

Proof. The assertion is obviously true in case (A )(i). In case (A )(ii) with p, q ρ,σ ρ,σ ∈ (1, ) we have for all δ (0, 1) and u C∞(U) that ∞ 1 ∈ ∈ 0 p/q 2 1 2 δ1 1 2 f ρ− u dx A + u σ dx, (1.15) ≤ p qδ Z 1 Z 8 where 2p p p/q 2 A := f ρ− σ− u dx (1.16) Z and where we used that a1/pb1/q a/p + b/q for a, b (0, ). Setting ≤ ∈ ∞ p p+1 p/q 1/2 f0 := f (ρ σ )− (1.17)

we obtain by H¨older’sinequality with q (2, ), 1 + 1 = 1 , and (1.10) that ∈ ∞ q d 2 A f 2 ρ1/2u 2 ≤ k 0kd k kq 2 2 1/2 1 1/2 2 λ f ρ u + uρ− ρ dx ≤ k 0kd k ∇ 2 ∇ k (1.18) Z ° B ± λ2 f 2 2 u 2ρ dx + , ≤ k 0kd k∇ k 2 ° Z ± where 2 1 2 B := ρ ρ− u dx. (1.19) k∇ k Z 1/2 If ρ (ρσ)− is bounded, the assertion obviously follows. If k∇ k p p +1 p /q 1/2 d ρ := ρ 1 (ρ 1 σ 1 1 )− L (U; dx), (1.20) 0 k∇ k ∈ applying what we have proved so far with f := ρ we obtain for all δ (0, 1) and k∇ k 2 ∈ u C∞(U) that ∈ 0

p1/q1 δ2 2 2 B 1 2 B ρ0 d 2 u ρ dx + + u σ dx. (1.21) ≤ p1 k k k∇ k 2 q1δ2 ° Z ± Z Solving for B we get for δ2 small enough that

p /q 1 p /q δ 1 1 − 2δ 1 1 1 B 1 2 ρ 2 2 ρ 2 u 2ρ dx + u2σ dx , (1.22) ≤ − 2p k 0kd p k 0kd k∇ k q δ ² 1 ³ ² 1 Z 1 2 Z ³ 1 and resubstitution in (1.15) and (1.18) yields the assertion. The case where ρ ρ− d k∇ k ∈ L (U; dx) is similar. If f has property (Aρ,σ)(ii) with p = 1, q = , we have as in (1.18), since ∞ f = 1 f(ρσ) 1/2 k f + fk (1.23) { − ≤ } with fk := 1 f(ρσ) 1/2>k f, k IN, that { − } ∈

2 1 2 2 2 1 2 1/2 2 f ρ− u dx 2k u σ dx + 2 f ρ− ρ u . (1.24) ≤ k k kd k kq Z Z 1 Noting that by assumption fkρ− d 0 as k , we obtain the assertion also in this case by the same argumentsk as before.k → → ∞

9 Let , denote the Euclidean inner product on IRd. h i Lemma 1.6. Consider the situation of Theorem 1.2. Then for any ¯ (0, 1) there ∈ exists α [α , ) such that for all u C∞(U), ∈ 0 ∞ ∈ 0

u, β u dx ¯ (u, u) u, β u dx . (1.25) |h∇ i | ≤ Eα − h∇ i Z ° Z ± Proof. We ﬁrst note that for all u C∞(U), ∈ 0

(u, u) u, β u dx Eα0 − h∇ i Z 1 = u 2 dx + [ u2, γ + d + 2(c + α σ)u2] dx (1.26) k∇ ka˜ 2 h∇ i 0 Z Z u 2 dx. ≥ k∇ ka˜ Z

Furthermore, by (1.5) and Lemma 1.5 for all δ0, δ (0, 1) and u C∞(U), ∈ ∈ 0

1 2 1 2 1 2 u, β u dx (δ0ρ u + β ρ− u ) dx |h∇ i | ≤ 2 k∇ k δ k k Z Z 0 (1.27) 1 δ 2 η 2 (δ0 + ) u dx + u σ dx, ≤ 2 δ k∇ kα˜ 2δ 0 Z 0 Z for some η (0, ). Now the assertion follows by (1.26). ∈ ∞ Proof of Theorem 1.2. Let ¯, α be as in Lemma 1.6. Since ¯ < 1 the positive deﬁniteness 2 of ( ,C∞(U)) is obvious by 1.7. To prove closability of ( ,C∞(U)) on L (U; σdx) ﬁrst Eα 0 Eα 0 note that by Lemma 1.6 for all u C∞(U) ∈ 0

1 1 (1 + ¯)− (u, u) (u, u) u, β u dx (1 ¯)− (u, u). (1.28) Eα ≤ Eα − h∇ i ≤ − Eα Z Hence it suffices to consider the case β 0. By [MR 92; Chapter II, Subsection 2b)] we ≡ know that if for u, v C∞(U), ∈ 0 d ∂u ∂v a˜(u, v) := a˜ dx, (1.29) E ∂x ∂x ij i,j i j X=1 Z a˜ 2 then ( ,C0∞(U)) is closable on L (U; σdx). Let µ be the positive Radon measure on (U) definedE by B d ∂(d + b ) µ := 2(c + ασ)dx i i (1.30) − ∂x i i X=1 (cf. (1.7), (1.9) and recall that β 0). ≡ 10 Claim 1.7. 2 Let un C0∞(U), n N, with un 0 in L (U; σdx) as n , and un 0 in 2 ∈ ∈ → → ∞ k∇ k → L (U; ρdx) as n . Then there exists a subsequence (unk )k N with unk 0 µ-a.e. as k .→ ∞ ∈ → → ∞ Before we prove the claim, for the convenience of the reader we repeat the (modified)

argument from [MR 92; page 51] that it implies closability. So, let un C0∞(U), n IN 2 ∈ ∈ such that un 0 in L (U; σdx) as n , and α(un um, un um) 0 as n, m . → a˜ → ∞ E − − → a˜ → ∞ Then by (1.26), (un um, un um) 0 as n, m , and, since ( ,C0∞(U)) is 2 E − − → a˜ → ∞ E closable in L (U; σdx), we therefore obtain that (un, un) 0 as n and by (1.5) 2 E → → ∞ that un 0 as n in L (U; ρdx). If (unk )k IN is as in the claim, Fatou’s lemma impliesk∇ thatk → for all n →IN ∞ ∈ ∈

a˜ 1 2 α(un, un) (un, un) + lim inf (un un ) dµ (1.31) E ≤ E 2 k − k →∞ Z (cf. (1.26)). Hence

α(un, un) lim inf α(un un , un un ) (1.32) E ≤ k E − k − k →∞

which can be made arbitrarily small for large enough n. Hence ( α0 ,C0∞(U)) is closable 2 E on L (U; σdx). To prove the claim, replacing u by u v for any v C∞(U), v 0 we n n ∈ 0 ≥ may assume that supp [un] K for some compact set K U and all n IN. By the Cauchy-Schwarz inequality we⊂ obtain that ⊂ ∈

1 u2 dµ = 1 d + b, u u dx + u2 (c + ασ) dx 2 n K h ∇ ni n n Z Z Z (1.33) 1/2 1/2 2 1 d + b ρ− u u ρ + 1 (c + ασ)u . ≤ K k k n 2 k k∇ k k2 k K nk1 Using Lemma 1.5 we see that for some δ, η (0, ) ∈ ∞

1/2 2 2 2 1 d + b ρ− u δ u ρ dx + η u σ dx. (1.34) K k k n 2 ≤ k∇ nk n Z Z and 1 (c + ασ)u2 δ u 2ρ dx + (η + α) u2 σ dx. (1.35) K n 1 ≤ k∇ nk n Z Z Now the claim follows.

2 To prove the sector condition of ( α,D( α)) on L (U; σdx) we recall that by [MR 92; Chapter I, 2.1(iv)] it suﬃces to showE that thereE exists K (0, ) such that for all u, v ∈ ∞ ∈ C0∞(U) ˇ (u, v) K (u, u)1/2 (v, v)1/2. (1.36) |Eα | ≤ Eα Eα 11 where ˇ (u, v) := 1 ( (u, v) (v, u)) is the anti-symmetric part of ( ,D( )). But Eα 2 Eα − Eα Eα Eα for all u, v C∞(U) ∈ 0 d ∂u ∂v ˇ (u, v) aˇ dx + d b, u v dx |Eα | ≤ ∂x ∂x ij |h − ∇ i | i,j i j X=1 Z Z 1 1/2 1/2 (1.37) sup aˇ (x)ρ− (x) u ρ v ρ ij 2 2 ≤ x,i,j | | k∇ k k∇ k 1/2 1/2 + d b ρ− v u ρ . k − k 2 k∇ k 2 By (1.8) and Lemma 1.5 the second summand is dominated by

1/2 1/2 δ v 2ρ dx + η v2σ dx u 2ρ dx . (1.38) k∇ k k∇ k ² Z Z ³ ²Z ³

Hence by (1.5) and (1.6) we can ﬁnd a constant K0 such that for all u, v C∞(U) ∈ 0

2 2 1/2 2 2 1/2 ˇ (u, v) K0 u dx + u σ dx v dx + v σ dx , (1.39) |Eα | ≤ k∇ ka˜ k∇ ka˜ Z Z Z Z ¡ ¡ which by (1.26) and (1.28) implies (1.36). From the proof we see that α = α0 if β 0. The semi-Dirichlet property (resp. the Dirichlet property if β 0), is proved by exactly≡ ≡ the same arguments as those on pages 48 and 49 in [MR 92]. The form ( α,D( α)) is by definition regular (i.e., satisfies (0.1)). E E The semi-Dirichlet property (resp. the Dirichlet property if β = 0), is proved similarly to [MR 92; Chapter I, Proposition 4.7]. We need that for every ¯ > 0 there exists φ¯ : [ ¯, [ such that φ (t) = t for all t [0, [, 0 φ (t ) φ (t ) t t if t t , → − ∞ ¯ ∈ ∞ ≤ ¯ 1 − ¯ 2 ≤ 2 − 1 1 ≤ 2 φ¯ u D( ), sup¯>0 (φ¯ u, φ¯ u) < and lim sup¯ 0 (u + φ¯ u, u φ¯ u) 0. ◦ ∈ E E ◦ ◦ ∞ → E ◦ − ◦ ≥ The boundedness sup¯>0 (φ¯ u, φ¯ u) < is proved by using inequality (1.28) and reducing to the case β = 0.E The◦ limsup◦ statement∞ follows from arguing directly as in [MR 92; Chapter I, Proposition 4.7]. The existence of the corresponding diffusions now follows from [MOR 93; Theorem 3.8] (see also [CaMe 75]) resp. [MR 92; Chapter IV, Theorem 3.5] and [MR 92; Chapter V, Theorem 1.5] (see also [O 88] and the references quoted in [MR 92; Chapter IV, Section 7 and Chapter V, Section 3] as well as [AMR 93a,b].) Note that the proof of Theorem 1.5 in [MR 92; Chapter V] can be carried over to the case of semi-Dirichlet forms. Now the proof is complete.

