Ergodic Decomposition of Dirichlet Formsvia Direct Integrals

Home , Direct integral

arXiv:2003.01366v2 [math.FA] 5 Sep 2021 olwn are following EC rn o 117 wre oPo.D.JnMaas). Jan Dr. Fund Prof. Science to Austrian awarded 716117, the No. by grant position (ERC, Collaborativ current the his by of supported funding Research Bonn. of University edblt ftepaper. the of readability ei seilygaeu oa nnmu eiwrfrtheir for Reviewer anonymous an to grateful especially is he estoso h ujc ftepeetwr,adfrrespec for and work, present the of subject the on versations earglrDrcltfr on form Dirichlet regular a be ih process right ihsaespace state with fteform the If † ewrsadphrases: and Keywords ∗ Introduction. 1. S 00sbetclassiﬁcations: subject 2010 MSC • • • • • ag ato hswr a rte hl h uhrwsamem a was author the while written was work this of part large A h uhri rtflt rfsosSri leei n An and Albeverio Sergio Professors to grateful is author The A A A A A is is is is is M E E M E - -invariant; -invariant -invariant -invariant; quasi-clopen µ ( esnilyeuvln,seRk . below: 3.5 Rmk. see equivalent, -essentially E, X ROI EOPSTO FDRCLTFORMS DIRICHLET OF DECOMPOSITION ERGODIC D life-time , ( I IETITGASADAPPLICATIONS AND INTEGRALS DIRECT VIA Ω hscs,terpeetto sol rjcieyunique. projectively ope only champ is du representation carré the admitting case, and this space, measure spac Radon Dirichlet of Luzin, local strongly integral quasi-regular to each direct underlying for same a holds the as for quasi-regular representation spaces, unique Dirichlet over a space Dirichlet admits quasi-regular space each that show We forms. E A )) ietitga,egdcdcmoiin roi theorem, ergodic decomposition, ergodic integral, direct as f f.31blw if below) 3.1 Dfn. cf. (also eg 2,Dn V61)i hr exists there if IV.6.1]) Dfn. [23, (e.g. c Let esuysproiin n ietitgaso udai a quadratic of integrals direct and superpositions study We E ⊃ sadtoal togylcl hnteprocess the then local, strongly additionally is .. simultaneously i.e., , ( ,g f, n ( ,τ, X, ω ξ n cemetery and , = ) ( ∈ ,τ, X, 73,scnay 12,6J5 60J35 60J25, 31C25, secondary: 37A30, : Ω X E M µ , yLrnoDloSchiavo Dello Lorenzo By ( A X 1 : ) = c A µ , ealclycmatPls ao esr pc,and space, measure Radon Polish compact locally a be ∩ f, ) Ω { si swell-known, is it As . etme ,2021 7, September 1 M , F A s , ∂ S Austria IST g ( E ( ω o a For . + ) M qaioe and -quasi-open : ) 1 t ) E eycrflraigadtersgetoswihipoe t improved which suggestions their and reading careful very A ieypitn u h eeecs[] 1] n 2,3.Fi 3]. [20, and [13], [1], references the out pointing tively s 1 t eerhCne 00 h uhrgaeul acknowledges gratefully author The 1060. Center Research e ≥ ( ∈ f FF rn 6,adb h uoenRsac Council Research European the by and F65, grant (FWF) 1 0 µ A ∈ , [0 maual subset -measurable ( ra bre n oD.KhiSzk,frfutu con- fruitful for Suzuki, Kohei Dr. to and Eberle, dreas c t , P D f, x ] e fteIsiu ü neadeMteai fthe of Mathematik angewandte für Institut the of ber Ω ( 6 } ) 1 E x A = A ∈ ) c c X o any for ∅ F ∈ ( g ∂ E E, ) ξ , qaicoe seeg,[5 .70]). p. [15, e.g., (see -quasi-closed o some for ,g f, , ∗ vrametrizable a over e D with ooy h same The pology. ( E iihe forms. Dirichlet M f probability a )) P dDirichlet nd ∈ saMro iuin n the and diﬀusion, Markov a is irreducible x spoel soitdwt a with associated properly is D A 0 Ω ao.In rator. ∈ ≤ ( A ⊂ E c D ξ < t ) X 0 = ( and E esythat say we , † ) o every for o ; ( x E, ∈ D A nally, ( and E he )) 2 L. DELLO SCHIAVO

We say that a set A X is µ-trivial if it is µ-measurable and either µA =0 or µAc =0. The process M ⊂ is irreducible if every M-invariant set is µ-trivial. When M is not irreducible, it is natural — in the study of the pathwise properties of M — to restrict our attention to “minimal” M-invariant subsets of X. In the local case, thanks to the quasi-topological characterization of M-invariance, such sets may be thought of as the “connected components” of the space X as seen by M. This description is in fact purely heuristic, since it may happen that all such “minimal” M-invariant sets are µ-negligible. The question arises, whether these ideas for the study of M-invariance can be made rigorous by resorting to the Dirichlet form (E, D(E)) associated with M. More precisely, we look for a D D decomposition (Eζ , (Eζ )) ζ∈Z of (E, (E)) over some index set Z, and we require that • (E , D(E )) is a Dirichlet form on (X, τ) additionally irreducible (Dfn. 3.1) for every ζ Z; ζ ζ ∈ D D • we may reconstruct (E, (E)) from (Eζ , (Eζ )) ζ∈Z in a unique way. Because of the ﬁrst property, such a decomposition — if any — would deserve the name of ergodic decomposition of (E, D(E)). For instance, let us consider the standard Dirichlet form Eg on a (second-countable) Riemannian manifold (M,g), i.e. the one generated by the (negative halved) Laplace–Beltrami operator ∆g and properly associated with the Brownian motion on M. In this case, we expect that Z is a discrete space, indexing the connected components of M, and that

g g E = Eζ , ζM∈Z g D g where (Eζ , (Eζ )) is but the standard form of the connected component of index ζ. This simple example suggests that, in the general case of our interest, we should expect that (E, D(E)) is recovered from the D decomposition (Eζ , (Eζ )) ζ∈Z as a “direct integral”, ⊕ E = Eζ . ZZ Our purpose is morefold:

• to introduce a notion of direct integral of Dirichlet forms, and to compare it with the existing notions of superposition of Dirichlet forms [5, §V.3.1] (also cf. [15, §3.1(2◦), p. 113] and [29]), and of direct integral of quadratic forms [3]; • to discuss an Ergodic Decomposition Theorem for quasi-regular Dirichlet forms, a counterpart for Dirichlet forms to the Ergodic Decomposition Theorems for group actions, e.g. [16, 6, 9]; • to provide rigorous justiﬁcation to the assumption — quite standard in the literature about (quasi-) regular Dirichlet forms —, that one may consider irreducible forms with no loss of generality; • to establish tools for the generalization to arbitrary (quasi-regular) Dirichlet spaces of results currently available only in the irreducible case, e.g. the study [22] of invariance under order- isomorphism, cf. [11].

For strongly local Dirichlet forms, our ergodic decomposition result takes the following form.

Theorem. Let (X, τ, ,µ) be a metrizable Luzin Radon measure space, and consider a quasi-regular X strongly local Dirichlet form (E, D(E)) on L2(µ) admitting carré du champ operator. Then, there exist

(i) a σ-ﬁnite measure space (Z, ,ν); Z ERGODIC DECOMPOSITION OF DIRICHLET FORMS 3

(ii) a family of measures (µ ) so that (X, τ, ,µ ) is a (metrizable Luzin) Radon measure space ζ ζ∈Z X ζ for ν-a.e. ζ Z, the map ζ µ A is ν-measurable for every A and ∈ 7→ ζ ∈ X

µA = µζ A dν(ζ) , A , ZZ ∈ X

⊗2 ′ ′ and for ν -a.e. (ζ, ζ ), with ζ = ζ , the measures µ and µ ′ are mutually singular; 6 ζ ζ 2 (iii) a family of quasi-regular strongly local irreducible Dirichlet forms (Eζ , D(Eζ )) on L (µζ ); so that

⊕ ⊕ 2 2 L (µ)= L (µζ ) dν(ζ) and E = Eζ dν(ζ) . ZZ ZZ Additionally, the disintegration is (ν-essentially) projectively unique, and unique if µ is totally ﬁnite.

Plan of the work. Firstly, we shall discuss the notion of direct integral of (non-negative deﬁnite) quadratic forms on abstract Hilbert spaces, §2.3, and specialize it to direct integrals of Dirichlet forms, §2.5, via disintegration of measures. In §3.2 we show existence and uniqueness of the ergodic decomposition for regular and quasi-regular (not necessarily local) Dirichlet forms on probability spaces. The results are subsequently extended to strongly local quasi-regular Dirichlet forms on σ-ﬁnite spaces and admitting carré du champ operator, §3.3. Examples are discussed in §3.4; an application is discussed in 3.5.

Bibliographical note. Our reference of choice for direct integrals of Hilbert spaces is the monograph [12] by J. Dixmier. For some results we shall however need the more general concept of direct integrals of Banach spaces, after [17, 9]. For the sake of simplicity, all such results are conﬁned to the Appendix.

2. Direct Integrals. Every Hilbert space is assumed to be separable and a real vector space.

2.1. Quadratic forms. Let (H, ) be a Hilbert space with scalar product . By a quadratic k·k h·|·i form (Q,D) on H we shall always mean a symmetric positive semi-definite — if not otherwise stated, densely defined — bilinear form. To (Q,D) we associate the non-relabeled quadratic functional Q: H → R + defined by ∪{ ∞} Q(u,u) if u D Q(u) :=  ∈ , u H. + otherwise ∈ ∞  Additionally, for every α> 0, we set

Q (u, v) := Q(u, v)+ α u v , u,v D, α h | i ∈ Q (u) := Q(u)+ α u 2 , u H. α k k ∈ 1/2 For α > 0, we let D(Q)α be the completion of D, endowed with the Hilbert norm Qα . The following result is well-known.

Lemma 2.1. Let (Q,D) be a quadratic form on H. The following are equivalent:

(i) (Q,D) is closable, say, with closure (Q, D(Q)); (ii) the identical inclusion ι: D H extends to a continuous injection ι : D(Q) H satisfy- → α α → ing ι α−1; k αkop ≤ (iii) Q is lower semi-continuous w.r.t. the strong topology of H; 4 L. DELLO SCHIAVO

(iv) Q is lower semi-continuous w.r.t. the weak topology of H.

Proof. (i) (iii) is [18, Lem. VIII.3.14a, p. 461]. (i) (ii) is noted in [23, Rmk. I.3.2.(ii)]. ⇐⇒ ⇐⇒ For (iii) (iv) note that every convex subset of a Hilbert space is weakly closed if and only if it is ⇐⇒ strongly closed and apply this fact to the sublevel sets of Q: H R + . → ∪{ ∞}

To every closed quadratic form (Q, D(Q)) we associate a non-negative self-adjoint operator L, with − domain deﬁned by the equality D(√ L)= D(Q), such that Q(u, v)= Lu v for all u, v D(L). We − h− | i ∈ tL denote the associated strongly continuous contraction semigroup by Tt := e , t > 0, and the associated strongly continuous contraction resolvent by G :=(α L)−1, α> 0. By Hille–Yosida Theorem, e.g. [23, α − p. 27],

(2.1a) Q (G u, v)= u v , u H , v D(Q) , α α h | iH ∈ ∈ tα(αGα−1) (2.1b) Tt =H- lim e . α→∞ (β) (2.1c) Q(u, v) = lim Q (u, v) := lim βu βGβ u v , u,v H. β→∞ β→∞ h − | iH ∈

2.2. Direct integrals. Let (Hζ )ζ∈Z be a family of Hilbert spaces indexed by some index set Z. If Z is at most countable, the direct sum of the Hilbert spaces Hζ is deﬁned as

2 (2.2) Hζ := (hζ ) : hζ Hζ for all ζ Z and hζ <  . ζ∈Z ∈ ∈ k kHζ ∞ ζM∈Z  ζX∈Z    The direct integral of a family of Hilbert spaces is a natural generalization of the concept of direct sum of Hilbert spaces to the case when the indexing set Z is more than countable. In this case, the requirement 2 of ℓ2-summability in the definition of direct sum is replaced by a requirement of L -integrability (see below for the precise definitions), which implies that Z should be taken to be a measure space (Z, ,ν). Z When Z is an at most countable discrete space, and ν is the counting measure, then the direct integral of the Hilbert spaces (Hζ )ζ is isomorphic to their direct sum (2.2). Since their introduction by J. von Neumann in [24, §3], direct integrals have become a main tool in operator theory, and in particular in the classification of von Neumann algebras. In order to make the definition of direct integral precise, let us first introduce some measure-theoretical notions.

Definition 2.2 (Measure spaces). A measurable space (X, ) is X • separable if contains all singletons in X, i.e. x for each x X; X { } ∈ X ∈ • separated if separates points in X, i.e. for every x, y X there exists A with x A X ∈ ∈ X ∈ and y Ac; ∈ • countably separated if there exists a countable family of sets in separating points in X; X • countably generated if there exists a countable family of sets in generating as a σ-algebra; X X • a standard Borel space if there exists a Polish topology τ on X so that coincides with the Borel X σ-algebra induced by τ.

For any subset X of a measurable space (X, ), the trace σ-algebra on X is X := A X : A . 0 X 0 X ∩ 0 { ∩ 0 ∈X} A σ-ﬁnite measure space (X, ,µ) is standard if there exists X , µ-conegligible and so that (X , X 0 ∈ X 0 X ∩ X ) is a standard Borel space. We denote by (X, µ, µˆ) the (Carathéodory) completion of (X, ,µ). A 0 X X ERGODIC DECOMPOSITION OF DIRICHLET FORMS 5

[ , ]-valued function is called µ-measurable if it is measurable w.r.t. µ. For measures µ , µ we −∞ ∞ X 1 2 write µ µ to indicate that µ and µ are equivalent, i.e. mutually absolutely continuous. A σ-ideal 1 ∼ 2 1 2 N of a measure space (X, ,µ) is any sub-σ-algebra of closed w.r.t. , i.e. satisfying A N when- X X ∩ ∩ ∈ N ever A and N . In particular, the family of µ-negligible subsets of (X, ) is always a σ-ideal ∈ X ∈ N Nµ X of (X, µ, µˆ). X For functions f,g : X R we denote by f + :=0 f, resp. f − := (0 f), the positive, resp. → ∪ {±∞} ∨ − ∧ negative part of f, and by f g, resp. f g, the pointwise minimum, resp. maximum, of f and g. ∧ ∨ We now recall the main deﬁnitions concerning direct integrals of separable Hilbert spaces, referring to [12, §§II.1, II.2] for a systematic treatment.