12 2. Perturbations of a quasi-regular Dirichlet form. We begin with the result which says that the notion of quasi-regularity is equivalent for two equivalent Dirichlet forms 0 and . E E Proposition 2.1. Suppose ( ,D( )) is a Dirichlet form on L2(E; m) and D is an 1/2 E E ˜ -dense linear subspace of D( ). Let 0 be a positive deﬁnite, bilinear form on D E1 E E such that for some c > 0

(1/c) (u) 0 (u) c (u), for all u D. (2.1) E1 ≤ E1 ≤ E1 ∈ 2 Then the form ( 0,D) is closable in L (E; m) and the closure ( 0,D( 0)) satisﬁes E E E (2.1) on all of D( 0) = D( ). If for some constant K > 0, we have E E 1/2 1/2 0 (u, v) K (u) (v), for all u, v D, (2.2) |E1 | ≤ E1 E1 ∈ 2 then ( 0,D( 0)) is a coercive closed form on L (E; m). Furthermore, if ( 0,D( 0)) is E E E E a Dirichlet form, then ( 0,D( 0)) is quasi-regular if and only if ( ,D( )) is quasi- E E E E regular.

Proof. The proof of all but the final sentence can be found in [MR 92; Chapter I, Proposition 3.5]. To prove the final sentence, we note that for any closed set F , the spaces D( 0)F and D( )F coincide, as both are simply the set of u D( ) which vanish m-a.e. E E 1/2∈ E 1/2 outside of the set F . From (2.1) we see that the norms ˜ and ( ˜0) are equivalent E1 E 1 on D( ), so for any increasing sequence (Fk)k IN of closed sets, the subspace kD( )Fk is 1/2 E 1/2 ∈ ∪ E ˜ -dense if and only if it is ( ˜0) -dense. In other words, (Fk)k IN is an -nest if and E1 E 1 ∈ E only if it is an 0-nest. Therefore the notions of quasi-continuity and exceptional set are E also equivalent for the two forms and 0. Since D( ) = D( 0), it follows that ( 0,D( 0)) is quasi-regular if and only if ( ,DE ( ))E is quasi-regular.E E E E E E In the rest of this section we let ( ,D( )) be a fixed quasi-regular Dirichlet form on L2(E; m), and we consider a number ofE methodsE for getting a new quasi-regular form from the given one. One way to get a new Dirichlet form is to perturb by adding a killing term. This idea was used by, for instance, S. Albeverio and Z.M. MaE [AM 91a] to obtain a quasi-regular Dirichlet form whose domain contains no non-zero continuous function and is therefore far from being regular. This example can also be found in [MR 92; Chapter II, Example 2(e)]. In Proposition 2.3 below, we show that adding a reasonable killing term does not affect the quasi-regularity of . E Definition 2.2. A positive measure µ on (E, (E)) is said to be -smooth if µ(A) = 0 B E for all -exceptional sets A (E), and there exists an -nest (Fk)k IN of compact sets such thatE µ(F ) < for all∈k B IN. E ∈ k ∞ ∈ The relation between smooth measures and Radon measures has recently been clarified in [AMR 93c].

13 Since ( ,D( )) is quasi-regular, every u D( ) has an -quasi-continuous m-version E E ∈ E E u˜ of u [MR 92; Chapter IV, Proposition 3.3(ii)]. Ifu ˜ andu ˜0 are two -quasi-continuous E m-versions of u, thenu ˜ =u ˜0 -q.e. [MR 92; Chapter IV, Proposition 3.3(iii)] and, since E µ µ is smooth,u ˜ =u ˜0 µ-a.e. Thus, it makes sense to deﬁne D( ) in the following way: D( µ) consists of all m-classes in D( ) whose -quasi-continuousE m-versions are µ-square integrable.E Set E E µ (u, v) = (u, v) + (˜u, v˜) 2 (2.3) E E L (E;µ) for u, v D( µ). The following generalizes [MR 92; Chapter IV, Theorem 4.6] (which was taken from∈ [AME 91b] and only proved for ( ,D( )) regular). E E Proposition 2.3. If µ is an -smooth measure, then ( µ,D( µ)) is a quasi-regular E E E Dirichlet form on L2(E; m).

µ ˜1/2 Proof. We begin by showing that D( ) is 1 -dense in D( ). Let (Fk)k IN be an - nest that corresponds to the measure µE, as inE the definition ofE a smooth measure.∈ SinceE ˜1/2 (Fk)k IN is an -nest we know that k∞ D( )Fk is -dense in D( ), and furthermore, ∈ E ∪ =1 E E1 E by truncation (cf. [MR 92; Chapter I, Proposition 4.17(i)]) we see that k∞=1(L∞(E; m) 1/2 ∪ ∩ D( ) ) is ˜ -dense in D( ). Now, for any u L∞(E; m) D( ) we haveu ˜ = 0 -q.e. E Fk E1 E ∈ ∩ E Fk E on E Fk and u˜ u L∞(E;m) -q.e. on E. Since µ(Fk) < we find thatu ˜ is µ-square integrable\ and| so|u ≤ k Dk( µ). E ∞ ∈ E In particular D( µ) is also L2-dense in L2(E; m) and so µ is densely defined. By [MR 92; Chapter I, ExerciseE 2.1 (iv)] and [MR 92; Chapter III, PropositionE 3.5] it is easy to see that ( µ,D( µ)) is a closed coercive form with the Markov property, and so is a Dirichlet form. E E µ µ Now we show that any -nest (Fk)k IN is also an -nest. Let u D( ) and suppose E ∈ E ∈ E that u 0 m-a.e. so thatu ˜ 0 -q.e. Let un kD( )Fk be a sequence which converges to ˜1≥/2 ≥ E ∈ ∪ E ˜1/2 u in 1 -norm. We can replace un by vn := (u un) 0 without affecting 1 -convergence (cf. theE proof of 4.17 in [MR 92; Chapter I]), and∧ by∨ taking a subsequenceE we may assume thatv ñ u˜ -q.e. Thusv ñ u˜ µ-a.e. and also 0 vñ u˜ µ-a.e. so thatv ñ u˜ in 2 → E → ˜1/2 ≤ ≤ ˜µ 1/2 → L (E; µ). Since vn already converges to u in 1 we have vn u in ( )1 -norm. Also µ E → E vn = 0 wherever un = 0 and so vn D( )Fk for some k IN. For an arbitrary element u D( µ), we apply this argument∈ separatelyE to the positive∈ and negative parts, u+ and ∈ E µ µ µ u−, and conclude that kD( )F is dense in D( ). Thus (Fk)k IN is an -nest. ∪ E k E ∈ E Since an -nest is also an µ-nest, the properties (QR1) and (QR2) for ( µ,D( µ)) follow immediatelyE from (QR1) andE (QR2) for ( ,D( )). By [MR 92; ChapterE IV, PropositionE 3.4(i)] there is a countable collection u˜ nE INE of -quasi-continuous functions in D( µ) { n | ∈ } E E that separates the points of E N, where N is -exceptional. But then uñ n IN are also µ-quasi-continuous and N\ is also µ-exceptionalE so (QR3) holds for{ ( µ| ,D∈( µ)).} E E E E The following definition extends one given in [RW 85] (cf. also [FST 91]).

Deﬁnition 2.4. Let D D( ). A positive measure µ on (E) is called a D-proper ⊆ E B speed measure for ( ,D( )), if it does not charge any Borel -exceptional set, and for all E E E 14 -quasi-continuous m-versionsv ˜ of v D, E ∈ v˜ = 0 µ-a.e. impliesv ˜ = 0 -q.e. (2.4) E Note that because µ does not charge any -exceptional set, the implication in (2.4) is E independent of which -quasi-continuous m-version of v is chosen. Let PS(D) denote the set of all D-properE speed measures. M The next result shows that, if D is large enough, replacing m by a D-proper speed measure does not aﬀect the quasi-regularity of ( ,D( )). E E

Proposition 2.5. Let µ PS(D) and assume, for simplicity, that is symmetric. ∈ M1/2 ˜ E Suppose D D( ) is an 1 -dense Stone lattice such that D consists of µ-square ⊆ E E 2 integrable functions, and that ( , D˜) is closable on L (E; µ) with closure ( 0,D( 0)). E E E Then 2 (i) ( 0,D( 0)) is a symmetric Dirichlet form on L (E; µ). E E (ii) Every -nest is an 0-nest. E E (iii) ( 0,D( 0)) is quasi-regular. E E Proof. First we note that condition (2.4) guarantees that is well defined on D˜ regarded as a subspace of L2(E; µ). We want to show that it is alsoE densely defined. By [MR 92; 1/2 Chapter IV, Proposition 3.3(i)], the space D( ) is separable with respect to the 1 -norm. E 1/2 E Therefore we can find a sequence (un)n IN in D which is an -dense set. By [MR 92; ∈ E1 Chapter IV, Proposition 3.4(i)] if we fix a sequence of -quasi-continuous m-versionsu ñ, then we have E

uñ n IN separates the points in E N, where N is an -exceptional set in (E). { ∈ } \ E B (2.5) ¬ Also, there¬ exists an h D( ) with an -quasi-continuous m-version h˜ which is strictly pos- ∈ E E itive -quasi-everywhere and a sequence (hn)n IN in D˜ which converges -quasi-uniformly E ∈ E to h. Let (Fk)k IN be an -nest so that, on each Fk, hn is continuous, h˜ is strictly positive, and h h˜ uniformly.∈ E By taking an even smaller nest we may also assume thatu ˜ is n → n continuous on Fk for every n, k 1, and that uñ n IN separates the points in kFk. Define for each k 1, ≥ { | ∈ } ∪ ≥

(Fk) := f C(Fk) f =u ˜ Fk for some -quasi-continuous m-versionu ˜ of u D . A { ∈ | E ∈ (2} .6) ¬ From what we have proved¬ so far, we know that (F ) separates points and contains a A k strictly positive function. Since (Fk) is a subspace lattice, the Stone-Weierstrass theo- A ˜ 2 rem tells us that it is uniformly dense in C(Fk). We conclude that D Fk is L -dense in 2 ˜ 2 2 | L (Fk; µ Fk ) for every k, and hence D is L -dense in L (E; µ). Now we| prove the Markov property. For u D˜ we have u+ 1 in D˜, so ∈ ∧ + + 0(u 1) = (u 1) (u) = 0(u). (2.7) E ∧ E ∧ ≤ E E This proves (0.2) for u D˜ and now we apply [MR 92; Chapter I, Proposition 4.10] to get ∈ the Markov property for 0 on all of D( 0). E E 15 In order to prove (ii) we need a preliminary result which says that if u D˜ with u 0 ∈ ≥ m-a.e., and if v D( ) with v 0 m-a.e., then u v˜ D( 0). Suppose we are given ∈ E ≥ ∧ ∈ E such u and v and let v D so that v v in 1/2-norm, andv ˜ v˜ -q.e. By n ∈ n → E1 n → E replacing v with v+ we may suppose v 0. Now u v u v in 1/2-norm, in n n n ≥ ∧ n → ∧ E1 particular (u vn)n IN is -Cauchy. Also u v˜n u v˜ µ-a.e. and hence, by dominated ∧ ∈ E ∧ →2 ∧ convergence, the convergence also holds in the L (µ) sense. Since ( 0,D( 0)) is a closed 1/2 E E form, we conclude that u v˜ D( 0) and u v˜n u v˜ in ( 0)1 -norm. This result can be localized by noting that∧ for∈ anyE closed set∧F →E, ∧ E ⊆ D( ) = v D( ) v = 0 m-a.e. on F c E F { ∈ E | } = v D( ) v˜ = 0 -q.e. on F c (2.8) { ∈ E | E } v D( ) v˜ = 0 µ-a.e. on F c . ⊆ { ∈ E | } Therefore, if u D˜ with u 0 m-a.e., and if v D( ) with v 0 m-a.e., then ∈ ≥ ∈ E F ≥ u v˜ D( 0) . ∧ ∈ E F Now let (Fk)k IN be an -nest of compact sets in E. For u D˜ let un kD( )F so ∈ E ∈ ∈ ∪ E k that u u in 1/2-norm and, without loss of generality,u ˜ u -q.e. Set n → E1 n → E + + v := (˜u u ) (˜u− u−). (2.9) n n ∧ − n ∧ 1/2 Then vn kD( 0)Fk and, arguing as above we have vn u in ( 0)1 -norm. Since D ∈ ∪ 1/2E → E is already ( 0)1 -dense in D( 0), this shows that (Fk)k IN is also an 0-nest, and (ii) is proven. E E ∈ E By (ii) any -quasi-continuous function is 0-quasi-continuous, hence (QR2) holds for E E ( 0,D( 0)). Since (QR 1,3) hold by (ii) and (2.5) respectively, ( 0,D( 0)) is quasi-regular. E E E E

Remark 2.6. (i) If ( ,D( )) is transient in the sense of [F 80] it can be proved as in [RW 85] that ( ,ED˜) isE always closable on L2(E; µ). For a nice necessary and suﬃcient condition onE µ for the closability of ( , D˜) on L2(E; µ) in the locally compact regular case provided µ is a Radon measureE of full support we refer to [FST 91].

(ii) The condition that D is a Stone lattice may be replaced by the condition that D is closed under composition with smooth maps on IR which vanish at the origin.

Since ( ,D( )) is a quasi-regular Dirichlet form, we may as in Definition 0.5(i), define a subspaceE of (E ,D( )) by setting, for any Borel set B, E E D( ) := u D( ) u˜ = 0 -q.e. on Bc . (2.10) E B { ∈ E | E } This is more general than Definition 0.5(i) in that the set B need not be closed. It follows from [MR 92; Chapter IV, Proposition 3.3 (iii)] that these two definitions are consistent. Now D( ) is a closed subspace of ( ,D( )) and it is closed under normal contractions. E B E E 16 2 c Also, D( )B is a subspace of u L (E; m) u = 0 m-a.e. on B which can be identified with L2(EB; m ) in the obvious{ way.∈ Therefore,| if } |B D( ) is L2-dense in L2(B; m ), (2.11) E B |B then ( ,D( )) is a Dirichlet form on L2(B; m ), where we define as the restriction EB EB |B EB of to D( )B. We shall prove that if (2.11) holds, then ( B,D( B)) is a quasi-regular DirichletE form,E but first we need a few lemmas. E E

Lemma 2.7. There exists an increasing sequence (Ek)k IN of compact subsets of B, ∈ so that D( ) is ˜1/2-dense in D( ) . ∪k E Ek E1 E B Proof. Fix u D( ) and for every ¯ > 0, let u(¯) := u (( ¯) u) ¯. Letu ˜ be an ∈ E B − − ∨ ∧ -quasi-continuous m-version of u. By definition,u ˜ =u ˜1B -q.e. sou ˜1B is also -quasi- continuous.E So, without loss of generality, we will assume thatE u ˜(z) = 0 for all Ez Bc. ∈ Let (Fk)k IN be a nest of compacts so thatu ˜ Fk is a continuous function for each k as in Definition∈ 0.5(iv). Then | F ¯ := z E u˜(z) ¯ F (2.12) k { ∈ | | ≥ } ∩ k is a compact subset of B. ¬ ¬ For any f D( ) consider the function f c D( ) as defined in [MR 92; Chapter Fk ∈ E ∈ E c III, Proposition 1.5]. Then f c 0 m-a.e, f c f m-a.e. on F , and f c 0 weakly Fk Fk k Fk in (D( ), ) as k (cf. [MR≥ 92, page 79]).≥ Therefore it is possible to→ extract a E E1 → ∞ subsequence (kn)n IN so that ∈ 1 N wf,N := fF c (2.13) N kn n X=1 converges strongly to zero as N (cf. [MR 92; Chapter I, Lemma 2.12]). Once again we have w 0 m-a.e. and w → ∞f m-a.e. on F c . f,N f,N kN Now let k ≥be a common subsequence≥ so that, as above, { n}

wu(¯),N 0 and w( u(¯)),N 0 (2.14) → − → (¯) + (¯) c strongly as N . Let g := (u w (¯) ) . Since w (¯) u m-a.e. on F N u ,N u ,N kN → ∞ c − ≥ (¯) we see that gN vanishes m-a.e. on F . Also, w (¯) 0 m-a.e. and u = 0 m-a.e. on kN u ,N ≥ z E u˜(z) < ¯ so gN also vanishes m-a.e. there. This means that gN = 0 m-a.e. on the{ ∈ set | | } ¬ ¬ z E u˜(z) < ¯ F c = (F ¯ )c. (2.15) { ∈ | | } ∪ kN kN (¯) + Thus gN D( )F ¯ . Similarly hN¬ := ( u w( u(¯)),N ) D( )F ¯ . Thus gN hN ∈ E kN ¬ − − − ∈ E kN − ∈ (¯)+ D( )F ¯ and using the strong convergence in (2.14) we conclude that gN hN u E kN − → − u(¯)− = u(¯) as N . → ∞

The above argument demonstrates the existence of a sequence (Fk)k IN of compact subsets (¯) ∈ of B depending on u and ¯, so that u belongs to the closure of kD( )Fk . We need to ﬁnd a single sequence of compacts that works simultaneously for all∪ u ED( ) . ∈ E B 17 Now the metric space D( ) is separable and hence so is the subspace D( )B. Let 1/2 E E (ui)i IN be ˜ -dense in D( )B, and let (vi)i IN be an enumeration of the double sequence ∈ 1 ∈ (1/j) E E (¯) i (ui )i,j IN which is again dense because ui ui as ¯ 0. For each i 1, let (Fj )j IN ∈ → → ≥ ∈ be an increasing sequence of compact subsets of B so that vi belongs to the closure of i jD( )F i , and deﬁne Ek = i,j kFj . This new (Ek)k IN is an increasing sequence of ∪ E j ∪ ≤ ∈ compact subsets of B. For any u D( ) and any δ > 0, let v so v u δ/2 and ∈ E B i k i − k ≤ i take j 1 and w D( )F so that vi w δ/2. Then w D( )Ei j and u w δ. ≥ ∈ E j k − k ≤ ∈ E ∨ k − k ≤ This shows that D( ) is dense in D( ) . ∪k E Ek E B

Remark 2.8. For future reference we note that each of the Ek sets deﬁned above is the ﬁnite union of sets of the type shown in (2.12). It follows that for each k, there exists u ∈ D( )B with -quasi-continuous m-versionu ˜ so that for all z Ek we haveu ˜(z) ¯ > 0. ByE truncatingE from above with ¯, and from below with 0, and∈ then multiplying by≥ 1/¯, we may even assume that 0 u˜(z) 1 everywhere on E, andu ˜(z) = 1 identically on E . ≤ ≤ k

Lemma 2.9. If (Fk)k IN is an -nest, then kD( )B F is dense in D( )B. ∈ E ∪ E ∩ k E

Proof. Let f D( )B and choose sequences (un)n IN ,(vn)n IN kD( )Fk so that + ∈ E ∈ ∈ ∈ ∪ E u f and v f −. Then n → n → + + + wn := f (un) f − (vn) kD( )B F (2.16) ∧ − ∧ ∈ ∪ E ∩ k ¡ ¡ and w f as n . n → → ∞

Corollary 2.10. If (Fk)k IN is an -nest and (Ek)k IN is an B-nest, then (Ek ∈ E ∈ E ∩ Fk)k IN is also an B-nest. Therefore, an -quasi-continuous m-version u˜ of u ∈ E E ∈ D( ) is also -quasi-continuous. E B EB Proof. Fix k and let u D( ) . Applying the previous lemma with B = E , we can ∈ E Ek k ﬁnd k0 k and g D( )Ek F D( )E F so u g ¯. This gives the ﬁrst part of k0 k0 k0 the result,≥ because∈ weE already∩ know⊆ thatE ∩ D( ) k is− densek ≤ in D( ) . ∪k E Ek E B For u D( )B choose an -quasi-continuous m-versionu ˜ and an -nest (Fk)k IN so ∈ E E E ∈ thatu ˜ Fk is continuous for each k 1. Then (Ek Fk)k IN is an B-nest andu ˜ Ek Fk is continuous| for all k 1. Sou ˜ is ≥-quasi-continuous.∩ ∈ E | ∩ ≥ EB Proposition 2.11. If (2.11) holds, then the Dirichlet form ( ,D( )) is quasi- EB EB regular.

Proof. Lemma 2.7 tells us that (QR1) holds, and Corollary 2.10 shows that (QR2) holds for ( ,D( )). So it only remains to show (QR3). EB EB Let (un)n IN be a sequence in D( ) with -quasi-continuous m-versions (˜un)n IN , and ∈ E E ∈ an -nest (Fk)k IN so that u˜n n IN separates points in kFk. Now let (Ek)k IN be E ∈ { | ∈ } ∪ ∈ an B-nest as constructed in Lemma 2.7, and for each k let vk D( )B with an B-quasi- continuousE m-versionv ˜ so that 0 v˜ (z) 1 for all z B, and∈ v ˜E(z) = 1 for allE z E k ≤ k ≤ ∈ k ∈ k (see Remark 2.8). Then the sequence (Fk Ek)k IN is an B-nest, and the doubly indexed ∩ ∈ E sequence (unvk)n,k IN in D( )B has B-quasi-continuous m-versions (˜unv˜k)n,k IN . which ∈ E E ∈ 18 separate points in k(Fk Ek). Since B k(Fk Ek) is B-exceptional, this gives us (QR3). ∪ ∩ \ ∪ ∩ E

By Theorem 0.1, Proposition 2.11 has a corresponding probabilistic counterpart, i.e., it means that the restriction of an m-sectorial standard process to a Borel subset is again an m-sectorial standard process.For the deﬁnition of m-sectorial for standard processes we refer to [MR 92; Chapter IV, Section 6]. We have proved the quasi-regularity of the form ( B,D( )B) provided that it is a Dirich- let form, that is, provided (2.11) holds. The followingE lemmaE gives conditions under which (2.11) will hold, in particular, it holds for every non-empty open set in E (cf. [F 80]).