Definition 2.3 (Measurable fields, [12, §II.1.3, Dfn. 1, p. 164]). Let (Z, ,ν) be a σ-finite measure Z space, (Hζ )ζ∈Z be a family of separable Hilbert spaces, and F be the linear space F := ζ∈Z Hζ . We say that ζ H is a ν-measurable field of Hilbert spaces (with underlying space S) if thereQ exists a linear 7→ ζ subspace S of F with

(a) for every u S, the function ζ u is ν-measurable; ∈ 7→ k ζ kζ (b) if v F is such that ζ u v is ν-measurable for every u S, then v S; ∈ 7→ h ζ | ζ iζ ∈ ∈ (c) there exists a sequence (u ) S such that (u ) is a total sequence1 in H for every ζ Z. n n ⊂ n,ζ n ζ ∈ Any such S is called a space of ν-measurable vector ﬁelds. Any sequence in S possessing property (c) is called a fundamental sequence.

Proposition 2.4 ([12, §II.1.4, Prop. 4, p. 167]). Let be a subfamily of F satisfying both Deﬁni- S tion 2.3(a) and (c) with in place of S. Then, there exists exactly one space of ν-measurable vector S ﬁelds S so that S. S ⊂

Definition 2.5 (Direct integrals, [12, §II.1.5, Prop. 5, p. 169]). Let ζ H be a ν-measurable ﬁeld of 7→ ζ Hilbert spaces with underlying space S. A ν-measurable vector ﬁeld u S is called (ν-)square-integrable ∈ if

1/2 2 (2.3) u := uζ ζ dν(ζ) < . k k ZZ k k ∞ Two square-integrable vector ﬁelds u, v are called (ν-)equivalent if u v = 0. The space H of k − k equivalence classes of square-integrable vector ﬁelds, endowed with the non-relabeled quotient norm , k·k is a Hilbert space [12, §II.1.5, Prop. 5(i), p. 169], called the direct integral of ζ H (with underlying 7→ ζ space S) and denoted by

S ⊕ (2.4) H = Hζ dν(ζ) . ZZ The superscript ‘S’ is omitted whenever S is apparent from context.

In the following, it will occasionally be necessary to distinguish an element u of H from one of its representatives modulo ν-equivalence, say uˆ in S. In this case, we shall write u = [ˆu]H . When the speciﬁcation of the variable ζ is necessary, given u H, resp. uˆ S, we shall write ζ u in place of u, ∈ ∈ 7→ ζ resp. ζ uˆ in place of uˆ. In most cases, the distinction between u and uˆ is however immaterial, similarly 7→ ζ 1 A sequence (un)n in a Banach space B is called total if the strong closure of its linear span coincides with B. 6 L. DELLO SCHIAVO to the distinction between the class of a function in L2(ν) and any of its ν-representatives. Therefore in most cases we shall simply write u in place of both u and uˆ.

Lemma 2.6. Let (Z, ,ν) be a σ-ﬁnite countably generated measure space. Then, the space H in (2.4) Z is separable.

Proof. It suﬃces to note that L2(ν) is separable, e.g. [14, 365X(p)]. Then, the proof of [12, §II.1.6, Cor., p. 172] applies verbatim.

Remark 2.7. In general, the space H in (2.4) depends on S, cf. [12, p. 169, after Dfn. 3]. It is nowadays standard to define the direct integral of ζ H as the one with underlying space S generated 7→ ζ (in the sense of Proposition 2.4) by an algebra of ‘simple functions’, see e.g. [17, §6.1, p. 61] or the S Appendix. Here, we prefer the original definition in [12], since we shall make a (possibly) different choice of S, more natural when addressing direct integrals of Dirichlet forms.

Let H be a direct integral Hilbert space defined as in (2.4). We now turn to the discussion of bounded operators in (H) and their representation by measurable fields of bounded operators. B Definition 2.8 (Measurable fields of bounded operators, decomposable operators). A field of bounded operators ζ B (H ) is called ν-measurable (with underlying space S) if ζ B u H is a 7→ ζ ∈ B ζ 7→ ζ ζ ∈ ζ ν-measurable vector field with underlying space S for every ν-measurable vector field u with underlying space S. A ν-measurable vector field of bounded operators is called ν-essentially bounded if ν- esssup B < . In this case, the direct integral operator B : H H of ζ B given by ζ∈Z k ζ kop,ζ ∞ → 7→ ζ (2.5) B : [û] [ζ B uˆ ] H 7−→ 7→ ζ ζ H is well-defined (in the sense that it does not depend on the choice of the representative uˆ S of [û] H), ∈ H ∈ and a bounded operator in (H). Its operator norm B satisfies B = ν- esssup B , [12, B k kop k kop ζ∈Z k ζ kop,ζ §II.2.3, Prop. 2, p. 181]. Conversely, a bounded operator B (H) is called decomposable, [12, §II.2.3, ∈ B Dfn. 2, p. 182], if B is represented by a ν-essentially bounded ν-measurable field of bounded operators ζ 7→ Bζ in the sense of (2.5), in which case we write

⊕ B = Bζ dν(ζ) . ZZ The next statement is readily deduced from e.g. [8, Thm. 2] or [21, Thm. 1.10]. For the reader’s convenience, a short proof is included.

Lemma 2.9. Let H be deﬁned as in (2.4), B (H) be decomposable, and D be the closed disk of ∈B B radius B in the complex plane. Then, for every ϕ (D ) the continuous functional calculus ϕ(B) k kop ∈C B of B is decomposable and

⊕ (2.6) ϕ(B)= ϕ(Bζ ) dν(ζ) . ZZ Proof. Well-posedness follows by [12, §II.2.3, Prop. 2, p. 181]. The proof is then a straightforward application of [12, §II.2.3, Prop. 3, p. 182] and [12, §II.2.3, Prop. 4(ii), p. 183] by approximation of ϕ with suitable polynomials, since σ(B) is compact.

Remark 2.10. Arguing with Tietze Extension Theorem, it is possible to show that the direct-integral representation (2.6) of ϕ(B) only depends on the values of ϕ on the spectrum σ(B) of B. ERGODIC DECOMPOSITION OF DIRICHLET FORMS 7

2.3. Direct integrals of quadratic forms. The main object of our study are direct integrals of quadratic forms. Before turning to the case of Dirichlet forms on concrete Hilbert spaces (L2-spaces), we give the main deﬁnitions in the general case of quadratic forms on abstract Hilbert spaces.

Definition 2.11 (Direct integral of quadratic forms). Let (Z, ,ν) be a σ-finite countably generated Z measure space. For ζ Z let (Q ,D ) be a closable (densely defined) quadratic form on a Hilbert ∈ ζ ζ space H . We say that ζ (Q ,D ) is a ν-measurable field of quadratic forms on Z if ζ 7→ ζ ζ (a) ζ H is a ν-measurable field of Hilbert spaces on Z with underlying space S ; 7→ ζ H (b) ζ D(Q ) is a ν-measurable field of Hilbert spaces on Z with underlying space S ; 7→ ζ 1 Q (c) SQ is a linear subspace of SH under the identification of D(Qζ ) with a subspace of Hζ granted by Lemma 2.1.

We denote by

SQ ⊕ Q = Qζ dν(ζ) ZZ the direct integral of ζ (Q , D(Q )), i.e. the quadratic form deﬁned on H as in (2.4) given by 7→ ζ ζ

D(Q) := [ˆu] :u ˆ SQ, Qζ,1(ˆuζ ) dν(ζ) < , H ∈ Z ∞ (2.7) Z

Q(u, v) := Qζ (uζ , vζ ) dν(ζ) , u,v D(Q) . ZZ ∈

Remark 2.12 (Separability). It is implicit in our definition of ν-measurable field of Hilbert spaces that H is separable for every ζ Z. Therefore, when considering ν-measurable fields of domains as in ζ ∈ D 1/2 Definition 2.11(b), (Qζ )1 is taken to be (Qζ )1 -separable by assumption.

Proposition 2.13. Let (Q, D(Q)) be a direct integral of quadratic forms. Then,

(i) Q, D(Q) is a densely defined closed quadratic form on H; (ii) ζ G , ζ T are ν-measurable fields of bounded operators for every α,t > 0; 7→ ζ,α 7→ ζ,t (iii) Q has resolvent and semigroup respectively defined by

SH ⊕ Gα := Gζ,α dν(ζ) , α> 0 ; Z (2.8) Z SH ⊕ Tt := Tζ,t dν(ζ) , t> 0 . ZZ

Proof. (i) Since ζ H is a ν-measurable family of Hilbert spaces by Deﬁnition 2.11(a), the map ζ 7→ ζ 7→ u is ν-measurable for every u S by Deﬁnition 2.3(a). Analogously, the map ζ Q1/2(u ) k ζ kζ ∈ H 7→ ζ,1 ζ is ν-measurable for every u S . Together with the polarization identity for D(Q) , this yields the ∈ Q 1 measurability of the maps

ζ Q (u , v ) , u,v D(Q) , α> 0 . 7→ ζ,α ζ ζ ∈ As a consequence, ζ D(Q ) is a ν-measurable ﬁeld of Hilbert spaces (on Z, with underlying space S ) 7→ ζ α Q for every α> 0. Thus, it admits a direct integral of Hilbert spaces

SQ ⊕ Dα := D(Qζ )α dν(ζ) , α> 0 . ZZ 8 L. DELLO SCHIAVO

α For α > 0 let (un)n be a fundamental sequence of ν-measurable vector fields for Dα and (un)n be a fundamental sequence of ν-measurable vector fields for H. Since (Qζ ,Dζ ) is closable on Hζ for every ζ Z, the extension of the canonical inclusion ι : D(Q ) H is injective and non-expansive2 ∈ ζ,1 ζ 1 → ζ for every ζ Z by Lemma 2.1. By Definition 2.11, D and H are defined on the same underlying space S. ∈ α Therefore, the maps

ζ ι uα u = uα u , i,j N , α> 0 , 7→ ζ,α i,ζ j,ζ ζ i,ζ j,ζ ζ ∈ are ν-measurable. Together with the uniform boundedness of ι in ζ Z, this yields the decomposability ζ,α ∈ of the operator ι : D H defined by α α → SQ ⊕ ια := ιζ,α dν(ζ) , α> 0 . ZZ By [12, §II.2.3, Example, p. 182] and the injectivity of ι for every ζ Z and every α > 0, the ζ,α ∈ operator ι : D H is injective. In particular, the composition of ι with the inclusion of D(Q) α α → 1 into H is injective, thus Q is closed. Finally, since uα is Q1/2-total in D(Q ) for every ζ Z n,ζ n ζ,α ζ α ∈ by Definition 2.3(c), it is additionally H -total for every ζ Z by H -density of D(Q ) in H . As a ζ ∈ ζ ζ ζ α D consequence, un n is fundamental also for H, thus (Q) is H-dense in H. (ii) For fixed α> 0 consider the field of linear operators ζ G . The map (cf. (2.1a)) 7→ ζ,α ζ Q (G uα ,uα )= uα uα 7→ ζ,α ζ,α i,ζ j,ζ i,ζ j,ζ ζ is ν-measurable for every i, j N since uα is a ν-measurable vector field. Since G α−1 and (uα) ∈ n k α,ζ kop ≤ n n is a fundamental sequence of ν-measurable vector fields for H, then ζ G is a ν-measurable field of 7→ ζ,α bounded operators by [12, §II.2.1, Prop. 1, p. 179] and the operator Gα defined in (2.8) is decomposable for every α> 0.

By Lemma 2.9 any image of Gα via its continuous functional calculus is itself decomposable. For every ζ Z one has T = lim etβ(βGζ,β−1) strongly in H by (2.1b), hence ∈ ζ,t β→∞ ζ

α α tβ(βGζ,β−1) α α ζ Tζ,tui,ζ uj,ζ = lim e ui,ζ uj,ζ 7→ ζ β→∞ ζ D E is a pointwise limit of ν-measurable functions, thus it is ν-measurable, for every i, j N and every t> 0. ∈ As a consequence, ζ T is a ν-measurable ﬁeld of bounded operators for every t > 0, again by [12, 7→ ζ,t §II.2.1, Prop. 1, p. 179].

(iii) It suffices to show (2.1) for (Q, D(Q)), Gα and Tt defined in (2.8). Now, by definition of (Q, D(Q)) one has for every α> 0

Qα(Gαu, v)= Qζ (Gαu)ζ , vζ dν(ζ)+ α (Gαu)ζ vζ dν(ζ) Z Z h | iζ Z Z

= Qζ,α (Gαu)ζ , vζ dν(ζ) Z Z

= Qζ,α Gα,ζ uζ , vζ dν(ζ) . Z Z By [12, §II.2.3, Cor., p. 182] and decomposability of G , one has G = G for ν-a.e. ζ Z, whence, α α,ζ ζ,α ∈ by (2.1a) applied to (Qζ , D(Qζ )) and Gζ,α,

Qα(Gαu, v)= uζ vζ ζ dν(ζ)= u v , ZZ h | i h | i 2 We say that a linear map ι: H1 → H2 between Hilbert spaces (H1, k · k1) and (H2, k · k2) is non-expansive if kιhk2 ≤ khk1 for every h ∈ H1. ERGODIC DECOMPOSITION OF DIRICHLET FORMS 9 which concludes the proof of (2.1a) for Gα.

Let us show (2.1b) for Tt. Deﬁne the operators

⊕ (β) tβ(βGζ,β−1) Tt := e dν(ζ) , β,t> 0 , ZZ and note that sup T (β) < for every t > 0. By (2.1b) applied to T for every ζ Z and β>1 t op ∞ ζ,t ∈ tβ(βGζ,β−1) every t > 0 we have that T ζ,t = limβ→∞ e strongly in Hζ . On the one hand, we may now apply [12, §II.2.3, Prop. 4(ii), p. 183] to conclude that

⊕ ⊕ (β) tβ(βGζ,β−1) (2.9) H- lim Tt = Hζ - lim e dν(ζ)= Tζ,t dν(ζ)=: Tt . β→∞ ZZ β→∞ ZZ strongly in H. On the other hand, by Lemma 2.9 we have that

⊕ (β) tβ(βGζ,β−1) tβ(βGβ−1) (2.10) Tt := e dν(ζ)= e , β,t> 0 . ZZ Taking the strong H-limit as β in (2.10) yields the assertion by comparison with (2.9). → ∞

Remark 2.14 (cf. [12, §II.1.3, Rmk. 3 p. 166 and §II.1.4, Rmk. p. 168]). Each of the above statements holds with identical proof if one substitutes ‘ν-measurable’ with ‘measurable’.

Remark 2.15. Under the assumptions of Proposition 2.13, assertion (i) of the same Proposition implies that the space of ν-measurable vector ﬁelds SH is uniquely determined by SQ as a consequence of Proposition 2.4. Thus, everywhere in the following when referring to a direct integral of quadratic forms we shall — with abuse of notation — write S in place of both SH and SQ.

The next proposition completes the picture, by providing a direct-integral representation for the generator of the form (Q, D(Q)) in (2.7). Since we shall not need this result in the following, we omit an account of direct integrals of unbounded operators, referring the reader to [21, §1]. Once the necessary deﬁnitions are established, a proof is straightforward.