Lemma 2.12. Suppose that U is a Borel subset of E with m(U) > 0, and so that for some -nest (Fk)k IN we have U Fk is open in Fk for all k. Then (2.11) holds so E ∈ ∩ that ( ,D( ) ) is a quasi-regular Dirichlet form. EU E U

Proof. Applying Remark 2.8 to B = E, we see that there exists an -nest (Fk)k IN so that for each k, the space E ∈

(Fk) := f C(Fk) f =u ˜ Fk for some -quasi-continuous m-versionu ˜ of u D( ) A { ∈ | E ∈ (2E.17)} ¬ separates points and contains¬ the function 1. Since (F ) is a subspace lattice, the Stone- A k Weierstrass theorem tells us that it is uniformly dense in C(Fk). Using the hypothesis on U, by taking the nest even smaller we may assume that U Fk is open in Fk for every k. Let K and K be any two disjoint closed subsets of F ∩and, let f C(F ) so f = 2 1 2 k ∈ k on K1 and f = 1 on K2. Choose u D( ) with an -quasi-continuous m-versionu ˜ so − ∈ E1 E thatu ˜ Fk is continuous and u˜(z) f(z) 2 for all z Fk. Then v = (u 0) 1 has an -quasi-continuous| m-version| (˜u −0) 1| ≤which is continuous∈ on F , is equal∧ to∨ 1 on K , E ∧ ∨ k 1 and equal to 0 on K2. Fix ¯ > 0 and let K be any compact subset of U with with m(K) < . Since m(K) = ∞ m(K ( kFk)) it follows that 1 1 ¯ for some k . Here, and in the remainder K K Fk0 0 ∩ ∪ k − ∩ k ≤ 2 of the proof, the norm will refer to the norm in L (U; m U ). Since (Fk)k IN is an k · k | ∈ -nest, we may choose k1 k0, and u D( )Fk so that u 1K Fk < ¯. Setting E c≥ ∈ E 1 k − ∩ 0 k K1 = K Fk0 and K2 = U Fk1 we may use the result in the previous paragraph to ﬁnd v D( )∩ with -quasi-continuous∩ m-versionv ˜ so that 0 v˜(z) 1 for all z E,v ˜ = 1 on∈K EF , andE v ˜ = 0 on U c F . Finally set w = uv, so≤w D≤( ) and ∈ ∩ k0 ∩ k1 ∈ E U (w(z) 1 (z))2 m(dz) − K1 Z = (uv(z) 1 (z))2 m(dz) − K1 Z = (uv(z) 1)2 m(dz) + (uv(z))2 m(dz) (2.18) K − Kc Z 1 Z 1 (u(z) 1)2 m(dz) + (u(z))2 m(dz) ≤ K − Kc Z 1 Z 1 = u 1 ¯. k − K1 k ≤ 19 Then w 1 w 1 + 1 1 ¯+¯ = 2¯, and since ¯ is arbitrary, we conclude k − K k ≤ k − K1 k k K − K1 k ≤ that 1K can be approximated from within D( )U to any degree of accuracy. Now D( )U E 2 E is a linear space, and the linear span of such 1K is dense in L (U; m U ), so therefore we see that D( ) is dense in L2(U; m ). | E U |U We may now use Lemma 2.12 to prove a generalization of Proposition 2.3, where we do not assume that the measure µ is smooth. In the classical case where ( ,D( )) is associated with Brownian motion, the set U in the statement of Proposition 2.13E belowE can be chosen in a canonical way. We refer to recent work [Stu 93] by K.T. Sturm. For a more functional analytic approach to the problem of perturbations of Dirichlet forms by not necessarily smooth measures, while addressing the problem of quasi-regularity, we refer to Theorem 4.1 in the paper by P. Stollmann and J. Voigt ([StoV 93], see also [Sto 93]).

Proposition 2.13. Suppose that µ is a positive measure on (E, (E)) so that µ(A) = 0 B for all -exceptional sets A (E). Deﬁne the form µ as in (2.3). If D( µ) E ∈ B E E contains at least one non-zero function, then ( µ,D( µ)) is a quasi-regular form on E E some Borel subset U of E.

µ 2 Proof. Recall that D( ) = u D( ) u˜(z) µ(dz) < . Let (un)n IN be a sequence ∈ µ 1/2E { ∈ µE | ∞} 2 µ in D( ) which is ˜ -dense in D( ), and so that (˜un)n IN is L (E; µ)-dense in D( ). E E1 E R ∈ E Fix -quasi-continuous Borel m-versions (˜un)n IN of (un)n IN , and define E ∈ ∈ F := z E u˜ (z) = 0 for all n . (2.19) { ∈ | n } µ For every u D( ) we can find a subsequence (unk )k IN so thatu ñk (z) u˜(z) -quasi- everywhere on∈ E.E Thereforeu ˜(z) = 0 -q.e. on F , and∈ since µ does not charge→ theE Borel -exceptional set (˜u(z) = 0) F , alsou ˜(Ez) = 0 µ-a.e. on F . Define U := E F . Then D( µ) E 6 ∩ 2 µ µ \ µ E µ can be identified with a subspace of L (U; m U ), and D(( U ) ) = D( ) with ( U ) = . Since we assume that D( µ) is non-trivial, it| follows thatE m(U) > 0.E E E E Now let (Fk)k IN be an -nest so thatu ñ is continuous on Fk for each n, k 1. Then ∈ E ≥ U Fk = n uñ = 0 Fk is an open set in Fk so we may apply Lemma 2.12 and conclude ∩ ∪ { 6 } ∩ 2 that ( U ,D( U )) is a quasi-regular Dirichlet form on L (U; m U ). UsingE PropositionE 2.3 we will be able to get the desired conclusion| provided we can show that µ (more precisely µ U ) is U -smooth. Let (Ek)k IN be an U -nest of compact subsets | E ∈ E of U, and without loss of generality assume that Ek Fk for all k. By construction, at each point z E , there exists u D( µ) with an m⊆-versionu ˜ that is continuous on E ∈ k ∈ E k and so thatu ˜(z) > 0. In fact, u is one of the members of the sequence (un)n IN . Now ∈ z Ek uñ(z) > 1/j n,j∞ =1 is an open cover of Ek and so has a finite subcover. That { ∈ | } N means there exist indices nl l=1 and ¯ > 0 so thatv ˜ :=u ñ1 uñ2 uñN satisfies 2{ } ∨ ∨ · · · ∨ v˜ > ¯ on Ek. Since v˜(z) µ(dz) < , this proves that µ(Ek) < , which means that µ is -smooth. ∞ ∞ EU R

20 3. Quasi-regularity of square field operator Dirichlet forms. In this section we prove a general quasi-regularity result for Dirichlet forms which are made up of a square field operator part plus a jump part and a killing part. Our proof of quasi-regularity, in particular of (QR1), will use the nest obtained using Lemma 0.6. Let (E, ρ) be a complete, separable metric space equipped with its Borel σ-algebra (E). Let m, µ, and k be positive σ-finite measures on (E, (E)) and J a symmetric finite positiveB measure on E E. B Now we start× with a core D of functions. Suppose D is a linear space of bounded, continuous, real-valued functions on E, so that D separates points in E and D is closed under composition with smooth functions which vanish at the origin. We assume that each member u of D is square integrable with respect to the measures m, µ, and k. We also assume that for u, v D, if u = v m-a.e., then u = v (which is the case, for example, if supp[m]=E). Thus∈D, L2(E; m) is a one-to-one map and so we may regard D as a subspace of L2(E; m). Finally,→ we assume that D does not vanish identically at any point in E, and combined with the fact that D is a point separating algebra, this implies that D is in fact dense in L2(E; m). Next we assume that we are given a (generalized) square field operator Γ. This means that Γ : D D L1(E; µ) is a positive bilinear mapping, where positivity means that for each u D×we have→ Γ(u) := Γ(u, u) 0 µ-a.e. We now∈ define a bilinear form on≥ the core D by setting, for u, v D, E ∈

(u, v) = Γ(u, v) dµ + (u(z1) u(z2))(v(z1) v(z2)) J(dz1dz2) + uv dk. (3.1) E E E E − − E Z Z × Z We assume that ( ,D) is closable in L2(E; m) and that its closure ( ,D( )) is a Dirichlet form (see Section 4E below for examples). Then the map Γ : D D LE 1(EE; µ) is continuous 1/2 × → in the 1 -norm and so Γ extends to a continuous bilinear map on D( ). For notational convenienceE we will continue to denote this map as Γ. E We want to give conditions on Γ under which the form ( ,D( )) is quasi-regular. Since the space D consists of continuous functions, the condition (QR2)E E is automatically fulfilled. Since E E is a separable metric space, it is strongly Lindelöfand since D separates points × in E, we conclude that there is a countable set un n IN in D that separates points in E. Thus, (QR3) is also automatically fulfilled,{ so in| order∈ } to prove the quasi-regularity of ( ,D( )) we only need to show (QR1), that is, we need to find an -nest (Fk)k IN consistingE E of compact sets. E ∈ In proving the quasi-regularity of ( ,D( )), it will be convenient for us to change the base measure. Consider ( ,D) as a bilinearE form,E not over L2(E; m), but over L2(E; m+µ+k). E This form is again closable and its closure ( 0,D( 0)) is a Dirichlet form. If we could E E prove that ( 0,D( 0)) is quasi-regular, then the measure m is certainly a D-proper speed E E measure for 0, and applying Proposition 2.5 we find that ( ,D( )) is also quasi-regular. Therefore, inE proving quasi-regularity we could take the baseE measureE to be m + µ + k. But for notational convenience we will relabel this new base measure as m, and assume, without loss of generality, that µ + k m. In the next lemma we obtain a substitute≤ for the representation (3.1), as this representation may not hold on the complete domain D( ). E 21 Lemma 3.1. For u D( ), we have ∈ E

(u) Γ(u) dµ + 4J(E E) u 2 + u2 dk. (3.2) E ≤ × k kL∞(E;m) Z Z

Proof. We begin by assuming that u 2 < , since the inequality is trivial other- k kL∞(E;m) ∞ wise. From (3.1) we see that the inequality (3.2) is true for u D. Now let un D so that (u u ) 0, and let ψ be a smooth function on IR with∈ bounded derivative∈ so that E1 − n → supx IR ψ(x) u L∞(E;m) + ¯, and ψ(x) = x for x u L∞(E;m). Setting vn := ψ(un) ∈ | | ≤ k k | | ≤ k k we see that vn D and from Remark 0.4 we know that 1(u vn) 0. In particular, Γ(v ) Γ(u) in∈L1(E; µ) and v u in L2(E; k). Now pluggingE − v into→ (3.2) gives n → n → n

(v ) Γ(v ) dµ + 4J(E E)( u + ¯)2 + v2 dk, (3.3) E n ≤ n × k kL∞(E;m) n Z Z and letting n and then ¯ 0 gives us the required result (3.2). → ∞ → The fact that m, µ, and k are not necessarily finite measures causes problems for our calculations. In order to overcome this difficulty we will use a scaling technique that was used in [ALR 93]. Since m is σ-finite we can find a function 0 < ψ 1 so that ψ2dm < . ≤ ∞ Let h = G1ψ. It follows that 0 < h 1 m-a.e. and h is 1-excessive (cf. [MR 92; Chapter ≤ h h R 2 2 III] for a discussion of excessive functions). Define 1 on D( 1 ) := u L (E; h m): uh D( ) by E E { ∈ ∈ E } h(u, v) := (uh, vh), u, v D( h), (3.4) E1 E1 ∈ E1 2 2 h h considered as a form over L (E; h m). Since h is 1-excessive, ( 1 ,D( 1 )) is a Dirichlet hE hE form, and the map u uh defines a bijective isometry between ( 1 ,D( 1 )) and ( 1,D( )). Since h > 0 m-a.e., it→ follows that, for any closed set F E, theE subspacesE D(E h) Eand ⊆ E1 F D( )F correspond to each other under this isometry. Consequently, a sequence (Fk)k IN is anE -nest if and only if it is an h-nest. So to prove (QR1) for ( ,D( )), it suffices∈ E h h E h h E E to prove it for ( 1 ,D( 1 )). For u, v D( 1 ), let us define Γ (u, v) := Γ(uh, vh). Notice that from LemmaE 3.1,E and the fact that∈ hEis bounded and belongs to both L2(E; m) and L2(E; k), we see that there exists c > 0 so that for every u D( h), ∈ E1

h(u) Γh(u) dµ + c u 2 . (3.5) E1 ≤ k kL∞(E;m) Z

There is one more condition that we must impose on the operator Γh: we suppose that if h u, v D( 1 ), then ∈ E Γh u v Γh(u) Γh(v) µ-a.e. (3.6) ∨ ≤ ∨ The following lemma gives a way in¡ which to check condition (3.6). Lemma 3.2. Suppose that if u, v D, then ∈ Γ(u, φ(v)) Γ(u, v) µ-a.e., (3.7) | | ≤ | | 22 whenever φ is a smooth function on IR with φ(0) = 0 and φ0(x) 1. Then (3.6) | | ≤ holds for the operator Γ on D( ), and hence also for Γh on D( h). E E1 Proof. Let φ be a sequence of smooth functions on IR satisfying φ (0) = 0 and φ0 (x) n n | n | ≤ 1, and such that φn(x) x as n . Just as in [MR 92; Chapter I, Proposition 4.17] → | | → ∞ 1/2 we can show that for any u D( ), φn(u) u in 1 -norm as n . For u, v D, we apply (3.7) to φ and then∈ letEn to→ obtain | | E → ∞ ∈ n → ∞ Γ(u, v ) Γ(u, v) µ-a.e. (3.8) | | | | ≤ | | But since φ(x) = x is a normal contraction, we can use Remark 0.4 to conclude that (3.8) can be extended| | to all of D( ). Now we use the inequality (3.8), the fact that Γ is a bilinear form, and the equation xE y = (1/2) (x + y) + x y to obtain (3.6) for Γ on D( ). ∨ { | − |} E Γ(u v) = (1/4) Γ(u + v) + 2 Γ(u + v, u v ) + Γ( u v ) ∨ { | − | | − | } (1/4) Γ(u + v) + 2 Γ(u + v, u v ) + Γ( u v ) ≤ { | | − | | | − | } (1/4) Γ(u + v) + 2 Γ(u + v, u v) + Γ(u v) ≤ { | − | − } = (1/4) Γ(u + v) + 2 Γ(u) Γ(v) + Γ(u v) { | − | − } = (1/4) Γ(u) + 2Γ(u, v) + Γ(v) + 2 Γ(u) Γ(v) + Γ(u) 2Γ(u, v) + Γ(v) { | − | − } = (1/2) (Γ(u) + Γ(v)) + Γ(u) Γ(v) { | − |} = Γ(u) Γ(v). ∨ (3.9) h If u, v D( 1 ), then using the positivity of h and applying inequality (3.6) for Γ on D( ) to uh and∈ vhE , we obtain (3.6) for Γh. E

Remark 3.3. For any u, v D( h), applying (3.6) to the pair u and v gives us ∈ E1 − − Γh u v Γh(u) Γh(v) µ-a.e. (3.10) ∧ ≤ ∨ ¡ We recall that a metric ρ1 on E is called uniformly equivalent to ρ if the identity from (E, ρ) to (E, ρ1) and its inverse are uniformly continuous.

Theorem 3.4. Suppose that for some countable dense set x i 1 in (E, ρ), there { i | ≥ } exists a countable collection f i 1, j 1 of functions in D( h) satisfying { ij | ≥ ≥ } E1 h 1 sup Γ (fij ) =: ϕ L (E; µ), (3.11) i,j ∈

and ˜ h ρ1(z, xi) = sup fij(z) 1 -q.e. z E for all i IN, (3.12) j E ∈ ∈ where ρ is a bounded metric on E uniformly equivalent to ρ, and f˜ is an h-quasi- 1 ij E1 continuous h2m-version of f . Then the Dirichlet form ( h,D( h)) satisﬁes (QR1), ij E1 E1 23 which implies that ( ,D( )) also satisﬁes (QR1). We conclude that ( ,D( )) is E E E E quasi-regular.

Proof. Fix the index i and for each n 1 deﬁne the function ≥ n un(z) := sup f˜ij(z). (3.13) j=1

h h Then un D( 1 ) and un is 1 -quasi-continuous. Furthermore, from (3.5), (3.6), and ∈ E E h h 1/2 (3.10) we see that the sequence (un)n IN is bounded in D( 1 ) in the ( 1 ) -norm, and h ∈ 1 E E even more, Γ (un) is dominated by ϕ L (E; µ). By the Banach-Saks theorem, there ∈ 1 N exists a subsequence unk whose averages N k=1 unk converge strongly in the Hilbert h h space (D( 1 ), 1 ). From Lemma 0.6 we know that a further subsequence of these averages E E h h P must converge 1 -quasi-everywhere to an 1 -quasi-continuous m-version of a function in h E E h D( 1 ). But, on the other hand, the original sequence un already converges 1 -q.e. to the E h E h limit ρ1(z, xi). Thus the function z ρ1(z, xi) is 1 -quasi-continuous, belongs to D( 1 ) and satisﬁes Γh(ρ ( , x )) ϕ L1(µ7→). E E 1 · i ≤ ∈ Now for each n 1, deﬁne the function ≥ n wn(z) = inf ρ1(z, xi). (3.14) i=1

Arguing as above we ﬁnd that the averages yN of a subsequence of (wn)n IN converge strongly in (D( h), h), that is, ∈ E1 E1

1 N y = w y in D( h). (3.15) N N nk → E1 kX=1

On the other hand, the sequence (wn)n IN converges pointwise to zero on E and so the limit y in (3.15) is the zero function. ∈ h By Lemma 0.6 we know that a subsequence of (yN )N IN converges 1 -quasi-uniformly ∈ E to zero on E. But since the sequence wn is decreasing, it follows that (wn)n IN itself h h ∈ converges 1 -quasi-uniformly to zero on E. This means that there is an 1 -nest (Fk)k IN E E ∈ so that, for each k IN, wn converges to zero uniformly on Fk. Fix any k IN and δ > 0, then choose N so w∈ δ on F . Then inf n ρ (z, x ) δ on F , or ∈ N ≤ k i=1 1 i ≤ k

N F B(x , δ), (3.16) k ⊆ i i [=1 where B(δ) = y E ρ (x, y) δ is the ball centered at x, with radius δ. Since this is { ∈ | 1 ≤ } possible for every δ > 0, we conclude that Fk is totally bounded and so, since (E, ρ1) is h h complete, Fk is compact. This gives (QR1) for ( 1 ,D( 1 )) and hence for ( ,D( )), Since (QR2) and (QR3) already hold, we conclude thatE ( ,DE ( )) is a quasi-regularE E Dirichlet form. E E

24 Now that we know ( ,D( )) is quasi-regular we may use the existence results [MR 92; Chapter IV, Theorem 3.5]E toE construct an associated strong Markov process.

Deﬁnition 3.5. A right process M with state space E and transition semigroup (pt)t>0 2 is called properly associated with ( ,D( )) if for all f Bb(E) L (E; m) and all t > 0 we have E E ∈ ∩ p f is an -quasi-continuous m-version of T f, (3.17) t E t where (T ) is the semigroup on L2(E; m) generated by ( ,D( )). t t>0 E E Corollary 3.6. There exists a right process M properly associated with ( ,D( )). E E Remark 3.7. The process M can, in fact, be taken to be an m-tight special standard process. We refer the reader to [MR 92; Chapter IV] for deﬁnitions and more details.