Proposition 2.16. Let (Q, D(Q)) be deﬁned as in (2.7). Then, its generator (L, D(L)) has the direct-integral representation

⊕ (2.11) L = Lζ dν(ζ) . ZZ

Remark 2.17 (Comparison with [3]). As for quadratic forms, (2.11) is understood as a direct-integral representation of the Hilbert space D(L), endowed with the graph norm, by the measurable ﬁeld of Hilbert spaces ζ D(L ), each endowed with the relative graph norm. The set-wise identiﬁcation of D(L) as a 7→ ζ linear subspace of H as in (2.4) is already shown in [3, Prop. 1.6].

2.4. Dirichlet forms. We recall a standard setting for the theory of Dirichlet forms, following [23].

Assumption 2.18. The quadruple (X, τ, ,µ) is so that X (a) (X, τ) is a metrizable Luzin space with Borel σ-algebra ; X (b)µ ˆ is a Radon measure on (X, τ, µ) with full support. X 10 L. DELLO SCHIAVO

By [14, 415D(iii), 424G] any space (X, ,µ) satisfying Assumption 2.18 is, in particular, σ-finite X standard. The support ofa (µ-)measurable function f : X R (possibly defined only on a µ-conegligible → set) is defined as the measure-theoretical support supp[ f µ]. Every such f has support, independent of | | · the µ-representative of f, cf. [23, p. 148]. A closed positive semi-definite quadratic form (Q, D(Q)) on L2(µ) is a (symmetric) Dirichlet form if

(2.12) f D(Q) = f + 1 D(Q) and Q(f + 1) Q(f) . ∈ ⇒ ∧ ∈ ∧ ≤ We shall denote Dirichlet forms by (E, D(E)). A Dirichlet form (E, D(E)) is

• local if E(f,g)=0 for every f,g D(E) with supp[f] supp[g]= ∅; ∈ ∩ • strongly local if E(f,g)=0 for every f,g D(E) with g constant on a neighborhood of supp[f]; ∈ • regular if (X, τ) is (additionally) locally compact and there exists a core for (E, D(E)), i.e. a C subset D(E) (τ) both E1/2-dense in D(E) and uniformly dense in (τ). C ⊂ ∩C0 1 C0 On spaces that are not necessarily locally compact, the interplay between a Dirichlet form (E, D(E)) and the topology τ on X is speciﬁed by the following deﬁnitions. For an increasing sequence (Fk)k of Borel subsets F X set k ⊂ D(E, (F ) ) := f D(E): f 0 µ-a.e. on F c for some k N . k k { ∈ ≡ k ∈ } D D The sequence (Fk)k is called an E-nest if each Fk is closed and (E, (Fk)k) is dense in (E)1. A set N X is E-exceptional if N F c for some E-nest (F ) . A property of points in X holds E-quasi- ⊂ ⊂ ∩k k k k everywhere if it holds for every point in N c for some E-exceptional set N. A function f : X R is → E-quasi-continuous if there exists an E-nest (Fk)k so that the restriction of f to Fk is continuous for every k N. Finally, a form (E, D(E)) is ∈ 1/2 • quasi-regular on (X, τ) if there exist: (qr1) an E-nest (Fk)k of compact sets; (qr2) an E1 -dense

subset of D(E)1 the elements of which all have an E-quasi-continuous µ-version; (qr3) an E-exceptional set N X and a countable family (f ) of E-quasi-continuous functions f D(E) ⊂ n n n ∈ separating points in N c.

We refer to [7] or [15, §A.4] for the notion of quasi-homeomorphism of Dirichlet forms. We say that (E, D(E)) has carré du champ operator (Γ, D(E)), if Γ: D(E)⊗2 L1(µ) is a non-negative → deﬁnite symmetric continuous bilinear operator so that

(2.13) E(f,gh)+ E(fh,g) E(fg,h)=2 h Γ(f,g) dµ, f,g,h D(E) L∞(µ) . − ZX ∈ ∩ Finally, let D(E) be the linear space of all functions u L0(µ) so that there exists an E1/2-Cauchy e ∈ sequence (u ) D(E) with lim u = u µ-a.e. We denote by (D(E) , E) the space D(E) endowed n n ⊂ n n e e with the extension of E to D(E)e called the extended Dirichlet space of (E, D(E)). For proofs of well- posedness in this generality, see [19, p. 693]. If (E, D(E)) has semigroup T : L2(µ) L2(µ), we denote • → as well by T : L∞(µ) L∞(µ) the extension of the semigroup to L∞(µ). We say that (E, D(E)) is • → • conservative if T 1 = 1 µ-a.e. for all t 0; t ≥ • transient if D(E)e is a Hilbert space with inner product E; • recurrent if 1 D(E) and E(1)=0. ∈ e These deﬁnitions are equivalent to the standard ones (e.g. [15, p. 55]) by [15, Thm.s 1.6.2, 1.6.3, p. 58], a proof of which may be adapted to the case of spaces satisfying Assumption 2.18. ERGODIC DECOMPOSITION OF DIRICHLET FORMS 11

2.5. Direct-integral representation of L2-spaces. In order to introduce direct-integral representations of Dirichlet forms, we need to construct direct integrals of concrete Hilbert spaces in such a way to additionally preserve the Riesz structure of Lebesgue spaces implicitly used to phrase the sub-Markovianity property (2.12). To this end, we shall need the concept of a disintegration of measures.

Disintegrations. Let (X, ,µ) and (Z, ,ν) be (non-trivial) measure spaces. A map s: (X, ) (Z, ) X Z X → Z is inverse-measure-preserving if s µ := µ s−1 = ν. Hereafter, we ﬁx an inverse-measure-preserving ♯ ◦ map s: (X, ) (Z, ). X → Z

Definition 2.19 (Disintegrations, [14, 452E]). A pseudo-disintegration of µ over ν is any family of non-zero measures (µ ) on (X, ) so that ζ µ A is ν-measurable and ζ ζ∈Z X 7→ ζ

µA = µζ A dν(ζ) , A . ZZ ∈ X A pseudo-disintegration is

• separated if there exists a family of pairwise disjoint sets (A ) µ so that A is µ -conegligible ζ ζ∈Z ⊂ X ζ ζ for ν-a.e. ζ Z, henceforth called a separating family for (µ ) . ∈ ζ ζ∈Z

A disintegration of µ over ν is a pseudo-disintegration additionally so that µζ is a sub-probability measure for every ζ Z. A disintegration is ∈ • ν-essentially unique if the measures µ are uniquely determined for ν-a.e. ζ Z; ζ ∈ • consistent with s if

−1 (2.14) µ A s (B) = µζ A dν(ζ) , A , B ; ∩ Z ∈ X ∈ Z B • strongly consistent with s if s−1(ζ) is µ -conegligible for ν-a.e. ζ Z. ζ ∈

If (µζ )ζ∈Z is a pseudo-disintegration of µ over ν, then

(2.15) g dµ = g(x) dµζ (x) dν(ζ) ZX ZZ ZX whenever the left-hand side makes sense, [14, 452F]. We note that a disintegration (µζ )ζ∈Z of µ over ν −1 strongly consistent with a map s is automatically separated, with separating family s (ζ) ζ∈Z . Direct integrals and disintegrations. Let (X, ,µ) be σ-ﬁnite standard, (Z, ,ν) be σ-ﬁnite countably X Z generated, and (µζ )ζ∈Z be a pseudo-disintegration of µ over ν. Denote by • 0(µ) the space of µ-measurable real-valued functions (not: µ-classes) on X; L • ∞(µ) the space of uniformly bounded (not: µ-essentially uniformly bounded) functions in 0(µ); L L • p(µ) the space of p-integrable functions in 0(µ). L L For a family 0(µ), let [ ] denote the family of the corresponding µ-classes. A ⊂ L A µ Let now F := L2(µ ). We denote by δ : 2(µ) F the diagonal embedding of 2(µ) into F , ζ∈Z ζ L → L regarded up to µQ-classes, viz. δ : f (ζ δ(f) ), where ζ 7→ 7→ ζ [f] if f 2(µ ) µζ ζ (2.16) δ(f)ζ :=  ∈ L . 0 2  L (µζ ) otherwise  12 L. DELLO SCHIAVO

In general, it does not hold that f 2(µ ) for every ζ Z, thus we need to adjust the obvious definition ∈ L ζ ∈ of δ(f) as above in such a way that δ(f) F , that is δ(f) 2(µ ) for every ζ Z. Note that δ is thus ∈ ζ ∈ L ζ ∈ 2 not linear. However, since f (µ), then δ(f)ζ = [f] for ν-a.e. ζ Z by (2.15). It will be shown in ∈ L µζ ∈ Proposition 2.25 below that δ is well-defined as linear morphism mapping µ-classes to H-classes. Further let be satisfying A (2.17) is a linear subspace of 2(µ), and [ ] is dense in L2(µ). A L A µ Since [ ] is dense in L2(µ) and the latter is separable, then there exists a countable family so A µ U ⊂ A 2 that [ ] is total in L (µζ ) for ν-a.e. ζ Z. Thus for every as in (2.17) there exists a unique space U µζ ∈ A of ν-measurable vector fields S = S containing δ( ), generated by δ( ) in the sense of Proposition 2.4. A A A We denote by H the corresponding direct integral of Hilbert spaces

S ⊕ 2 (2.18) H := L (µζ ) dν(ζ) . ZZ Since S is unique, it is in fact independent of . Indeed, let , be satisfying (2.17) and note A A0 A1 that := satisﬁes (2.17) as well. Thus, δ( ), δ( ) S , and so S = S = S by A A0 ⊕ A1 A0 A1 ⊂ A A A0 A1 uniqueness.

Remark 2.20 (cf. [17, §7.2, p. 84]). The direct integral H constructed in (2.18) is a Banach lattice (e.g. [14, 354A(b)]) for the order

h 0 h 0 2 for ν-a.e. ζ Z. ≥ H ⇐⇒ ζ ≥ L (µζ ) ∈ In particular, for every g,h H, the fields h+, h−, g h, and g h, respectively defined by ∈ ∧ ∨ h± : ζ h± , g h: ζ (g h ) , g h: ζ (g h ) , 7→ ζ ∧ 7→ ζ ∧ ζ ∨ 7→ ζ ∨ ζ are ν-measurable fields representing elements of H. In the following, we shall occasionally write — here, 1H is merely a shorthand —

0 h 1 H ≤ ≤ H to indicate that

0 h 1 µ -a.e. for ν-a.e. ζ Z. ≤ ζ ≤ ζ ∈

Remark 2.21. For arbitrary measurable spaces, the standard choice for is the algebra of µ- A integrable simple functions. If (X, τ, ,µ) were a locally compact Polish Radon measure space, one X might take for instance = (τ), the algebra of continuous compactly supported functions. In fact, for A Cc the purposes of the present section, we might as well choose = 2(µ), as the largest possible choice, A L or a countable Q-vector subspace of 1(µ) ∞(µ), as a smallest possible one. When dealing with A L ∩ L direct integrals of regular Dirichlet forms however, the natural choice for is that of a special standard A core for the resulting direct-integral form. C

Remark 2.22 (Comparison with [3]). We note that for every as in (2.17), [ ] is a determining A A µ class in the sense of [3, p. 402]. Conversely, every determining class L0 is contained in a minimal linear space of functions [ ] satisfying (2.17). A µ ERGODIC DECOMPOSITION OF DIRICHLET FORMS 13

Remark 2.23 (Caveat). Whereas the space H does not depend on , in general it does depend A on SA, cf. Rmk. 2.7. Furthermore, H depends on the chosen pseudo-disintegration too, and thus H need not be isomorphic to L2(µ), as shown in the next example.

Example 2.24. Let denote the one-point space, set µ :=2δ , and note that L2(µ) = R. On the {∗} ∗ ∼ other hand, if Z :=( 0, 1 ,ν) is the two-point space with uniform measure ν, and µ := δ for ζ Z, { } ζ ∗ ∈ then (µ ) is a (pseudo-)disintegration of µ, yet H = L2(µ ) L2(µ ) = R2. ζ ζ∈Z ∼ 0 ⊕ 1 ∼ Proposition 2.25. Let (X, ,µ) be σ-ﬁnite standard, (Z, ,ν) be σ-ﬁnite countably generated, X Z and (µζ )ζ∈Z be a pseudo-disintegration of µ over ν. Then, the morphism

S ⊕ 2 2 ι: L (µ) H := L (µζ ) dν(ζ) (2.19) −→ ZZ [f] [δ(f)] µ 7−→ H

(i) is well-deﬁned, linear, and an isometry of Hilbert spaces, additionally unitary if (µζ )ζ∈Z is separated; (ii) is a Riesz homomorphism (e.g. [14, 351H]). In particular,

2 • for each f (µ), it holds that (ι [f] ) 0 2 for ν-a.e. ζ Z if and only if f 0 µ-a.e.; ∈ L µ ζ ≥ L (µζ ) ∈ ≥ • for each f,g 2(µ), it holds that (ι [f g] ) = (ι [f] ) (ι [g] ) for ν-a.e. ζ Z. ∈ L ∧ µ ζ µ ζ ∧ µ ζ ∈ Proof. As usual, we denote by the norm on H, and by , resp. , the norm on L2(µ), k·k k·k2 k·k2,ζ resp. L2(µ ). Let be satisfying (2.17), and define a map ˆι: H by ˆι: f [δ(f)] . ζ A A → 7→ H By definition of ˆι and δ, by definition (2.3) of , and by the property (2.15) of the disintegration, k·k 2 2 ˆιf1 ˆιf2 = f1 f2 2,ζ dν(ζ) k − k ZZ k − k 2 f ,f . = f1 f2 dµζ dν(ζ) 1 2 ∈ A ZZ ZX | − | = f f 2 k 1 − 2k2 As a consequence, ˆι: H descends to a linear isometry ι:[ ] H, and the latter extends to the A → A µ → non-relabeled (linear) isometry (2.19) by density of [ ] in L2(µ). A µ Assume now that (µ ) is separated with separating family (A ) , and fix h (im ι)⊥. Let hˆ S ζ ζ∈Z ζ ζ∈Z ∈ ∈ be an H-representative of h. For each ζ Z, let h˜ 2(µ ) be a representative of hˆ , and define a ∈ ζ ∈ L ζ ζ function h˜ : X R by →

h˜ζ (x) if x Aζ , ζ Z, h˜(x) :=  ∈ ∈ . 0 otherwise  This deﬁnition is well-posed since the sets Aζ ’s are pairwise disjoint.