25 4. Applications. (a) Quasi-regular gradient-type Dirichlet forms on Banach space. Let E be a (real) separable Banach space, and µ a ﬁnite measure on (E) which charges every weakly open set. Deﬁne a linear space of functions on E by B

m C∞ = f(l , . . . , l ) m IN, f C∞(IR ), l , . . . , l E0 . (4.1) F b { 1 m | ∈ ∈ b 1 m ∈ } m m Here Cb∞(IR ) denotes the space of all infinitely differentiable functions on IR with all partial derivatives bounded. By the Hahn-Banach theorem, Cb∞ separates the points of F 2 E. The support condition on µ means that we can regard Cb∞ as a subspace of L (E; µ), F 2 and a monotone class argument shows that it is dense in L (E; µ). Define for u Cb∞ and k E, ∈ F ∈ ∂u d (z) := u(z + sk) , z E. (4.2) ∂k ds |s=0 ∈

Observe that if u = f(l1, . . . , lm), then

∂u m ∂f = (l , . . . , l ) l , k , (4.3) ∂k ∂x 1 m E0 h i iE i i X=1 which shows us that ∂u/∂k is again a member of Cb∞. Also let us assume that there is a separable real Hilbert space (H, , ) denselyF and continuously embedded into E. h iH Identifying H with its dual H0 we have that

E0 H E densely and continuously, (4.4) ⊂ ⊂ and , restricted to E0 H coincides with , . Observe that by (4.3) and (4.4), for E0 h iE × h iH u Cb∞ and ﬁxed z E, the map k (∂u/∂k)(z) is a continuous linear functional on H∈. Deﬁne F u(z) H ∈by → ∇ ∈ ∂u u(z), k = (z), k H. (4.5) h∇ iH ∂k ∈

Deﬁne a bilinear form on C∞ by F b

(u, v) = u(z), v(z) µ(dz). (4.6) Eµ h∇ ∇ iH Z

Assumption 4.1. We assume that the form in (4.6) is closable in L2(E; µ). Eµ Let (H) denote the set of all bounded linear operators on H with operator norm L∞ . Suppose z A(z), z E is a map from E to (H) such that z A(z)h1, h2 H isk k∞(E)-measurable→ for all h∈, h H. Furthermore,L assume∞ that → h i B 1 2 ∈ there exists α (0, ) such that A(z)h, h α h 2 for all h H, (4.7) ∈ ∞ h iH ≥ k kH ∈ 26 1 1 1 and that A˜ L (E; µ) and Aˇ L∞(E; µ), where A˜ := (A + Aˆ), Aˇ := (A Aˆ) k k∞ ∈ k k∞ ∈ 2 2 − and Aˆ(z) denotes the adjoint of A(z), z E. Let c L∞(E; µ) and b, d L∞(E H; µ) ∈ ∈ ∈ → such that for all u C∞ with u 0, ∈ F b ≥

( d, u + cu) dµ 0 and ( b, u + cu) dµ 0. (4.8) h ∇ iH ≥ h ∇ iH ≥ Z Z Deﬁne the constant k = b + d . For u, v C∞ let, k kL∞(E;µ) ∈ F b

(u, v) = A(z) u(z), v(z) µ(dz), (4.9) EA h ∇ ∇ iH Z and

(u, v) = (u, v) + u d, v dµ + b, u v dµ + uvc dµ. (4.10) E EA h ∇ iH h ∇ iH Z Z Z Then Example 3e of [MR 92; Chapter II] shows us that the forms ( , C∞) and ( , C∞) EA F b E F b are closable and that their closures ( A,D( A)) and ( ,D( )) are Dirichlet forms. We would like to show that theseE two formsE are equivalentE E in the sense of Proposition 2.1. To begin with we note that for u C∞ we have, for µ-almost every z E, ∈ F b ∈ (b + d)(z), u(z) u(z) (b + d)(z) u(z) u(z) |h ∇ iH | ≤ k kH k∇ kH | | 2k ( u(z) 2 + u(z) 2) (4.11) ≤ k∇ kH | | 2k ((1/α) A(z) u(z), u(z) + u(z) 2) ≤ h ∇ ∇ iH | | and c(z)u(z)2 c u(z)2. (4.12) | | ≤ k kL∞(E;µ) Therefore, using (4.10) and integrating with respect to µ we see that for u C∞, ∈ F b (u) (u) k ( ) (u), (4.13) EA ≤ E ≤ 1 EA 1 where the constant k can be taken to be max (2/α)k+1, 2k+ c . By Proposition 1 { k kL∞(E;µ)} 2.1, the domains D( ) and D( A) coincide and the form ( ,D( )) is quasi-regular if and only if the form ( ,DE ( )) is.E E E EA EA In order to prove quasi-regularity of ( A,D( A)) (cf. [RS 92]), we shall use Theorem 3.4, and since the measure µ is already ﬁniteE we doE not require the re-scaling, that is, we take

h 1. The square ﬁeld operator is given on Cb∞ by Γ(u, v)(z) = A(z) u(z), v(z) H , which≡ clearly satisﬁes (3.7), and by LemmaF 3.2, also condition (3.6).h Now∇ since∇ E isi a

separable Banach space, there exists a countable set (lj)j IN in E0 such that lj E0 1 for all j IN, and z = sup l (z) for all z E. Let ϕ be∈ a bounded, smoothk functionk ≤ ∈ k kE j j ∈ on IR such that ϕ(0) = 0, ϕ is strictly increasing, and ϕ0 is decreasing and bounded by 1. Then ρ (z, x) := ϕ( z x ) is a bounded metric on E that is uniformly equivalent with 1 k − kE the usual metric ρ(z, x) := z x E. Let (xi)i IN be a countable dense subset of E, and deﬁne for every i, j IN, k − k ∈ ∈ f (z) := ϕ(l (z x )). (4.14) ij j − i 27 Then f C∞ for every i, j IN, and ij ∈ F b ∈

f (z) = ϕ0(l (z x ))l , (4.15) ∇ ij j − i j and so for µ-a.e. z E, ∈

sup Γ(fij )(z) = sup A(z) fij (z), fij(z) H i,j i,j h ∇ ∇ i 2 = sup(ϕ0(l (z x ))) A(z)l , l j − i h j jiH i,j (4.16) 2 ˜ = sup(ϕ0(lj(z xi))) A(z)lj, lj H i,j − h i A˜(z) . ≤ k k∞

Since A˜ belongs to L1(E; µ) we see that (3.11) is satisﬁed. On the other hand, for every ﬁxedk k∞i IN, we have ∈

sup fij(z) = sup ϕ(lj(z xi)) = ϕ(sup lj(z xi)) j j − j − (4.17) = ϕ( z x ) = ρ (z, x ), k − ikE 1 i for every z E and so (3.12) is also fulfilled. Therefore Theorem 3.4 applies and we conclude that∈ ( ,D( )), and hence ( ,D( )) is quasi-regular. EA EA E E (b) An intrinsic quasi-regular Dirichlet form on the free loop space. The results of this subsection have been first proved in [ALR 93]. Our purpose here is to show that they also be obtained from the general Theorem 3.4. Let g := (gij ) be a uniformly elliptic Riemannian metric with bounded derivatives d 1/2 ∂ 1/2 ij ∂ over IR and ∆ := (det g)− (det g) g the corresponding Laplacian. Let g ∂xi ∂xj d pt(x, y), x, y IR , t 0, be theP associated¢ heat kernel£ with respect to the Riemannian volume element.∈ Let ≥W (IRd) denote the set of all continuous paths ω : [0, 1] IRd and let (IRd) := ω W (IRd) ω(0) = w(1) , i.e., (IRd) is the free loop space over→ IRd. Let x L { ∈ | } L d P1 be the law of the bridge defined on ω (IR ) ω(0) = ω(1) = x coming from the d { ∈ L | } diffusion on IR generated by ∆g and let

x µ := P1 p1(x, x) dx (4.18) Z be the Bismut measure on (IRd) which is σ-finite but not finite. We consider (IRd) equipped with the Borel σ-algebraL coming from the uniform norm on (IRd)L which makes it a Banach space. The tangent space T (IRd) at a loop ω k( kIR∞d) wasL introduced ωL ∈ L in [JL 91] as the space of periodical vector fields Xt(ω) = τt(ω) h(t), t [0, 1], along ω. Here τ denotes the stochastic parallel transport associated with the Levi-Civita∈ connection

28 d of (IR , g) and h belongs to the linear space H0 consisting of all absolutely continuous maps h : [0, 1] T IRd IRd such that → ω(0) ≡ 1 1 2 2 (h, h) := h0(s) ds + h(s) ds < (4.19) H0 | | | | ∞ Z0 Z0 (where v 2 := g (v, v)) and τ (ω)h(1) = h(0) with τ (ω) = holonomy along ω (cf. | | ω(0) 1 1 [JL 91] for details). Note that if we consider (IRd) as continuous maps from S1 to IRd this notion is invariant by rotations of S1 andL that (4.19) induces an inner product on d Tω (IR ) which turns it into a Hilbert space. Below we shall also need the Hilbert space H L (IRd)( T (IRd)) with inner product ( , ) which is constructed analogously but ωL ⊃ ωL H without the holonomy condition, i.e., H0 is replaced by H which denotes the linear space d d of all absolutely continuous maps h : [0, 1] T IR IR satisfying (4.19). Let ∞ → ω(0) ≡ FC0 denote the linear span of the set of all functions u : (IRd) IR such that there exists d k L → k IN, f C∞((IR ) ), t , . . . , t [0, 1] with ∈ ∈ 0 1 k ∈ u(ω) = f(ω(t ), . . . , ω(t )), ω (IRd). (4.20) 1 k ∈ L 2 2 d Note that 0∞ is dense in L (µ) := (real) L ( (IR ); µ). Let ∞, Cb∞ be deﬁned FC d k d Lk FC d kF correspondingly with C∞((IR ) ) resp. Cb∞((IR ) ) replacing C0∞((IR ) ). We deﬁne the d directional derivative of u C∞, u as in (4.20), at ω (IR ) with respect to X(ω) H (IRd) by ∈ F ∈ L ∈ ωL

∂hu(ω) := ∂X u(ω) : = dif(ω(t1), . . . , ω(tk))Xti (ω) (4.21) i X=1 k = g ( f(ω(t ), . . . , ω(t )), τ (ω)h(t )) ω(ti) 5i 1 k ti i i X=1

where h H with X(ω) = (τt(ω)h(t))t [0,1] and i resp. di denotes the gradient (with ∈ ∈ 5 respect to g) resp. the diﬀerential relative to the i-th coordinate of f. We extend ∂h to all d of C∞ by linearity. Note that if we consider u as a function on W (IR ) then F

d d ∂X u(ω) = u(ω + sX(ω)) s=0, ω (IR ). (4.22) ds | ∈ L

Hence ∂X u is well-deﬁned by (4.21) (i.e., independent of the special representation of u).

d Let for u C∞ and ω (IR ), Du(ω) be the unique element in H such that (Du(ω), h) ∈ F ∈ L H = ∂ u(ω) for all h H and let Du(ω) be its projection onto H . Deﬁne for u, v C∞ h ∈ 0 ∈ F 0 e e (u, v) = (Du, Dv) dµ. (4.23) E H0 (IRZ d) L 29 (cf. [AR 89,90,91] for the ﬂat case). By our assumptions on g and (4.29) below it follows

that (u, u) < for all u C0∞. By [L 92,93], [ALR 93] the densely deﬁned quadratic E ∞ ∈ F 2 form ( , C∞) is closable on L (µ). Clearly, the closure ( ,D( )) is of the type discussed E F 0 E E in the preceding section (see (3.1)) with core C∞ and F 0 Γ(u, v) = (Du, Dv) , u, v D( ) (4.24) H0 ∈ E where we denote the closure of D with domain D( ) also by D. We note that D satisﬁes the chain rule, in particular, E