Claim: h˜ 0 µ-a.e. With slight abuse of notation, set δ(h˜) :=[h˜] for ζ Z, and δ(h˜) :=(ζ δ(h˜) ). ≡ ζ µζ ∈ 7→ ζ By construction, δ(h˜)= hˆ, therefore δ(h˜) S, and so ∈ 2 0= h ι [f] = hf˜ dµζ dν(ζ) , f (µ) , µ Z Z ∈ L Z X where the right-hand side is well-deﬁned since hˆ S. As a consequence, ∈

f hf˜ dµζ dν(ζ) 7→ ZZ ZX 14 L. DELLO SCHIAVO is the 0-functional on L2(µ). By the Riesz Representation Theorem for L2(µ), and by arbitrariness of [f] L2(µ), we thus have h˜ 0 µ-a.e. µ ∈ ≡ As a consequence of the claim, h = [δ(h˜)] = ˆι(h˜) = 0 . By arbitrariness of h (im ι)⊥, we may H H ∈ conclude that (im ι)⊥ = 0 , i.e. that ι is surjective. { H }

We show the first assertion in (ii). A proof of the second assertion is similar, and therefore it is omitted. 2 − 0 2 Argue by contradiction that there exists f (µ) with f 0 µ-a.e., yet such that (ι [f]µ)ζ = L (µζ ) ∈ L ≥ − 6 for all ζ in some B with νB > 0. In particular, since ζ (ι [f] ) is a ν-measurable field by ∈ Z 7→ µ ζ Remark 2.20, the following integral is well-defined and strictly positive

− 2 (ι [f] )ζ dν(ζ) > 0 . µ 2 ZB L (µζ )

Then, by (2.15),

2 2 + 2 + 2 + f dµ = (f ) dµ = (f ) dµζ dν(ζ)= f dν(ζ) µζ 2 Z Z Z Z Z L (µζ ) X X Z X Z 2 2 + − < f dν(ζ)+ (ι [f] )ζ dν(ζ) . µζ 2 µ 2 ZZ L (µζ ) ZB L (µζ )

Since [f +] = [f]+ for every ζ Z, and by deﬁnition of ι, continuing from the previous inequality, we µζ µζ ∈ have that

2 2 2 + − f dµ< f dν(ζ)+ (ι [f] )ζ dν(ζ) µζ 2 µ 2 ZX ZZ L (µζ ) ZB L (µζ ) + 2 − 2 = [f] dν(ζ)+ [f] dν(ζ) µζ 2 µζ 2 ZZ L (µζ ) ZB L (µζ ) + 2 − 2 [f] + [f] dν(ζ) µζ 2 µζ 2 ≤ ZZ L (µζ ) L (µζ ) 2 = [f]µ 2 dν(ζ) Z ζ L (µζ ) Z 2 = ι [f]µ H

by deﬁnition of H. The inequality contradicts the fact, shown in (i), that ι: L2(µ) H is an isometry, → 2 2 and therefore [f]µ L2(µ) = ι [f]µ H .

2.6. Direct integrals of Dirichlet forms. Let (X, ,µ) be σ-finite standard, (Z, ,ν) be σ-finite count- X Z ably generated, and (µ ) be a pseudo-disintegration of µ over ν. Further let ζ (Q , D(Q )) be a ζ ζ∈Z 7→ ζ ζ 2 ν-measurable field of quadratic forms, each densely defined in L (µζ ) with separable domain, and denote by (Q, D(Q)) their direct integral in the sense of Definition 2.11.

Definition . D 2.26 We say that (Q, (Q)) is compatible with the pseudo-disintegration (µζ )ζ∈Z if the space S underlying ζ D(Q ) is of the form S for some as in (2.17) and additionally Q 7→ ζ 1 A A satisfying D(Q). A⊂

Note that, if S is of the form S for D(Q) and satisfying (2.17), then S is of the form S as Q A A⊂ H A well by Remark 2.15. ERGODIC DECOMPOSITION OF DIRICHLET FORMS 15

Definition 2.27 (Diagonal restriction). Let (Q, D(Q)) be a direct integral of quadratic forms compatible with a pseudo-disintegration (µζ )ζ∈Z . The form

S ⊕ 2 Qres = Q := Qζ dν(ζ) , D(Qres) := D(Q) ι(L (µ)) ZZ ∩ is a closed (densely deﬁned) quadratic form on ι(L2(µ)), called the diagonal restriction of (Q, D(Q)).

Remark 2.28 (Comparison with [3]). We note that the form (Qres, D(Qres)) coincides with the form (E ,D(E )) deﬁned in [3, Thm. 1.2]. As a consequence, at least in this case, the closability of E in [3, Thm. 1.2] follows from our Proposition 2.13.

Our ﬁrst result on direct integrals of concrete quadratic forms is as follows.

Proposition 2.29. Let (X, ,µ) be σ-finite standard, (Z, ,ν) be σ-finite countably generated, X Z D and (µζ )ζ∈Z be a separated pseudo-disintegration of µ over ν. Further let (E, (E)) be a direct integral of closed quadratic forms ζ (E , D(E )) compatible with (µ ) . Then, (E, D(E)) is a Dirichlet 7→ ζ ζ ζ ζ∈Z form on L2(µ) if and only if (E , D(E )) is so on L2(µ ) for ν-a.e. ζ Z. ζ ζ ζ ∈ Remark . 2.30 Since (µζ )ζ∈Z is separated, the isometry ι in Proposition 2.25 is a unitary operator, thus there exists ι−1 : H L2(µ), where H is as in (2.19). For the sake of clarity, only in the proof of → Proposition 2.29 below, we distinguish between the quadratic form (E, D(E)) on H and the quadratic form (ι∗E, D(ι∗E)) on L2(µ) defined by

ι∗E(f,g) := E(ιf, ιg) , f,g D(ι∗E) := ι−1(D(E)) . ∈ By definition of ι∗E and since ι: L2(µ) H is unitary, we have that (ι∗E) = ι∗E . Therefore, → 1 1 (ι∗E, D(ι∗E)) is closed on L2(µ), since (E, D(E)) is closed on H by Proposition 2.13(i). Furthermore, ∗ the Hilbert spaces D(E)1 and D(ι E)1 are intertwined via the unitary isomorphism ι. In the statement of Proposition 2.29 above and everywhere after its proof — with a slight abuse of notation — these two quadratic forms are identified. Again for the sake of clarity, the statement of the proposition equivalently ∗ ∗ 2 reads as follows: (ι E, D(ι E)) is a Dirichlet form on L (µ) if and only if (Eζ , D(Eζ )) is a Dirichlet form on L2(µ ) for ν-a.e. ζ Z. ζ ∈ Proof of Proposition 2.29. By e.g. [15, Thm. 1.4.1], the closed quadratic form (ι∗E, D(ι∗E)) 2 2 on L (µ), resp. (Eζ , D(Eζ )) on L (µζ ), is a Dirichlet form if and only if the associated semigroup T ι : L2(µ) L2(µ), resp. T : L2(µ ) L2(µ ), is sub-Markovian, viz. • → ζ,ι ζ → ζ (2.20a) 0 T ιu 1 µ-a.e. , u L2(µ):0 u 1 µ-a.e. , t> 0 , ≤ t ≤ ∈ ≤ ≤ (2.20b) resp. 0 T v 1 µ -a.e. , v L2(µ ):0 v 1 µ -a.e. , t> 0 . ≤ ζ,t ζ ≤ ζ ζ ∈ ζ ≤ ζ ≤ ζ Thus, it suffices to show that T ι is sub-Markovian if and only if T is so for ν-a.e. ζ Z. • ζ,• ∈ Since (E, D(E)) and (ι∗E, D(ι∗E)) are intertwined by the unitary isomorphism ι, it is not difficult to ι show that their semigroups T• and T• are intertwined as well, viz.

T ι = ι∗T := T ι, t> 0 . t t t ◦ Furthermore, since ι: L2(µ) H is a Riesz homomorphism by Proposition 2.25(ii), → (2.21) T ι is sub-Markovian 0 T h 1 , h H : 0 h 1 , • ⇐⇒ H ≤ t ≤ H ∈ H ≤ ≤ H 16 L. DELLO SCHIAVO where, by deﬁnition of the Banach lattice structure on H in Remark 2.20,

0 (Tth)ζ 1 µζ -a.e. , t> 0 (2.22) 0H Tth 1H ≤ ≤ ≤ ≤ ⇐⇒ for ν-a.e. ζ Z, h H : 0 h 1 . ∈ ∈ H ≤ ≤ H By Proposition 2.13(iii) we have that (T h) = T h for ν-a.e. ζ Z for every h H. Thus, combin- t ζ ζ,t ζ ∈ ∈ ing (2.21) and (2.22) we may conclude that

0 T h 1 µ -a.e. , t> 0 ι ζ,t ζ ζ (2.23) T• is sub-Markovian ≤ ≤ ⇐⇒ for ν-a.e. ζ Z, h H : 0 h 1 . ∈ ∈ H ≤ ≤ H

The reverse implication in (2.23) together with (2.20b) immediately show that, if Tζ,• is sub-Markovian for ν-a.e. ζ Z, then T ι is sub-Markovian. The converse implication is not immediate, since the right-hand ∈ • side of (2.23), to be compared with (2.20b), contains the additional consistency constraint that ζ h 7→ ζ be a measurable ﬁeld representing an element h H. ∈ In order to show that, if T ι is sub-Markovian, then T is sub-Markovian for ν-a.e. ζ Z is sub- • ζ,• ∈ Markovian, we argue as follows. Since T : L2(µ ) L2(µ ) is bounded, and since the unit contraction ζ,t ζ → ζ operator v 0 v 1 operates continuously on L2(µ ), it suﬃces to show that, for any sequence vζ 7→ ∨ ∧ ζ n n ⊂ 2 2 L (µζ ) total in L (µζ ),

0 T (0 vζ 1) 1 , n N . ≤ ζ,t ∨ n ∧ ≤ ∈ Let (u ) H be a fundamental sequence of ν-measurable vector ﬁelds for H, and recall that (u ) n n ⊂ n,ζ n is total in L2(µ ) for every ζ Z by Deﬁnition 2.3(c). Applying (2.23) to each element u of this sequence ζ ∈ n proves the assertion.

Proposition 2.29 motivates the next deﬁnition. A simple example follows.

Definition 2.31. A quadratic form (E, D(E)) on L2(µ) is a direct integral of Dirichlet forms ζ 7→ (E , D(E )) on L2(µ ) if it is a direct integral of the Dirichlet forms ζ (E , D(E )) additionally ζ ζ ζ 7→ ζ ζ compatible with the separated pseudo-disintegration (µζ )ζ in the sense of Deﬁnition 2.26.

Example 2.32. Let X = R2 with standard topology, Borel σ-algebra, and the 2-dimensional Lebesgue measure Leb2. Consider a Dirichlet form measuring energy only in the ﬁrst coordinate, viz.

2 2 E(f) := ∂1f(x1, x2) dLeb (x1, x2) ZR2 | | with f L2(R2) and f( , x ) W 1,2(R) for Leb1-a.e. x R. Then, (E, D(E)) is the direct integral x ∈ · 2 ∈ 2 ∈ 2 7→ E , W 1,2(R) , where x ranges in Z = R the real line, (X, ,µ ) is again the standard real line for x2 2 X x2 every x R, and 2 ∈ 2 1 1 Ex2 (f) := df( , x2) dLeb , for Leb -a.e. x2 R . ZR | · | ∈

2.7. Superposition of Dirichlet forms. We recall here the definition of a superposition of Dirichlet forms in the sense of [5, §V.3.1]. Let (Z, ,ν) be σ-finite, (X, ,µ ) be σ-finite and (E , D(E )) be a Z X ζ ζ ζ Dirichlet form on L2(µ ), for every ζ Z. Assume that ζ ∈ (sp ) ζ µ A is ν-measurable for every A ; 1 7→ ζ ∈ X ERGODIC DECOMPOSITION OF DIRICHLET FORMS 17

(sp ) ζ E (fˆ) [0, ] is ν-measurable for every measurable fˆ: X [ , ]. 2 7→ ζ ∈ ∞ → −∞ ∞ Let us now consider

• a measure λ ν on (Z, ) so that µ := µ dλ(ζ) is σ-ﬁnite on (X, ) (therefore: (X, ,µ) is ≪ Z Z ζ X X standard); R • the subspace D of all functions f L2(µ) so that ∈

(2.24) (f) := Eζ (f) dν(ζ) < , E ZZ ∞ and let us further assume that

2 (sp3) D is dense in L (µ).

Then, it is claimed in [5, p. 214] that (2.24) is well-deﬁned and depends only on the µ-class of f, and it is shown in [5, Prop. V.3.1.1] that

Definition 2.33. ( , D) is a Dirichlet form on L2(µ), called the superposition of ζ E . E 7→ ζ Note that we may always choose λ = ν provided that the integral measure µ defined above is σ-finite. If this is not the case, we may recast the definition by letting ν := λ. In this way, we may always assume with no loss of generality that µ is given, and that (µζ ) is a pseudo-disintegration of µ over ν.

Remark 2.34. In fact, [5] requires all functions in (sp )-(sp ) to be -measurable, rather than only 1 2 Z ν-measurable. Here, we relax this condition to ‘ν-measurability’ in order to simplify the proof of the reverse implication in the next Proposition 2.35. Our deﬁnition of ‘superposition’ is equivalent to the one in [5].

Proposition 2.35. Let (X, ,µ) be σ-finite standard, (Z, ,ν) be σ-finite countably generated, X Z D 2 (µζ )ζ∈Z be a separated pseudo-disintegration of µ over ν, and (Eζ , (Eζ )) be a Dirichlet form on L (µζ ) for every ζ Z. Then, the following are equivalent: ∈ (i) there exists the superposition ( , D) of ζ E and the space D in (2.24) is 1/2-separable; E 7→ ζ E1 (ii) there exists a direct integral of Dirichlet forms (E, D(E)) of the forms ζ E . 7→ ζ Furthermore, if either one holds, then (E, D(E)) and ( , D) are isomorphic Dirichlet spaces. E Proof. We only show that (i) implies (ii). A proof of the reverse implication is similar, and it is therefore omitted. For simplicity, set throughout the proof H := L2(µ ), with norm , for every ζ Z. ζ ζ k·kζ ∈ Assume (i). It follows from (sp ) that ζ f is ν-measurable for every f D, and thus from (sp ) 1 7→ k kζ ∈ 2 that ζ E (f,g) is ν-measurable for every f,g D by polarization. By 1/2-separability of D, there 7→ ζ,1 ∈ E1 exists a countable Q-linear space 2(µ) so that [ ] is 1/2-dense in D, and dense in L2(µ) by (sp ). U ⊂ L U µ E1 3 2 Since (µζ ) is separated by assumption, it follows by Proposition 2.25 that [ ] is dense in L (µζ ) for ζ∈Z U µζ ν-a.e. ζ Z. As a consequence, the quadratic form (Eζ , [ ] ) is densely defined on Hζ . Since (Eζ , D(Eζ )) ∈ U µζ res res 2 is closed, the closure (E , D(E )) of (Eζ , [ ] ) is well-defined and a Dirichlet form on Hζ = L (µζ ). ζ ζ U µζ D Again since (µζ )ζ∈Z is separated, we may then construct a form (E, (E)) as the direct integral of Dirichlet forms (Dfn. 2.31) of the forms ζ (Eres, D(Eres)) with underlying space of measurable 7→ ζ ζ vector fields S = SU generated by δ( ) in the sense of Proposition 2.4. By construction, the pre-Hilbert 1/2 U 1/2 spaces [ ] , and [δ( )]D , E are linearly and latticially isometrically isomorphic. The U µ E1 U (E)1 1 18 L. DELLO SCHIAVO isomorphism extends to a unitary lattice isomorphism between D, 1/2 and D(E) . The last assertion E1 1 follows provided we show the following claim.

Claim: D(E )= D(Eres) for ν-a.e. ζ Z. Argue by contradiction that there exists B ν , with νB > ζ ζ ∈ ∈ Z 0 and so that D(Eres) ( D(E ) for every ζ B. We may assume with no loss of generality that B . ζ ζ ∈ ∈ Z Furthermore, since (Z, ,ν) is σ-finite countably generated, we may and shall assume that νB < . Z ∞ D res ⊥ζ 1/2 D res D Denote by (Eζ )1 the (Eζ )1 -orthogonal complement of (Eζ )1 in (Eζ )1. By the axiom of choice, res ⊥ζ there exists hˆ :=(ζ hˆ ) with hˆ D(E ) and hˆ D = 1 for all ζ B and hˆ := 0D 7→ ζ ζ ∈ ζ 1 k ζ k (Eζ)1 ∈ ζ (Eζ) for ζ Bc. By closability of (E , D(E )), the domain D(E ) embeds identically via ι into H , thus hˆ ∈ ζ ζ ζ 1 ζ,1 ζ ζ may be regarded as an element ι hˆ of H for every ζ Z. Since ι 1 for every ζ Z, then ζ,1 ζ ζ ∈ k ζ,1kop ≤ ∈ (2.25) 0 < ι hˆ 1 , ζ B. k ζ,1 ζ kζ ≤ ∈ Since (E, D(E)) is in particular a direct integral of quadratic forms, ζ ι is a ν-measurable field of 7→ ζ,1 bounded operators, and thus ζ ι hˆ is a ν-measurable field of vectors. 7→ ζ,1 ζ Now, let H be defined as in (2.18). Setting h¯ :=(ζ ι hˆ ), it follows by (2.25) that 7→ ζ,1 ζ ¯ 2 ˆ 2 ˆ 2 h := ιζ,1hζ ζ dν(ζ)= ιζ,1hζ ζ dν(ζ) (0,νB] . k k ZZ k k ZBk k ∈

In particular, for the equivalence class h :=[h¯] , we have that h = 0 . By Proposition 2.25, there exists h˜ H 6 H ∈ 2(µ) representing h H, and thus satisfying 0D = [h˜] D. On the other hand though, L ∈ 6 µ ∈

1([h˜]µ, [˜u]µ) := Eζ,1 [h˜]µ , [˜u] dν(ζ) E Z ζ µζ Z u˜ , ∈U = Eζ,1 hˆζ ,δ(˜u)ζ dν(ζ)=0 , Z Z 1/2 by deﬁnition of hˆ. By -density of [ ] in D, the latter implies that h = 0D, the desired contradiction. E1 U µ

Remark 2.36. If the disintegration in Proposition 2.35 is not separated, then ( , D) is still isomorphic, E as a quadratic form, to the diagonal restriction (Qres, D(Qres)) (Dfn. 2.27) of the direct integral of quadratic forms (E, D(E)).

3. Ergodic decomposition. Everywhere in this section, let (X, τ, ,µ) be satisfying Assump- X tion 2.18. We are interested in the notion of invariant sets for a Dirichlet form.

Definition 3.1 (Invariant sets, irreducibility, cf. [15, p. 53]). Let (E, D(E)) be a Dirichlet form on L2(µ). We say that A X is E-invariant if it is µ-measurable and any of the following equivalent3 ⊂ conditions holds.

(a) T (1 f)= 1 T f µ-a.e. for any f L2(µ) and t> 0; t A A t ∈ (b) T (1 f)=0 µ-a.e. on Ac for any f L2(µ) and t> 0; t A ∈ (c) G (1 f)=0 µ-a.e. on Ac for any f L2(µ) and α> 0; α A ∈ (d) 1 f D(E) for any f D(E) and A ∈ ∈

(3.1) E(f,g)= E(1 f, 1 g)+ E(1 c f, 1 c g) , f,g D(E) A A A A ∈ 3See [15, Lem. 1.6.1, p. 53], the proof of which adapts verbatim to our more general setting. ERGODIC DECOMPOSITION OF DIRICHLET FORMS 19

(e) 1 f D(E) for any f D(E) and (3.1) holds for any f,g D(E) . A ∈ e ∈ e ∈ e The form (E, D(E)) is irreducible if, whenever A is E-invariant, then either µA =0 or µAc =0.

As shown by Example 2.32, the form (E, D(E)) constructed in Proposition 2.29 is hardly ever irreducible, even if (E , D(E )) is so for every ζ Z. ζ ζ ∈

3.1. The algebra of invariant sets. Invariants sets of symmetric Markov processes on locally compact Polish spaces are studied in detail by H. Ôkura in [25]. In particular, he notes the following. For A ∈ X set

˜ µ 1 1 [A]E := A : A˜ is an E-quasi-continuous version of A . n ∈ X o For arbitrary A it can happen that [A] = ∅ or that A [A] . If however (E, D(E)) is regular, ∈ X E 6∈ E then [A] is non-empty for every E-invariant set A. Suppose now A , A and [A ] = ∅. Then, one E 0 1 ∈ X 0 E 6 has the following dichotomy, [25, Rmk. 1.1(ii)],

• [A ] = [A ] if (and only if) µ(A A )=0; 0 E 1 E 0△ 1 • [A ] [A ] = ∅ if (and only if) µ(A A ) > 0. 0 E ∩ 1 E 0△ 1 As a consequence, when describing an E-invariant set A of a regular Dirichlet form (E, D(E)), we may use interchangeably the E-class [A]E — i.e. the finest object representing A, as far as E is concerned — and the µ-class [A] representing A in the measure algebra of (X, ,µ). This motivates to allow A in our µ X definition of invariant set to be µ-measurable, rather than only measurable. We turn now to the study of invariant sets via direct integrals. We aim to show that, under suitable assumptions on µ, a Dirichlet form (E, D(E)) on L2(µ) may be decomposed as a direct integral ζ 7→ (E , D(E )) with (E , D(E )) irreducible for every ζ Z. To this end, we need to construct a measure ζ ζ ζ ζ ∈ space (Z, ,ν) “indexing” E-invariant sets. Let us start with a heuristic argument, showing how this Z cannot be done naïvely, at least in the general case when (X, ,µ) is merely σ-finite. X Let be the family of µ-measurable E-invariant subsets of X, and note that is a σ-sub-algebra X0 X0 of µ, e.g. [15, Lem. 1.6.1, p. 53]. Let µ be the restriction of µˆ to (X, ). The space (X, ,µ ) — our X 0 X0 X0 0 candidate for (Z, ,ν) — is generally not σ-finite, nor even semi-finite. For instance, in the extreme case Z when (E, D(E)) is irreducible and µX = , then is the minimal σ-algebra on X, the latter is an ∞ X0 atom, and thus µ is purely infinite. Since (X, ,µ) is σ-finite, every disjoint family of µ-measurable 0 X non-negligible subsets is at most countable [14, 215B(iii)], thus (X, ,µ ) has up to countably many X0 0 disjoint atoms. However, even in the case when (X, 0,µ0) has no atoms, µ0 might again be purely X 1 infinite. This is the case of Example 2.32, where = ∅, R B(R)Leb is the product σ-algebra of X0 { }⊗ the minimal σ-algebra on the first coordinate with the Lebesgue σ-algebra on the second coordinate, and where µ0 coincides with the µˆ-measure of horizontal stripes. This latter example shows that, again even when (X, ,µ ) has no atoms, the complete locally determined version [14, 213D] of (X, ,µ ) X0 0 X0 0 is trivial. Thus, in this generality, there is no natural way to make (X, ,µ ) into a more amenable X0 0 measure space while retaining information on E-invariant sets.

The situation improves as soon as (X, ,µ) is a probability space, in which case so is (X, ,µ ). The X X0 0 reasons for this fact are better phrased in the language of von Neumann algebras. 20 L. DELLO SCHIAVO

Remark 3.2 (Associated von Neumann algebras). Denote by M the space L∞(µ) regarded as the 2 ∞ (commutative, unital) von Neumann algebra of multiplication operators in B(L (µ)). Then, L (µ0) is a

(commutative) von Neumann sub-algebra of M , denoted by M0. Two key observations are as follows:

• since (X, 0,µ0) is now (semi-)ﬁnite, M0 is unital as well; X 2

• by Deﬁnition 3.1(d), the algebra M0 acts by multiplication also on D(E), and the action M0 L (µ) is compatible with the action M0 D(E) by restriction.

The next definition, borrowed from [4], encodes a notion of “smallness” of the σ-algebra w.r.t. µ. X Definition 3.3 ([4, Dfn. A.1]). Let ∗ be a countably generated σ-subalgebra. We say that: X ⊂ X • is µ-essentially countably generated by ∗ if for each A there is A∗ ∗ with µ(A A∗)=0; X X ∈ X ∈ X △ • is µ-essentially countably generated if it is so by some ∗ as above. X X By our Assumption 2.18, is countably generated, thus is µ -essentially countably generated by X X0 0 ∗ := . We denote by µ∗ the restriction of µ to ∗. As noted in [4, p. 418], atoms of ∗ are, X X ∩ X0 0 0 X X in general, larger (in cardinality, not in measure) than atoms of . It is therefore natural to pass to a X suitable quotient space. Following [4, Dfn. A.5], we define an equivalence relation on X by ∼ (3.2) x x if and only if x A x A for every A ∗ . 1 ∼ 2 1 ∈ ⇐⇒ 2 ∈ ∈ X Further let p: X Z := X/ be the quotient map, := B Z : p−1(B) ∗ be the quotient σ- → ∼ Z ⊂ ∈ X ∗ algebra induced by p, and ν := p♯µ0 be the quotient measure. Similarly to [4, p. 416], it follows by definition of that every A ∗ is p-saturated. In particular: ∼ ∈ X (3.3) ∅ = A p−1(p(x)) = A = p−1(p(x)) , A ∗ . 6 ⊂ ⇒ ∈ X As a consequence ∗ and are isomorphic and thus both are countably generated, since ∗ is by X Z X assumption. Furthermore, (Z, ) is separable by construction, and thus it is countably separated. Z 3.2. Ergodic decomposition of forms: probability measure case. We are now ready to state our main result, a decomposition theorem for Dirichlet forms over their invariant sets.

Theorem 3.4 (Ergodic decomposition: regular case). Let (X, τ, ,µ) be a locally compact Polish X probability space, and (E, D(E)) be a τ-regular Dirichlet form on L2(µ). Then, there exist

(i) a probability space (Z, ,ν) and a measurable map s: (X, ) (Z, ); Z X → Z (ii) a ν-essentially unique disintegration (µζ )ζ∈Z of µˆ w.r.t. ν, strongly consistent with s, and so that, when s−1(ζ) is endowed with the subspace topology and the trace σ-algebra inherited by (X, τ, µ), X then (s−1(ζ),µ ) is a Radon probability space for ν-a.e. ζ Z; ζ ∈ (iii) a ν-measurable ﬁeld ζ (E , D(E )) of τ-regular irreducible Dirichlet forms (E , D(E )) on L2(µ ); 7→ ζ ζ ζ ζ ζ so that ⊕ ⊕ 2 2 (3.4) L (µ)= L (µζ ) dν(ζ) and E = Eζ dν(ζ) . ZZ ZZ Proof. (i) Let (Z, ,ν) be the quotient space of (X, ∗,µ∗) deﬁned in §3.1, and recall that (Z, ) Z X 0 Z is countably separated. Note that id : (X, µ, µˆ) (X, µ0 , µˆ ) is inverse-measure-preserving [14, X X → X0 0 235H(b)], thus so is

(3.5) s := p id : (X, µ, µˆ) (Z, ,ν) . ◦ X X → Z ERGODIC DECOMPOSITION OF DIRICHLET FORMS 21

Since (X, τ, µ, µˆ) is Radon (in the sense of [14, 411H(b)]) and (Z, ,ν) is a probability space (in X Z particular: strictly localizable [14, 322C]), there exists a disintegration (µζ )ζ∈Z of µˆ over ν consistent with s, and so that (X, τ, ,µ ) is a Radon probability space [14, 452O, 452G(a)]. Since (Z, ) is countably X ζ Z separated, (µζ )ζ∈Z is in fact strongly consistent with s [14, 452G(c)]. By deﬁnition of strong consistency, −1 we may restrict µζ to s (ζ); the Radon property is preserved by this restriction [14, 416R(b)]. Since s factors through p, one has s−1(ζ) ∗ for every ζ Z. In particular, s−1(ζ) is a Borel subset of the ∈ X ⊂ X ∈ metrizable Luzin space (X, τ), and thus a metrizable Luzin space itself by [27, §II.1, Thm. 2, p. 95]. It follows that supp[µζ ], endowed with the subspace topology inherited from (X, τ) and the induced Borel σ-algebra, satisﬁes Assumption 2.18. The disintegration is ν-essentially unique, similarly to [4, Thm. A.7, Step 1, p. 420]. This shows (i)–(ii). The proof of (iii) is divided into several steps.

1. Measurable fields. Let be a special standard core [15, p. 6] for (E, D(E)), and N Z be a ν- C ⊂ c 2 negligible set so that (X, τ, ,µζ ) is Radon for every ζ N . Then, is dense in L (µζ ) for X ∈ C supp[µζ ] every ζ N c. In particular, since L2(µ) is separable, there exists a fundamental sequence (u ) , ∈ n n ⊂ C i.e. total in L2(µ ) for every ζ N c, and additionally total in L2(µ). Since (E, D(E)) is regular, D(E) ζ ∈ 1 is separable by [23, Prop. IV.3.3(i)], and therefore we can and will assume, with no loss of generality that (u ) is additionally E1/2-total in D(E) . Moreover, is an algebra, thus ζ f g = µ (fg) n n 1 1 C 7→ h | iζ ζ is ν-measurable by definition of disintegration for every f, g . As a consequence, by Proposition 2.4 ∈ C there exists a unique ν-measurable field of Hilbert spaces ζ L2(µ ) making ν-measurable all functions 7→ ζ of the form ζ µ f with f . We denote by S the underlying space of ν-measurable vector fields. 7→ ζ ∈ C C Everywhere in the following, we identify [f] with a fixed continuous representative f , thus writing f µζ ∈C in place of δ(f).

2 2. L -isomorphism. Since (µζ )ζ∈Z is strongly consistent with s, it is separated. Therefore, the ﬁrst 2 isomorphism in (3.4) follows now by (2.19) with underlying space SC. In the following, set H := L (µ) 2 and Hζ := L (µζ ).