Dφ(u) = φ0(u) D(u) for all u D( ). (4.25) ∈ E h h h Hence by Lemma 3.2, Γ and consequently also Γ satisfy (3.6). Here and below Γ , 1 , and D( h) are deﬁned as in Section 3. It is easy to see that E E1 h h 2 2 C∞ D( ) and Γ (u) 2(h Γ(u) + u Γ(h)). (4.26) F b ⊂ E1 ≤

Let ϕ C∞(IR) be an odd and increasing function such that ϕ 2, ϕ0 1, ϕ00 0 ∈ b | | ≤ ≤ ≤ on [0, ), and ϕ(x) = x for x [ 1, 1]. Let sk k IN be a dense set of [0,1] and fix ∞ d l d ∈ − { | ∈ } ω0 (IR ). Let x : IR IR, 1 l d, be the standard linear coordinates and define for ∈j LIN → ≤ ≤ ∈ l d uj(ω) := sup sup ϕ(x (ω(sk) ω0(sk))) , ω (IR ). (4.27) k j l d | − | ∈ L ≤ ≤ Applying first (3.6) and then (4.26) and the chain rule for D we obtain that for µ -a.e. ω (IRd) ∈ L Γh(u )(ω) 4 sup sup (h2 D(xl(ω(s ) ω (s ))) 2 + Dh(ω) 2 ). (4.28) j k 0 k H0 H0 ≤ k j l d k − k k k ≤ ≤

For u C∞ we have that Du Du and if u(ω) = f(ω(s ), . . . , ω(s )) then ∈ F k kH0 ≤ k kH 1 k k e 1 Du(ω)(s) = G(s, s ) τ (ω)− f(ω(s ), . . . , ω(s )) (4.29) i si 5i 1 k i X=1 e d2 where G is the Green function of dt2 + 1 with Neumann boundary conditions on [0,1], i.e., − e u+s 1 1 (u+s) u s 1 1 u s G(s, u) = e − + e − + e| − |− + e −| − | . (4.30) 2(e2 1) − ¡ Hence by our assumptions on g and by (4.29) there exists c ]0, [ such that for all d ∈ ∞ ω0 (IR ) ∈ L h 2 2 Γ (uj)(ω) c (h + Dh ) µ-a.e. (4.31) ≤ k kH0 1 But the function on the right hand side of (4.31) is in L (E; µ). Let now ωi i IN d { | ∈ } be a dense subset of (IR ) and define fij := uij 1 where uij is equal to uj with ω0 replaced by ω , i, j LIN. Then all assumptions of∧ Theorem 3.4 are fulfilled with the i ∈ 30 d metric ρ(ω, ω0) := 1 ω ω0 , ω, ω0 (IR ). Hence ( ,D( )) is quasi-regular. The corresponding Markov∧ k process− k is∞ in fact∈ a L diffusion whichE is invariantE by rotation of the loops. (cf. [ALR 93] for more details). Similarly, the tightness results in [DR 92] can be also derived from the general Theorem 3.4.

(c) Fleming-Viot processes. Let E := (S) be the space of probability measures on a Polish space S with Borel σ-algebra (MS). Let E be equipped with the topology of weak convergence. There exist B uniformly continuous functions (φi)i IN on S, such that φi 1, and the topology on E is generated by the metric ∈ k k∞ ≤

ρ(µ, ν) = sup φ dµ φ dν , µ, ν E, (4.32) i − i ∈ i ¬Z Z ¬ ¬ ¬ ¬ ¬ and (E, ρ) is complete. Set µ, φ :=¬ φ dµ for φ Cb(¬S) and µ a ﬁnite positive measure on (S) and let h i ∈ B R m C∞ := f( , ψ ,..., , ψ ) m IN, ψ C (S), 1 i m, f C∞(IR ) . (4.33) F b { h · 1i h · mi | ∈ i ∈ b ≤ ≤ ∈ b }

For u = f( , ψ ,..., , ψ ) C∞ and x S define h · 1i h · mi ∈ F b ∈ ∂u d (µ) := u(µ + s¯x) s=0, µ (S), (4.34) ∂¯x ds | ∈ M and ∂u u(µ) := (µ) . (4.35) ∇ ∂¯x x S ² ³ ∈ For µ (S) and f, g L2(S; µ) we also set ∈ M ∈ f, g := fg dµ f dµ g dµ. (4.36) h iµ ∫ − ∫ ∫ 2 Note that for u Cb∞, µ (S) the map x ∂u/∂¯x(µ) belongs to the space L (S; µ), i.e., u(µ) L2∈(S F; µ), hence∈ M if m is a finite positive7→ measure on the Borel sets (E) of E we can∇ define∈ B

(u, v) := u(µ), v(µ) m(dµ); u, v C∞. (4.37) E h∇ ∇ iµ ∈ F b Z Clearly, Cb∞ separates the points of E and therefore if supp[m] = S, then ( , Cb∞) is a denselyF defined positive definite symmetric bilinear form on L2(E; m). If itE isF closable, then its closure ( ,D( )) is clearly of the type studied in Section 3 with core C∞, and E E F b Γ(u, v)(µ) := u(µ), v(µ) , µ E. (4.38) h∇ ∇ iµ ∈ It is easy to check that ( ,D( )) is a Dirichlet form and by Lemma 3.2, Γ satisfies (3.6). E E Since ( (S), ρ) is separable we can find νi (S), i IN, which are dense in (S). DefiningM for i, j IN ∈ M ∈ M ∈ f (µ) := φ dµ φ dν , µ E, (4.39) ij ∫ j − ∫ j i ∈ 31 we see that f C∞, that for µ E ij ∈ F b ∈ Γ(f )(µ) = φ , φ ij h j jiµ (4.40) φ2 dµ 1 L1(E; m), ≤ j ≤ ∈ Z and that ρ(µ, νi) = sup fij (µ) for all µ E, i IN. (4.41) j ∈ ∈ Thus, Theorem 3.4 applies and ( ,D( )) is a quasi-regular Dirichlet form. The corresponding process is in fact a diffusion.E E

If m is the reversible invariant measure of the Fleming-Viot process on (S) (cf. e.g. M [EK 93]), then ( , Cb∞) is closable and the corresponding process is just the Fleming-Viot process. For moreE F details on this, a more general set-up including non-symmetric Dirich- let forms with state space (S) (i.e., Fleming-Viot process with generalized selection), and a thorough study of theM associated generating operators as well as the corresponding martingale problems we refer to [ORS 93].

32 5. Counterexamples. (a) A Dirichlet form ( ,D( )) that satisfies (QR1) and (QR2), but not (QR3). E E Our first example is a regular Dirichlet form that is not quasi-regular. This form is defined over a separable, compact space E, so this example shows that the assumption of metrizability in Proposition 0.9 cannot be dropped. All that this example requires is a pathological measure space. It really has very little to do with the Dirichlet form, in fact, we will take ( ,D( )) to be the “zero" form. Let X = [0, Ω] be the first uncountable ordinal space withE the orderE topology. Then X is a compact, Hausdorff space and (X) consists of all subsets that are countable or have countable complement. We define aB Borel measure µ by 0, if B is countable; µ(B) = (5.1) 1, otherwise. º A function u on X is measurable only if it is eventually constant on [0, Ω), that is, it is constant on an open set of the form (Λ, Ω) for some Λ X. We denote this left limit as u(Ω ). The space L2(X; µ) identifies those measurable∈ functions with the same left limit at Ω.− Everything is fine, except for the fact that X is not separable. So now we use the fact that X is completely regular and embed it into the separable, compact space E = [0, 1][0,1]. Let µ∗ be the image measure under this embedding and let (zj)j IN be a countable dense set in E. Define a Borel measure m on (E) by setting ∈ B

∞ j m = µ∗ + (δzj /2 ). (5.2) j X=1 Now let ( ,D( )) be the zero form on L2(E; m), that is, D( ) = L2(E; m) and (u, v) = 0 for all u,E v DE ( ). We now show that this Dirichlet formE satisﬁes (QR1) andE (QR2) but not (QR3).∈ SinceE E is compact, (QR1) is trivially satisﬁed. We will prove (QR2) by showing that every measurable function u on E has an -quasi-continuous m-version. Let u be measurable and set E u(Ω ), if z = Ω; u˜(z) = (5.3) u(z)−, otherwise. º Here [0, Ω] is regarded as a subset of E. Since m( Ω ) = 0, we see thatu ˜ is an m-version of u. Also, the sequence of closed sets { }

k F = [0, Ω] z (5.4) k ∪ { j} j [=1 is an -nest, and, since u is constant on a set of the form (Λ, Ω) E,u ˜ Fk is continuous for each kE. This means thatu ˜ is -quasi-continuous and so (QR2)⊆ holds.| Now we show that (QR3) fails. An exceptional setE N must always be contained in a Borel set of measure zero, c so N (Λ, Ω) for some Λ [0, Ω). This means that any sequence of functions (un)n IN ⊆ ∈ ∈ satisfying (QR3) must separate points in (Λ, Ω), for some Λ. Since each un is measurable, there exists Λ (Λ, Ω) so that u is constant on (Λ , Ω). Let Λ∗ = sup Λ . Because Ω n ∈ n n n n 33 is the ﬁrst uncountable ordinal, Λ∗ < Ω and so the sequence (un)n IN fails to separate the ∈ points of (Λ∗, Ω). Thus no countable collection un n IN can satisfy the conditions of (QR3). { | ∈ }

(b) A Dirichlet form ( ,D( )) that satisﬁes (QR2) and (QR3), but not (QR1). E E For our next example we take E = [0, 1) equipped with m= Lebesgue measure dz, and let ( ,D( )) be the Dirichlet form associated with reﬂecting Brownian motion on [0, 1]. ThatE is, E

2 D( ) = u u is absolutely continuous on (0, 1) and u0 L ((0, 1); dz) E { | ∈ } (5.5) (u, v) = (1/2) u0(z) v0(z) dz. E Z In order to explain this example we need the following result. / Lemma 5.1. If u u in 1 2-norm and m(z u (z) = 0) > 0 for all k, then there k → E1 | k exists z∗ [0, 1] so that the continuous version of u has a limit of zero at z∗. ∈ Proof. Since m(z uk(z) = 0) > 0, the continuous versionu ˜k of uk must vanish at some point z in [0, 1). Therefore,| for all z [0, 1), k ∈ z u˜k(z) = uk0(s) ds. (5.6) Zzk

Now by taking subsequences we may assume that zk z∗ [0, 1]. Since uk u in 1/2 2 → ∈ → -norm, the derivatives u 0 converge in L ((0, 1); dz) to u0 so E1 k z z u˜ (z) = u 0(s) ds u0(s) ds. (5.7) k k → Zzk Zz∗ The right hand side of (5.7) is the continuous version of u and it clearly has limit zero at z∗.