3. Semigroups. Let Tt be the semigroup associated to (E, D(E)) and consider the natural complexiﬁca- C ∞ tion T of T deﬁned on HC := H R C. For g L (ν) denote by t t ⊗ ∈

SC ⊕ 1 Mg := g(ζ) Hζ dν(ζ) ZZ the associated diagonalizable operator in B(H), [12, §II.2.4 Dfn. 3, p. 185]. For B set M := M1 . ∈ Z B B C C ∞ Claim: the commutator [T ,M ] vanishes for g LC (ν). By [12, §II.2.3, Prop. 4(ii), p. 183] and the t g ∈ norm-density of simple functions in L∞(ν), it suﬃces to show that [T ,M ]=0 for every B . To this t B ∈ Z end, recall the discussion [12, p. 165] on ν-measurable structures induced by ν-measurable subsets of Z. Since B , then A := p−1(B) µ0 , and A µ as well [14, 235H(c)]. Note further that, since H is ∈ Z ∈ X0 ∈ X reconstructed as a direct integral with underlying space S , for every h the representative h of h C ∈ C ζ in H may be chosen so that h = h for every ζ Z. Thus, for all f,g , ζ ζ ∈ ∈C

MBf g = 1B(ζ) 1H fζ gζ dν(ζ) h | iH Z ζ ζ Z

= fg dµζ dν(ζ)= fg dµ ZB ZX ZA = 1 f g . h A | iH 22 L. DELLO SCHIAVO

By density of in H and since M is bounded, it follows that M = 1 as elements of B(H). C B B A Thus, [Tt,MB] = [Tt, 1A]=0 for every t> 0 by Deﬁnition 3.1(a), since A is E-invariant.

By the characterization of decomposable operators via diagonalizable operators [12, §II.2.5, Thm. 1, p. 187], T is decomposable, and represented by a ν-measurable ﬁeld of contraction operators ζ T . t 7→ ζ,t Finally, in light of [12, §II.2.3, Prop. 4, p. 183], it is a straightforward veriﬁcation that Tζ,t, t > 0, is a strongly continuous symmetric contraction semigroup on H for ν-a.e. ζ Z, since so is T . Analogously ζ ∈ t to the proof of Proposition 2.29, T is sub-Markovian for ν-a.e. ζ Z, since so is T . ζ,t ∈ t 2 4. Forms: construction. Denote by (Eζ , D(Eζ )) the Dirichlet form on L (µζ ) associated to the sub- Markovian semigroup T for ν-a.e. ζ Z. Let further G , α> 0, be the associated strongly continuous ζ,t ∈ ζ,α contraction resolvent. We claim that D(E ) for ν-a.e. ζ Z. Firstly, note that ζ E (f,g) is ν-measurable, since it C ⊂ ζ ∈ 7→ ζ is the ν-a.e.-limit of the measurable functions ζ E(β)(f,g) := f βG f g as β by (2.1c). 7→ ζ h − ζ,β | iζ → ∞ By [23, p. 27],

∞ −βt Eζ (f,g) = lim f β e Tζ,tf dt g , f,g . β→∞ − Z ∈C 0 ζ

Now,

∞ −βt Eζ (f) dν(ζ)= lim f β e Tζ,tf dt f dν(ζ) Z Z β→∞ − Z Z Z 0 ζ ∞ −βt lim inf f β e Tζ,tf dt f dν(ζ) ≤ β→∞ Z − Z Z 0 ζ

by Fatou’s Lemma. It is readily checked that, since T 1, we may exchange the order of both k ζ,tkop ≤ integration and Hζ -scalar products by Fubini’s Theorem. Thus,

∞ −βt Eζ (f) dν(ζ) lim inf f β e Tζ,tf dt f dν(ζ) Z ≤ β→∞ Z − Z Z Z 0 ζ ∞ 2 − βt = f ζ dν(ζ) lim sup βe Tζ,tf f ζ dν(ζ) dt . ZZ k k − β→∞ Z0 ZZ h | i

By the representation of T via ζ T established in Step 3, t 7→ ζ,t ∞ 2 −βt Eζ (f) dν(ζ) f dµζ dν(ζ) lim sup βe Ttf f H dt . ZZ ≤ ZZ ZX − β→∞ Z0 h | i

Finally, by (2.14), [23, p. 27] and (2.1c),

∞ −βt Eζ (f) dν(ζ) lim inf β f e Ttf f dt Z ≤ β→∞ Z − H Z 0 ∞ −βt = lim inf β f e Ttf dt f = E(f) < . β→∞ − Z ∞ 0 H

This shows that E (f) < for every f for ν-a.e. ζ Z, thus D(E ) ν-a.e. ζ ∞ ∈C ∈ C ⊂ ζ Claim: is a core for (Eζ , D(Eζ )) for ν-a.e. ζ Z. It suﬃces to show that the inclusion D(Eζ ) 1/2 C ∈ C ⊂ is (Eζ )1 -dense for ν-a.e. ζ Z. Argue by contradiction that there exists a ν-measurable non-negligible ∈ 1/2 set B so that the inclusion D(E ) is not (E ) -dense for every ζ B, and let ⊥ζ be the C ⊂ ζ ζ 1 ∈ C (E )1/2-orthogonal complement of in D(E ). By the axiom of choice we may construct h H ζ 1 C ζ ∈ ζ∈Z ζ so that h ⊥ζ 0 for ζ B and h = 0 for ζ Bc. Further let (u ) be as inQStep 1. ζ ∈ C \{ } ∈ ζ ∈ n n ⊂ C ERGODIC DECOMPOSITION OF DIRICHLET FORMS 23

Then, ζ u h = 0 is ν-measurable for every n. As a consequence, ζ h is ν-measurable 7→ h n | ζ iζ 7→ ζ (i.e., it belongs to SC) by [12, §II.1.4, Prop. 2, p. 166]. By the ﬁrst isomorphism in (3.4), established in Step 1, ζ h represents an element h in H. Since (u ) is total in H, there exists n so that 7→ ζ n n

0 = un h H = un hζ ζ dν(ζ)=0 , 6 h | i ZZ h | i a contradiction. Since functions in are continuous, the form (E , D(E )) is regular for ν-a.e. ζ Z. In C ζ ζ ∈ particular, D(E ) is separable for ν-a.e. ζ Z by [23, Prop. IV.3.3]. ζ 1 ∈ We note that, by the above claim and [23, Prop. IV.3.3(i)], D(E ) is separable for every ζ Z, and ζ 1 ∈ so the observation in Remark 2.12 is satisﬁed.

5. Forms: direct integral. By Step 1, resp. Step 4, ζ L2(µ ), resp. ζ D(E ) , is a ν-measurable 7→ ζ 7→ ζ 1 field of Hilbert spaces with underlying space S . In particular, ζ (E , D(E )) satisfies Definition 2.11, C 7→ ζ ζ and we may consider the direct integral of quadratic forms

SC ⊕ (3.6) E˜(f) := Eζ (f) dν(ζ) ZZ deﬁned by (2.7). We claim that (E,˜ D(E˜)) = (E, D(E)). This is a consequence of Proposition 2.13(iii), since (2.8) was shown in Step 3 for T . Deﬁnition 2.26 holds with = by construction. t A C

6. Forms: irreducibility. Let Aζ be Eζ -invariant, with µˆζ Aζ > 0. With no loss of generality, we may and will assume that A . Up to removing a ν-negligible set of ζ’s, we have that A s−1(ζ), by strong ζ ∈ X ζ ⊂ consistency of the disintegration. Thus, by (2.14),

−1 (3.7) µAζ = µ Aζ s (ζ) = µζ Aζ dν(ζ)= µζ Aζ ν ζ . ∩ Z · { } {ζ} Claim: A . Assume ﬁrst that ν ζ = 0. Then, A is contained in the µ-negligible invariant ζ ∈ X0 { } ζ set s−1(ζ), hence, it is E-invariant, i.e. A . Assume now ν ζ > 0. By (3.7), µA > 0, thus A = ∅ ζ ∈ X0 { } ζ ζ 6 and 1 = 0 . By [12, §II.2.3, Prop. 3, p. 182] and the direct-integral representation of T established in Aζ 6 H t Step 3,

⊕ ′ 1 1 ′ Aζ Tt = Aζ Tζ ,t dν(ζ ) . ZZ

′ By strong consistency, A is µ ′ -negligible for every ζ = ζ, thus in fact ζ ζ 6 1 T = ν ζ 1 T , Aζ t { } Aζ ζ,t whence, by Eζ -invariance of Aζ ,

1 T = ν ζ 1 T = ν ζ T 1 = T 1 , Aζ t { } Aζ ζ,t { } ζ,t Aζ t Aζ and so A is E-invariant, and thus A . ζ ζ ∈ X0 Now, since A by assumption, then A ∗ := . Together with A s−1(ζ), this implies ζ ∈ X ζ ∈ X X ∩ X0 ζ ⊂ −1 −1 that either Aζ = ∅, or Aζ = s (ζ) by (3.3). Thus, it must be Aζ = s (ζ), since µˆζ Aζ > 0 by −1 assumption. Since s (ζ) is µˆζ -conegligible, this shows that (Eζ , D(Eζ )) is irreducible.

2 In the statement of Theorem 3.4, we write that each (Eζ , D(Eζ )) is a regular Dirichlet form on L (µζ ) with underlying space (X, τ, ,µ ) to emphasize that the topology of the space is the given one. As X ζ 24 L. DELLO SCHIAVO it is well-known however, in studying the potential-theoretic and probabilistic properties of a Dirichlet form (E, D(E)) on L2(µ), one should always assume that µ has full support, which is usually not the case −1 −1 for µζ on (X, τ). In the present case, the restriction of µζ to s (ζ) is however harmless, since s (ζ) is −1 c E-invariant, and therefore s (ζ) is also Eζ -exceptional.

Remark 3.5. As anticipated in §1, if (E, D(E)) is regular and strongly local, then every invariant set admits an E-quasi-clopen µ-modification [15, Cor. 4.6.3, p. 194]. This suggests that, at least in the local case, one may treat E-invariant sets as “connected components” of X. Our intuition can be made rigorous by noting that E-invariant subsets of X are in bijective correspondence to compact open subsets of the spectrum spec(M0) of the von Neumann algebra M0 (cf. Rmk. 3.2), endowed with its natural weak* topology. In particular, spec(M ) coincides with the Stone space of the measure algebra of (X, ,µ ), 0 X0 0 and is thus a totally disconnected Hausdorff space. Its singletons correspond to the “minimal connected components” sought after in §1. At this point, we should emphasize that (Z, ) and spec(M ) are different Z 0 measure spaces, the points of which index “minimal invariant sets” in X. However, points in Z index — via s — sets in ∗, whereas points in spec(M ) index sets in . In this sense at least, Z is minimal X 0 X0 with the property of indexing such “minimal invariant sets”, while spec(M0) is maximal. For this reason, one might be tempted to use spec(M ) in place of (Z, ) in Theorem 3.4. The issue is that spec(M ) is 0 Z 0 nearly always too large for the disintegration to be strongly consistent with the indexing map.

In the next result we show that the regularity of the Dirichlet form (E, D(E)) in Theorem 3.4 may be relaxed to quasi-regularity. As usual, a proof of this result relies on the so-called transfer method.

Let (X, τ, ,µ) and (X♯, τ ♯, ♯,µ♯) be measure spaces satisfying Assumption 2.18. We note en passant X X that a Hilbert isomorphism of Dirichlet spaces ι: D(E) D(E♯) , additionally preserving the L∞-norm 1 → 1 on D(E) L∞(µ), is automatically a lattice isomorphism, e.g. [15, Lem. A.4.1, p. 422]. ∩ Theorem 3.6 (Ergodic decomposition: quasi-regular case). The conclusions of Theorem 3.4 remain valid if (X, τ, ,µ) is a topological probability space satisfying Assumption 2.18 and “regular” is replaced X by “quasi-regular”.

Proof. By the general result [7, Thm. 3.7], there exist a locally compact Polish, Radon probability space (X♯, τ ♯, ♯,µ♯) and a quasi-homeomorphism X j : (X, τ, ,µ) (X♯, τ ♯, ♯,µ♯) X −→ X so that (E, D(E)) is quasi-homeomorphic, via j, to a regular Dirichlet form (E♯, D(E♯)) on (X♯, τ ♯, ♯,µ♯). X Applying Theorem 3.4 to (E♯, D(E♯)) gives a disintegration µ♯ of µ w.r.t. ν and a direct-integral rep- ζ ζ resentation

⊕ ♯ ♯ E = Eζ dν(ζ) , ZZ where (E♯, D(E♯)) is a regular Dirichlet form on L2(µ♯ ) for ν-a.e. ζ Z. ζ ζ ζ ∈

1. Forms. In the following, whenever (Fk)k is a nest, let us set F := k Fk. With no loss of generality by [15, Lem. 2.1.3, p. 69], we may and will always assume that every nestS is increasing, and regular w.r.t. a measure apparent from context. Let (F ) , resp. F ♯ , be an E-, resp. E♯-, nest, additionally so that j : F F ♯ restricts to a k k k k → homeomorphism j : F F ♯ for every k. Since (F ) is increasing, j : F F ♯ is bijective. Let N be k → k k k → 1 ERGODIC DECOMPOSITION OF DIRICHLET FORMS 25

ν-negligible so that (E♯, D(E♯)) is regular by Theorem 3.4. Let X := X ∂ , where ∂ is taken to be ζ ζ ∂ ∪{ } an isolated point in X . Since j may be not surjective, in the following we extend j−1 on X♯ j(F ) by ∂ \ setting j−1(x♯)= ∂. Note that this extension is ♯-to- -measurable (having care to extend on X in X X X ∂ the obvious way). Since j µ = µ♯ the set N := ζ Z : µ♯ j(F ) < 1 is ν-measurable, since j is measurable ♯ 2 { ∈ ζ } on F , and thus it is ν-negligible. In particular, j−1µ♯ ∂ =0, and j−1µ♯ ∂ =0 for every ζ N c. Set ♯ { } ♯ ζ { } ∈ 2 now N := N N . For ζ N c set µ := j−1µ♯ and denote by (E , D(E )) the image form of (E♯, D(E♯)) 1 ∪ 2 ∈ ζ ♯ ζ ζ ζ ζ ζ via j−1 on L2(µ ), cf. [7, Eqn. (3.2)]. For ζ N let (E , D(E )) be the 0-form on L2(µ). For f ♯ : X♯ R ζ ∈ ζ ζ → denote further by j∗f := f ♯ j : X R the pullback of f ♯ via j, and recall [7, Eqn. (3.3)]: ◦ → (3.8) G (j∗f ♯)= j∗G♯ f ♯ , f ♯ L2(µ♯) . α α ∈ 2. Nests. For ζ N c let F ♯ be a µ♯ -regular E♯-nest witnessing the (quasi-)regularity of the form, ∈ ζ,k k ζ ζ ♯ ♯ i.e. verifying [7, Dfn. 2.8]. With no loss of generality, up to intersecting Fζ,k with Fh if necessary, we may assume that for every k there exists h := h so that F ♯ F ♯. In particular, j−1 : F ♯ F := j(F ♯ ) is k ζ,k ⊂ h ζ,k → ζ,k ζ,k a homeomorphism onto its image. Let X♯ := supp µ♯ and note that F ♯ X♯ since F ♯ is µ♯ -regular. ζ ζ ζ,k ⊂ ζ ζ,k k ζ −1 −1 ♯ Denote by jζ the restriction of j to Xζ . Claim: j−1 is a quasi-homeomorphism for ν-a.e. ζ Z. It suﬃces to show that (F ) is an E -nest ζ ∈ ζ,k k ζ for ν-a.e. ζ Z, which holds by construction. ∈

Finally, set jζ := j −1 ♯ and note that, again by [7, Eqn. (3.3)], j (Xζ )

(3.9) G (j∗f ♯)= j∗G♯ f ♯ , f ♯ L2(µ♯ ) , ζ N c . ζ,α ζ ζ ζ,α ∈ ζ ∈ 3. Direct integral representation. By (2.8) for the resolvent applied to (E♯, D(E♯)),

⊕ ♯ ♯ (3.10) Gα = Gζ,α dν(ζ) . ZZ

By Step 1 and [7, Lem. 3.3(ii)], j∗ : L2(µ♯) L2(µ) is an isomorphism for ν-a.e. ζ Z, with in- ζ → ∈ −1 ∗ verse (jζ ) , and

⊕ ∗ ∗ (3.11) j = jζ dν(ζ) . ZZ Now, by a subsequent application of (3.8), (3.10), (3.11) and [12, §II.2.3, Prop. 3, p. 182], and (3.9),

⊕ ⊕ ∗ ∗ ♯ ∗ ♯ ∗ ♯ Gα j = j Gα = j Gζ,α dν(ζ)= jζ Gζ,α dν(ζ) ◦ ◦ ◦ ZZ ZZ ◦ ⊕ ∗ = Gζ,α jζ dν(ζ) . ZZ ◦ Thus, by a further application of [12, §II.2.3, Prop. 3, p. 182],

⊕ ∗ ∗ Gα j = Gζ,α dν(ζ) j . ◦ ZZ ◦ Cancelling j∗ by its inverse (j−1)∗, this yields the direct-integral representation of G via ζ G . α 7→ α,ζ By (2.8) for the resolvent, this shows

⊕ E = Eζ dν(ζ) . ZZ 26 L. DELLO SCHIAVO

4. Quasi-regularity and irreducibility. By Step 2 and [7, Thm. 3.7], the form (Eζ , D(Eζ )) is quasi- regular for ν-a.e. ζ Z. Again by Step 2, it is also irreducible, since it is isomorphic to the irreducible ∈ ♯ D ♯ form (Eζ , (Eζ )).

3.3. Ergodic decomposition of forms: σ-finite measure case. Under some additional assumptions, we may now extend the results in Theorem 3.4 to the case when µ is only σ-finite. The main idea — borrowed from [6] — is to reduce the σ-finite case to the probability case.

General strategy. Let (E, D(E)) be a strongly local regular Dirichlet form on a locally compact Polish, Radon measure space (X, τ, µ, µˆ) with full support. In particular, (X, ,µ) is σ-finite, [14, 415D(iii)]. X X Assume further that (E, D(E)) has carré du champ (Γ, D(E)). Let ϕ D(E), with ϕ > 0 µ-a.e. and ∈ ϕ 2 ϕ 2 =1, and set µ := ϕ µ. Here, we understand ϕ as a fixed E-quasi-continuous representative of k kL (µ) · its µ-class. Note that (X, ,µϕ) is a probability space and that µϕ is equivalent to µ. Therefore, µ-classes X and µϕ-classes coincide. On L2(µϕ) we define a bilinear form

(3.12) D(Eϕ) := (Γ(f)+ f 2) dµϕ < , Eϕ(f,g) := Γ(f,g) dµϕ . Z ∞ Z Suppose now that

(a) the form (Eϕ, D(Eϕ)) is a (closed) regular Dirichlet form on L2(µϕ).

Then, we may apply Theorem 3.4 to obtain

• a probability space (Z , ,ν ) and a measurable map s : X Z; ϕ Zϕ ϕ ϕ → • a ν -essentially unique disintegration µ(ϕ) of µϕ over ν , strongly consistent with s ; ϕ ζ ζ∈Z ϕ ϕ (ϕ) D (ϕ) 2 (ϕ) • a family of regular strongly local Dirichlet forms Eζ , (Eζ ) on L µζ ; satisfying the direct-integral representation

⊕ ϕ (ϕ) (3.13) E = Eζ dνϕ(ζ) . ZZϕ

For ζ Z let now µ[ϕ] := ϕ−2 µ(ϕ) be a measure on (X, ) and suppose further that ∈ ζ · ζ X (b) the form E(ϕ), D(E(ϕ)) has carré du champ operator Γ(ϕ), D(Γ(ϕ)) for ν -a.e. ζ Z ; ζ ζ ζ ζ ϕ ∈ ϕ (c) the form

D(E[ϕ]) := Γ(ϕ)(f)+ f 2 dµ[ϕ] < , ζ Z ζ ζ ∞ (3.14) E[ϕ](f,g) := Γ(ϕ)(f,g) dµ[ϕ] , f,g D(E[ϕ]) , ζ Z ζ ζ ∈ ζ

is a (closed) regular Dirichlet form on L2(µ[ϕ]) for ν -a.e. ζ Z. ζ ϕ ∈ Then, ﬁnally, we may expect to have a direct-integral representation

⊕ [ϕ] (3.15) E = Eζ dνϕ(ζ) . ZZϕ As it turns out, the properties of the Girsanov-type transformation (3.12) are quite delicate. Before discussing the technical details, let us note here that, provided we have shown the direct-integral representation in (3.15), it should not be expected that the latter is (essentially) unique, but rather merely essentially projectively unique — as it is the case for other ergodic theorems, e.g. [6, Thm. 2]. In the present setting, projective uniqueness is understood in the following sense. ERGODIC DECOMPOSITION OF DIRICHLET FORMS 27

Definition 3.7. We say that the direct integral representation (3.15) is essentially projectively unique if, for every ϕ, ψ as above:

(a) the measurable space (Z, ) :=(Z , ) = (Z , ) is uniquely determined; Z ϕ Zϕ ψ Zψ (b) the measures νϕ, νψ are equivalent (i.e., mutually absolutely continuous); (c) the forms E[ϕ], E[ψ] are multiple of each other for ν - (hence ν -)a.e. ζ Z. ζ ζ ϕ ψ ∈ As it is clear, the definition only depends on the σ-ideal of ν -negligible sets in . By condition (b), ϕ Z this ideal does not, in fact, depend on νϕ, hence the omission of the measure in the designation. The lack [ϕ] of uniqueness is shown as follows. Since µζ is merely a σ-finite (as opposed to: probability) measure, the family µ[ϕ] is merely a pseudo-disintegration (as opposed to: a disintegration). Thus, for every ζ ζ∈Z measurableg : Z (0, ), → ∞ [ϕ] [ϕ] −1 µA = µ A dνϕ(ζ)= g(ζ) µ A d g νϕ (ζ) , A . Z ζ Z · ζ · ∈ X Z Z Since g is defined on Z, the pullback function f :=(s )∗g is -measurable, i.e. all its level sets are ϕ X0 E-invariant; by strong locality of (E, D(E)), f is E-quasi-continuous, and therefore an element of the extended domain F of (E, D(E)). As soon as f L2(µ), then we have the direct-integral representation e ∈ ⊕ [ϕ] −1 [ϕ] (ϕ) [ϕ] E = g(ζ)E d g νϕ (ζ) , g(ζ) E (u)= Γ (u) d f µ . Z ζ · ζ Z ζ · ζ Zϕ X Proofs’ summary. The Girsanov-type transformations (3.12) are thoroughly studied by A. Eberle in [13], where (a) is proved. We shall therefore start by showing (b) above, Lemma 3.8 below. Informally, in the setting of Theorem 3.4, if (E, D(E)) has carré du champ Γ, then

⊕ (3.16) Γ= Γζ dν(ζ) , ZZ where Γζ is the carré du champ of (Eζ , D(Eζ )). Since the range of Γ is a Banach (not Hilbert) space, we shall need the concept of direct integrals of Banach spaces. In particular, we shall need an analogue of Proposition 2.25 for L1-spaces, an account of which is given in the Appendix, together with a proof of the next lemma.

Lemma 3.8. Under the assumptions of Theorem 3.4 suppose further that (E, D(E)) admits carré du champ operator (Γ, D(E)). Then, (Eζ , D(Eζ )) admits carré du champ operator (Γζ , D(Eζ )) for ν- a.e. ζ Z. ∈ Lemma 3.9. Under the assumptions of Theorem 3.4 suppose further that (E, D(E)) is strongly local. Then, (E , D(E )) is strongly local for ν-a.e. ζ Z. ζ ζ ∈ Proof. Note: In this proof we shall make use of results in [5]. We recall that a regular form is ‘strongly local’ in the sense of [15, p. 6] if and only if it is ‘local’ in the sense of [5, Dfn. I.V.1.2, p. 28]. This is noted e.g. in [28, §2, p. 78], after [26, Prop. 1.4]. In this respect, we always adhere to the terminology of [23, 15]. Let ( , D( )) be a regular Dirichlet form. By [5, Cor. I.5.1.4 and Rmk. I.5.1.5, p. 31], ( , D( )) E E E E is strogly local if and only if u D( ) and ( u )= (u) for every u D( ). Further note that u is a | | ∈ E E | | E ∈ E | | normal contraction of u D( ) for every u D( ). As a consequence, u D( ) and ( u ) (u) for ∈ E ∈ E | | ∈ E E | | ≤E every u D( ), see e.g. [15, Thm. 1.4.1]. In particular, a Dirichlet form ( , D( )) is not strongly local if ∈ E E E and only if there exists u D( ) with (u) > ( u ). ∈ E E E | | 28 L. DELLO SCHIAVO

Since µ X 1 for every ζ Z, it is not difficult to show, arguing by contradiction, that ζ ≤ ∈ (3.17) 1 D(E ) , E (1 )=0 for ν-a.e. ζ Z. X ∈ ζ ζ X ∈ Let (u ) be the fundamental sequence constructed in Step 1 in the proof of Theorem 3.4(iii). The n n ⊂C Dirichlet form (E , D(E )) on L2(µ ) is strongly local if and only if E (u )= E ( u ) for every n 1. ζ ζ ζ ζ n ζ | n| ≥ The same holds for (E, D(E)). Now, argue by contradiction that there exists a ν-measurable non-negligible set B Z so that the ⊂ form (E , D(E )) is not strongly local for each ζ B. Let B := ζ Z : E (u ) > E ( u ) and note ζ ζ ∈ n { ∈ ζ n ζ | n| } that B B is ν-measurable for every n 1 since (u ) . Since B = B and νB > 0, there exists n ⊂ ≥ n n ⊂C ∪n n some fixed n∗ so that νBn∗ > 0. Without loss of generality, up to relabeling, we may choose n∗ = 1. −1 Analogously to the proof of the Claim in Step 3 of Theorem 3.4(iii), set A := p (B1) and note that it is E-invariant. Thus, finally, 1 u D(E) and A 1 ∈

E(u1 1A)= Eζ (u1) dν(ζ) > Eζ ( u1 ) dν(ζ)= E( u1 1A)= E( u1 1A ) , ZB1 ZB1 | | | | | | which contradicts the strong locality of (E, D(E)).

Remark 3.10. The converse implications to Lemmas 3.8, 3.9 are true in a more general setting, viz.

(a) if (E, D(E)) is a direct integral of Dirichlet forms ζ (E , D(E )) each with carré du champ 7→ ζ ζ operator (Γζ , D(Γζ )), then (E, D(E)) has carré du champ given by (3.16), see [5, Ex. V.3.2, p. 216]; (b) if (E, D(E)) is a direct integral of strongly local Dirichlet forms ζ (E , D(E )), then (E, D(E)) 7→ ζ ζ is strongly local, see [5, Ex. V.3.1, p. 216].

We are now ready to prove the main result of this section.

Theorem 3.11 (Ergodic decomposition: σ-ﬁnite case). Let (X, τ, ,µ) be a locally compact Polish X Radon measure space, and (E, D(E)) be a regular strongly local Dirichlet form on L2(µ) with carré du champ operator (Γ, D(Γ)). Then, there exist

(i) a probability space (Z, ,ν) and a measurable map s: (X, ) (Z, ); Z X → Z −1 (ii) an essentially projectively unique family of measures (µζ )ζ∈Z so that, when s (ζ) is endowed with the subspace topology and the trace σ-algebra inherited by (X, τ, µ), then (s−1(ζ), µˆ ) is a Radon X ζ measure space for ν-a.e. ζ Z; ∈ (iii) a ν-measurable ﬁeld ζ (E , D(E )) of regular irreducible Dirichlet forms (E , D(E )) on L2(µ ); 7→ ζ ζ ζ ζ ζ so that

⊕ ⊕ 2 2 L (µ)= L (µζ ) dν(ζ) and E = Eζ dν(ζ) . ZZ ZZ

Proof. Let ϕ D(E) with 0 <ϕ< 1 µ-a.e. and ϕ 2 =1. Since (E, D(E)) is regular strongly local ∈ k kL on L2(µ) and admits carré du champ (Γ, D(E)), then the Girsanov-type transform (Eϕ, D(Eϕ)) deﬁned in (3.12) is a quasi-regular strongly local Dirichlet form on L2(µϕ) by [13, Thm.s 1.1 and 1.4(ii)], and admits carré du champ (Γ, D(Eϕ)) by construction. We note that we are applying the results in [13] in the context of [13, Example 1), p. 501]. In particular, Assumption (D3) in [13, p. 501] holds by deﬁnition. ERGODIC DECOMPOSITION OF DIRICHLET FORMS 29

1. Constructions. Let now be a core for (E, D(E)). Since ϕ 1 µ-a.e., then Eϕ E , and the C ≤ 1 ≤ 1 form (Eϕ, D(Eϕ)) is in fact regular, with same core . Since µϕ is a fully supported probability measure C by construction, we may apply Theorem 3.4 to obtain the direct integral representation (3.13). For νϕ- a.e. ζ Z , the form E(ϕ), D(E(ϕ)) is regular with core and irreducible by Theorem 3.4, strongly ∈ ϕ ζ ζ C (ϕ) D (ϕ) local by Lemma 3.9, and admitting carré du champ Γζ , (Eζ ) by Lemma 3.8. Claim: ϕ−2 n D(E(ϕ)) for ν -a.e. ζ Z , for every n 1. Since Eϕ E , one has ϕ D(Eϕ), ∨ ∈ ζ ϕ ∈ ϕ ≥ 1 ≤ 1 ∈ hence ϕ D(E(ϕ)) for ν -a.e. ζ Z as a consequence of (2.7). The claim then follows by strong locality ∈ ζ ϕ ∈ ϕ of (E , D(E )) for ν -a.e. ζ Z . ζ ζ ϕ ∈ ϕ Let now E[ϕ],n, D(E[ϕ],n) be defined analogously to (3.14) with µ[ϕ],n :=(ϕ−2 n) µ(ϕ) in place of µ[ϕ]. ζ ζ ζ ∨ · ζ ζ [ϕ],n D [ϕ],n By applying once more [13, Thm.s 1.1 and 1.4(ii)], the Girsanov-type transform Eζ , (Eζ ) defined in (3.14) is a regular Dirichlet form on L2(µ[ϕ],n) with core , strongly local, irreducible, and admitting ζ C carré du champ operator Γ(ϕ), D(E[ϕ],n) for ν -a.e. ζ Z , for every n 1. ζ ζ ϕ ∈ ϕ ≥ [ϕ] D [ϕ] 2 [ϕ] Claim: the quadratic form Eζ , (Eζ ) defined in (3.14) is a regular Dirichlet form on L (µζ ), strongly local, irreducible, and admitting carré du champ operator Γ(ϕ), D(E[ϕ]) for ν -a.e. ζ Z . ζ ζ ϕ ∈ ϕ [ϕ] [ϕ],n D [ϕ] D [ϕ],n [ϕ] D [ϕ] Firstly, note that Eζ = supn Eζ is well-defined on (Eζ ) = n≥1 (Eζ ), thus Eζ , (Eζ ) is a closable quadratic form by [23, Prop. I.3.7(ii)]. The Markov property,T the strong locality and the existence and computation of the carré du champ operator are straightforward. Note that D(E[ϕ]), C ⊂ ζ so that the latter is dense in L2(µ[ϕ]) for ν -a.e. ζ Z . By Dominated Convergence and (2.15) ζ ϕ ∈ ϕ

E(u)= Γ(u) dµ = lim Γ(u) (ϕ−2 n) dµϕ Z n Z · ∨ (ϕ) [ϕ],n = lim Γ (u) dµ dνϕ(ζ) n ζ ζ ZZϕ ZX

(ϕ) [ϕ] = Γζ (u) dµζ dνϕ(ζ) , ZZϕ ZX which establishes the direct integral representation (3.15), with underlying space SC. The regularity of the forms E[ϕ], D(E[ϕ]) , all with core , follows from the regularity of (E, D(E)), exactly as in the ζ ζ C Claim in Step 4 in the proof of Theorem 3.4.