One consequence of this lemma is that an increasing sequence of sets (Fk)k IN is an -nest only if m(F ) = 1 for some k. That is because the constant function 1 cannot∈ be E k approximated from within kD( )Fk otherwise. Since no compact subset of [0, 1) has full measure we see that (QR1)∪ fails forE ( ,D( )). The conditions (QR2) and (QR3) are easily seen to be satisﬁed. E E

(c) A Dirichlet form ( ,D( )) that satisﬁes (QR1) and (QR3), but not (QR2). E E In this example we take the same Dirichlet form ( ,D( )) as in Example (b), except we give E = [0, 1) the topology of the circle. Then E isE compactE so that (QR1) trivially holds. It is again quite easy to show that (QR3) holds, so we will only show that (QR2) fails. In fact, using Lemma 5.1 as before, we see that an increasing sequence (Fk)k IN can only be an -nest if F = E for some k. Thus an -quasi-continuous function must∈ be continuous E k E 34 everywhere on E. But D( ) contains a lot of functions that do not have a version that is continuous on the circle, forE example, u(z) = z. Therefore (QR2) fails.

In example (b), we saw a Dirichlet form which satisfies (QR2) and (QR3) but not (QR1) and so is not quasi-regular. In that example, the reason that quasi-regularity fails is that the space E = [0, 1) is adequate to define the form but not as a state space for reflecting Brownian motion. The boundary point 1 is missing from the space, and if we put it back, we get the usual quasi-regular form on{ }E = [0, 1] corresponding to reflecting Brownian motion. The following example shows that the problem of “missing boundary points" can even occur when E is a complete metric space. It is an example of a classical Dirichlet form, i.e., a form of gradient type defined on a complete linear space E, which, nevertheless, is not quasi-regular. Once again, it would be possible to embed E into an even larger space so that ( ,D( )) becomes quasi-regular, but we will not do it. E E (d) A classical Dirichlet form that is not quasi-regular. Let E be an infinite dimensional, separable Hilbert space and denote each point z E as ∈ z = (z1, z2, . . . , zi,...), (5.8) where (zi)i IN are the coordinates of z with respect to some fixed orthonormal basis. We ∈ 2 equip E with its Hilbert topology and its Borel σ-algebra (E). Let (σi )i IN be a sequence of strictly positive numbers such that σ2 < , and letB m be the Gaussian∈ measure on i i ∞ (E, (E)) so that (zi)i IN becomes a sequence of independent mean zero Gaussian random B 2 ∈ 2 P variables with E(zi ) = σi . Define a core of functions, dense in L2(E; m), by

k C∞ = u k 1, f C∞(IR ) such that u(z) = f(z , . . . , z ) . (5.9) F b { | ∃ ≥ ∈ b 1 k }

For i 1 and u C∞, with representation u(z) = f(z , . . . , z ), we deﬁne the function ≥ ∈ F b 1 k ∂u/∂zi by (∂f/∂x )(z , . . . , z ), for 1 i k; (∂u/∂z )(z) = i 1 k (5.10) i 0, otherwise.≤ ≤ º Now let (γi)i IN be a sequence of strictly positive constants and deﬁne the form by ∈ E ∞ (u, v) = γ (∂u/∂z )(∂v/∂z ) dm, (5.11) E i i i E i Z X=1

for u, v Cb∞. It can be shown that this is a well-deﬁned, symmetric, bilinear form with ∈ F 2 the Markov property and that ( , Cb∞) is closable in L (E; m). We denote its closure by ( ,D( )). The form ( ,D( )) isE anF example of a classical Dirichlet form of gradient type (cf.E [ARE 89] [AR 90] [SE 90]),E and because m is Gaussian, the form ( ,D( )) corresponds to an Ornstein-Uhlenbeck process. To get the required counterexampleE weE will show that for 2 some choice of constants (σi )i IN and (γi)i IN , the form ( ,D( )) fails to satisfy (QR1). What this means is that this Ornstein-Uhlenbeck∈ ∈ process cannotE E live on the space E, but must be modelled on a larger space where ( ,D( )) is quasi-regular. E E 35 Our analysis begins with the observation that because of the simple product structure of the measure space (E, (E), m), certain calculations can be reduced to one-dimensional problems. For a ﬁxed indexB j, we let z : E IR denote the map which sends z to its jth j → coordinate. Let Pju = E(u σ(zj)) be the conditional expectation with respect to σ(zj). | 2 The operator Pj is a projection in L (E; m) and we claim that it is also a contraction in 1/2 (D( ), 1 ). For u Cb∞, with representation u(z) = f(z1, . . . , zk), we have the explicit formulaE E ∈ F

(Pju)(z) = f(x1, . . . , xj 1, zj, xj+1, . . . , xk) mi(dxi), (5.12) ··· k 1 − IR − i=j Z Z Y6 2 where mi is the Gaussian measure on IR with mean zero and variance σi . We note that Pju is a function of zj only. This formula (5.12) also holds when f is not quite so smooth, k for example when f = g h where g, h C∞(IR ). Using this explicit formula we ﬁnd ∨ ∈ b that for u C∞, the function P u is again in C∞ and ∈ F b j F b

∂Pju/∂zi = δij Pj(∂u/∂zi). (5.13)

Consequently, ∞ (P u, P u) = γ (∂P u/∂z )2 dm E j j i j i i X=1 Z ∞ 2 = γiδij Pj(∂u/∂zj) dm i=1 Z X ¡ (5.14) 2 = γj Pj(∂u/∂zj) dm Z ¡ γ (∂u/∂z )2 dm ≤ j j Z (u, u). ≤ E Therefore, also 1(Pju, Pju) 1(u, u) on Cb∞ and by continuity this inequality extends to all of D( ). ThisE shows that≤ E the imageF of D( ) under P is the closure of E E j

P C∞ = f(z ) f C∞(IR) . (5.15) jF b { j | ∈ b } Thus P D( ) consists of all functions of the type f(z ) where f belongs to the closure of j E j Cb∞(R) with respect to the one-dimensional Dirichlet form γj u0v0 dmj + uv dmj. In particular, f must be absolutely continuous. Now let A be the open set z E z > 1 and deﬁne R R { ∈ | j| } = u ¬D( ) u 1 m-a.e. on A . (5.16) LA { ∈¬ E | ≥ } 1/2 The element in A with the smallest 1 -norm is written 1A and is called the reduite of the function 1 onL the set A (cf. [MR 92;E Chapter III, Section 1]). We would like to show that 1 P D( ), and to do so, it suﬃces to show that P maps 1 back into . A ∈ j E j A LA 36 First we note that since the map u u is norm-reducing in D( ), the function 1 → | | E A must be non-negative m-a.e. Now let g Cb∞(IR) be a function satisfying I(x 1+¯) ∈ ≥ ≤ g(x) I(x 1). Then the function v = g(zj) belongs to D( ) and v 1A m-a.e. Let un ≤ ≥ 1/2 E ≤ be a sequence in Cb∞ which converges to 1A in 1 -norm. Then the sequence un v 1/2F E ∨ converges in 1 -norm to 1A v = 1A. Using the fact that g(x) I(x 1+¯) and the E ∨ ≥ ≥ formula (5.12), we see that Pj(un v) 1 on z E zj > 1 + ¯ . As n , the sequence P (u v) converges to P ∨1 and≥ so this{ limit∈ also| must| be greater} than→ or ∞ equal j n ∨ j A ¬ to 1 on the set z E zj > 1 + ¯ . As this is true for¬ every ¯ > 0, we conclude that P 1 1 on A,{ in other∈ | words,| P 1 } . j A ≥ ¬ j A ∈ LA Since Pj1A 1/2 1¬A 1/2 and 1A is the unique norm-minimizing element in A, we k kE1 ≤ k kE1 L conclude that Pj1A = 1A. Thus 1A has an m-version which is of the form 1A(z) = f(zj) for some absolutely continuous function f. Now this function f is equal to 1 on the set x x 1 and for any point x ( 1, 1) we have { | | ≥ } ∈ − ¬ ¬ 1 1 f(x) = f(1) f(x) = f 0(y) dy. (5.17) − − Zx By Cauchy-Schwarz we get

1 2 2 (1 f(x)) (f 0(y)) ϕj(y) dy sup 2/ϕj(y) , (5.18) − ≤ 1 y 1 ²Z− ³²| |≤ ³

where ϕj is the density of the Gaussian measure mj on IR with mean zero and variance 2 σj . By the norm minimizing property of 1A we obtain

1 2 γj (f 0(y)) ϕj(y) dy = (1A, 1A) 1(1A, 1A) 1(1, 1) = 1, (5.19) 1 E ≤ E ≤ E Z−

and combined with the formula for ϕj(y) this leads to the bound

2 2 sup 1A(z) 1 √2π (2σj/γj) exp(1/2σj ). (5.20) z E | − | ≤ ∈ Proposition 5.2. If (σ /γ ) exp(1/2σ2) 0 as j , then the Dirichlet form j j j → → ∞ ( ,D( )) is not quasi-regular on L2(E; m). In fact, D( ) = 0 for any compact E E E K { } set K E and so (QR1) must fail to hold. ⊂ Note. Recalling that σ2 < , we see that choosing γ = exp(1/2σ2) will give us j ∞ j j coefficients satisfying the hypothesis of the proposition above. It is typical that γj must go to infinity very quicklyP in order to force ( ,D( )) not to be quasi-regular. E E Proof. For j 1 define Aj = z E zj > 1 and set ON = j∞=N Aj. Now if K is any compact subset≥ of E, then z { 0∈ uniformly| | on}K as j so∪ that K Oc for some N. j → ¬ → ∞ ⊆ N This implies that D( ) D( ) c and so it suffices to prove that the projection P in K ON ¬ 1/2 E ⊆ E (D( ), ) onto the space D( )Oc is the zero projection. Now since Aj ON E for E E1 E N ⊆ ⊆ 37 j N, we have 1 1 1 m-a.e. On the other hand, since (σ /γ ) exp(1/2σ2) 0 ≥ Aj ≤ ON ≤ j j j → we see from (5.20) that 1Aj 1 uniformly on E. Therefore 1ON = 1. Now for any other 1-excessive→ function h D( ) we have ∈ E

h dm = (h, 1) = (h, 1 ) = (h , 1) = h dm. (5.21) E1 E1 ON E1 ON ON Z Z Noting that h h m-a.e. we conclude that h = h in L2(E; m). Thus for every ≥ ON ON 1-excessive function h we have P h = h hON = 0. Therefore P is zero on the linear span of all 1-excessive functions, and since this− linear span is dense, we conclude that P = 0 on D( ) which proves that D( ) = 0 . E E K { } Acknowledgement. Part of this work was completed during a visit of B.S. to the Uni- versity of Bonn. He would like to thank his host, Michael R¨ockner, for hospitality and for providing a stimulating working environment. Both authors gratefully acknowledge the support of the Sonderforschungsbereich 256, Bonn, and the Natural Sciences and Engi- neering Research Council of Canada. In addition, we thank a very thorough referee for useful comments that improved this paper.

38 References

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