2. Projective uniqueness. By (3.13), E-invariant sets are also Eϕ-invariant. The reverse implication follows since µ and µϕ are equivalent. As a consequence, the σ-algebras and ∗ deﬁned w.r.t. µϕ as X0 X in §3.1 are independent of ϕ and thus so are the space (Z , ), henceforth denoted simply by (Z, ), ϕ Zϕ Z and the map s , henceforth denoted simply by s. Let now ϕ, ψ D(E), ϕ,ψ > 0 µ-a.e., and replicate ϕ ∈ every construction for ψ as well. Note that µϕ := ϕ2 µ and µψ := ψ2 µ are equivalent. · · Claim 1: ν ν . It suﬃces to recall that ν = s µϕ and analogously for ψ w.r.t. the same map s, ϕ ∼ ψ ϕ ♯ hence the conclusion, since µϕ µψ. ∼ It follows that the σ-ideal := of ν -negligible sets in does not in fact depend on ϕ. In the N Nϕ ϕ Z following, we write therefore “ -negligible” in place of “ν -negligible” and “ -a.e.” in place of “ν -a.e.”. N ϕ N ϕ Claim 2: µ(ϕ) µ(ψ) for -a.e. ζ Z. Argue by contradiction that there exist B and a ζ ∼ ζ N ∈ ∈Z\N family (A ) with A and, without loss of generality, µ(ϕ)A > µ(ψ)A = 0 for all ζ B. Set ζ ζ∈B ζ ∈ X ζ ζ ζ ζ ∈ further A˜ := A and let A be its measurable envelope [14, 132D]. Then, by (2.14) and strong ∪ζ∈B ζ ∈ X 30 L. DELLO SCHIAVO

(ϕ) consistency of (µζ )ζ∈Z with s,

ϕ ϕ −1 (ϕ) (ϕ) −1 µ A = µ A s (B) = µ A dνϕ(ζ)= µ A s (ζ) dνϕ(ζ) ∩ Z ζ Z ζ ∩ (3.18) B B (ϕ) µζ Aζ dνϕ(ζ) > 0 . ≥ ZB

ψ∗ ψ (ψ)∗ Let now µ be the outer measure of µ , and analogously for µζ . Note that, by Assumption 2.18, the Carathéodory measure induced by µψ∗ coincides with the completion measure µψ =µ ˆψ. Furthermore, (ψ) (ψ)∗ ˜ (ψ)∗ ˜ −1 (ψ)∗ by strong consistency of (µ )ζ∈Z with s, one has µ A = µ A s (ζc) = µ Aζ = 0 by ζ ζ ζ ∩ ζ assumption. In particular, the function ζ µ(ψ)∗A˜ 0 is measurable. Thus, by [14, 452X(i)], 7→ ζ ≡

(ψ)∗ ˜ ψ∗ ˜ ψ 0= µζ A = µ A = µ A, ZB which contradicts (3.18) since µϕ µψ by the previous claim. ∼ [ϕ] [ψ] [ϕ] By the last claim, µζ -classes and µζ -classes coincide. Therefore, the carré du champ operator Γζ = (ϕ) Γζ is independent of ϕ, and henceforth denoted by Γζ . Thus we have

[ϕ] [ψ] E(u)= Γζ (u) dµζ dνϕ(ζ)= Γζ (u) dµζ dνψ(ζ) , u , ZZ ZX ZZ ZX ∈C and, ﬁnally, it suﬃces to show the following.

Claim 3: µ[ϕ] = dνψ µ[ψ]. By construction, µ[ϕ] , resp. µ[ψ] , is a pseudo-disintegration of µ ζ dνϕ · ζ ζ ζ∈Z ζ ζ∈Z over ν , resp. ν . For ﬁxed f L1(µ)+ and every t > 0 set A := f/ϕ2 = t . By consistency of the ϕ ψ ∈ t { } ϕ disentegration of µ over νϕ,

∞ ∞ ϕ −1 (ϕ) µ At s (B) dt = µ At dνϕ(ζ) dt, B . Z ∩ Z Z ζ ∈ Z 0 0 B whence, by the level-set representation of the Lebesgue integral and Tonelli’s Theorem

[ϕ] (3.19) f dµ = f dµζ dνϕ(ζ) . Zs−1(B) ZB ZX

Note that the left-hand side does not depend on ϕ. Therefore, equating (3.19) with the same representation for ψ and using Claim 1 yields

[ϕ] [ψ] dνψ [ψ] f dµζ dνϕ(ζ)= f dµζ dνψ(ζ)= (ζ) f dµζ dνϕ(ζ) , ZB ZX ZB ZX ZB dνϕ ZX and the conclusion follows, since f and B were arbitrary.

3.4. Some examples. Here, we specialize the results in the previous sections to some particular cases.

In order to discuss the next example, we shall need the deﬁnition of 1-capacity capE of a regular Dirichlet form (E, D(E)) on a locally compact Polish space (X, τ), for which we refer the reader to [15,

§2.1]. We recall that capE is a Choquet capacity on X in the sense of e.g. [15, §A.1], see e.g. [15, Thm. 2.1.1]. Finally, we say that A X is E-capacitable if ⊂

capE(A) = sup capE(K) , K∈Kτ :K⊂A where Kτ denotes the family of all τ-compact subsets of X. ERGODIC DECOMPOSITION OF DIRICHLET FORMS 31

Example 3.12 (Ergodic decomposition of forms on product spaces). Let X = Y Z be a product × of locally compact Polish spaces endowed with a probability (hence Radon) measure µ, and (µζ )ζ∈Z be a disintegration of µ over ν := prZ µ strongly consistent with the standard projection prZ : X Z. ♯ → This includes the setting of Example 2.32. Indeed, let (Eζ , D(Eζ )) be regular irreducible Dirichlet forms on L2(µ ), all with common core (Y ), and assume that ζ (E , D(E )) is a ν-measurable field ζ C⊂C0 7→ ζ ζ of quadratic forms in the sense of Definition 2.11 with underlying ν-measurable field S = SC. Then, it is readily verified that

(i) the direct integral (E, D(E)) of quadratic forms ζ (E , D(E )) is a direct integral of Dirichlet 7→ ζ ζ forms; (ii) (E, D(E)) is a regular Dirichlet form on L2(µ) with core (Z) and semigroup C⊗C0

(T u)(y, ζ)= T id 2 u (y, ζ)= T u( , ζ) (y); t ζ,t ⊗ L (ν) ζ,t · (iii) if A Z is ν-measurable and U Y is E -capacitable for every ζ Z, then U A X satisﬁes ⊂ ⊂ ζ ∈ × ⊂

capE(U A) capEζ (U) dν(ζ) . × ≤ ZA

As a further example, we state here the ergodic decomposition theorem for mixed Poisson measures on the conﬁguration space over a connected Riemannian manifold. We refer the reader to [2] for the main deﬁnitions.

Example 3.13 (Mixed Poisson measures, [2]). Let (M,g) be a Riemannian manifold with infinite volume, and σ = ρ vol be a non-negative Borel measure on M with density ρ > 0 vol -a.e., and · g g satisfying ρ1/2 W 1,2(M). Let further Γ be the configuration space over M, endowed with the vague ∈ loc M topology and the induced Borel σ-algebra, and denote by πσ the Poisson measure on ΓM with intensity measure σ. Let now λ be a Borel probability measure on R :=(0, ) with finite second moment. The + ∞ mixed Poisson measure with intensity measure σ and Lévy measure λ is the measure

µλ,σ := πsσ dλ(s) . ZR+

In [2], Albeverio, Kondratiev and Röckner construct a canonical Dirichlet form , D( ) Eµλ,σ Eµλ,σ 2 on L (µλ,σ) and show that

• , D( ) is quasi-regular strongly local, [2, Thm. 6.1]; Eµλ,σ Eµλ,σ • , D( ) is irreducible if and only if λ = δ , i.e. µ = π , for some s 0, [2, Thm. 4.3]; Eµλ,σ Eµλ,σ s λ,σ sσ ≥ • π π for all r, s 0, r = s. sσ ⊥ rσ ≥ 6 Applying Theorem 3.6 to the form , D( ) yields the direct-integral representation Eµλ,σ Eµλ,σ ⊕

µλ,σ = πsσ dλ(s) , E ZR+ E where (Z, ,ν) = (R , B(R ), λ), and the disintegration of µ constructed in the theorem coincides Z + + λ,σ with (π ) . sσ s∈R+

Remark 3.14. Other examples are given by [1, Thm. 3.7] and [3], both concerned with strongly local Dirichlet forms on locally convex topological vector spaces, and by [10], concerned with a particular quasi-regular Dirichlet form on the space of probability measures over a closed Riemannian manifold. 32 L. DELLO SCHIAVO

3.5. Some applications. We collect here some applications of the direct-integral decomposition discussed in the previous sections.

Transience/recurrence. Let (X, τ, ,µ) be satisfying Assumption 2.18. For an invariant set A , we X ∈ X0 denote by µA the restriction of µ to A, and by (EA, D(EA)) the Dirichlet form

D(E ) := 1 f : f D(E) , E (f,g) := E(1 f, 1 g) , A { A ∈ } A A A

2 well-defined on L (µA) as a consequence of Definition 3.1(d). The next result is standard. In the generality of Assumption 2.18, a proof is readily deduced from the corresponding result for µ-tight Borel right processes, shown with different techniques by K. Kuwae in [20, Thm. 1.3], in the far more general setting of quasi-regular semi-Dirichlet forms.

Corollary 3.15 (Ergodic decomposition: transience/recurrence). Under the assumptions of Theo- rem 3.6, there exist E-invariant subsets Xc, Xd, and a properly E-exceptional subset N of X, so that

(i) X = X X N; c ⊔ d ⊔ (ii) the restriction (Ed, D(Ed)) of (E, D(E)) to Xd is transient;

(iii) the restriction (Ec, D(Ec)) of (E, D(E)) to Xc is recurrent.

As an application, we have the following proposition. Similarly to Remark 3.10, some implications hold for superpositions of arbitrary Dirichlet forms.

Proposition 3.16. Let (X, τ, ,µ) be a topological measure space satisfying Assumption 2.18, and X (Z, ,ν) be σ-ﬁnite countably generated. Further let (µ ) be a separated pseudo-disintegration of µ Z ζ ζ∈Z over ν, and (E, D(E)) be a direct integral of quasi-regular Dirichlet forms ζ (E , D(E )) on L2(µ ). 7→ ζ ζ ζ Then,

(i) (E, D(E)) is conservative if and only if (E , D(E )) is conservative for ν-a.e. ζ Z; ζ ζ ∈ (ii) (E, D(E)) is transient if and only if (E , D(E )) is transient for ν-a.e. ζ Z. Furthermore, one ζ ζ ∈ has the direct-integral representation of Hilbert spaces

⊕ (3.20) D(E)e = D(Eζ )e dν(ζ); ZZ (iii) if (E, D(E)) is recurrent, then (E , D(E )) is recurrent for ν-a.e. ζ Z. In the situation of Theo- ζ ζ ∈ rem 3.4 or Theorem 3.11, the converse implication holds as well.

Proof. Analogously to the proof of Theorem 3.6 we may restrict to the regular case by the transfer method. Thus we can and will assume with no loss of generality that (X, τ, µ, µˆ) is a locally compact X Polish Radon measure space with full support, and (E, D(E)) is a direct integral of Dirichlet forms ζ 7→ (E , D(E )) with underlying space of ν-measurable vector ﬁelds S generated by a core for (E, D(E)). ζ ζ C C By this assumption, the form (E, D(E)) and all forms (E , D(E )) are regular, with common core . ζ ζ C Analogously to the proof of Theorem 3.4, if u , then we may choose u as a representative for u , ∈ C ζ thus writing u in place of u for every ζ Z and every u . Without loss of generality, possibly up to ζ ∈ ∈C enlargement of , we may assume that is special standard (e.g., [15, p. 6]). In particular, is a lattice. C C C (i) Let (u ) (τ) with 0 u 1 and so that lim u 1 locally uniformly on (X, τ), and note n n ⊂Cc ≤ n ≤ n n ≡ that lim u = 1 both µ- and µ -a.e. for every ζ Z. By the direct-integral representation (2.8) of T n n ζ ∈ • ERGODIC DECOMPOSITION OF DIRICHLET FORMS 33 and by (2.19),

f L1(µ) L2(µ) , fTtun dµ = fζ Tζ,tun dµζ dν(ζ) , ∈ ∩ t> 0 . ZX ZZ ZX f = (ζ f ) , 7→ ζ Letting n to inﬁnity, it follows by several applications of the Dominated Convergence Theorem that

f L1(µ) L2(µ) , (3.21) fTt1 dµ = fζ Tζ,t1 dµζ dν(ζ) , ∈ ∩ t> 0 . ZX ZZ ZX f = (ζ f ) , 7→ ζ

Now, assume that T• is not conservative and argue by contradiction that Tζ,• is conservative for ν-a.e. ζ Z. Then, choosing f L1(µ) L2(µ), with f > 0 µ-a.e., in (3.21), ∈ ∈ ∩

f dµ> fTt1 dµ = fζ Tζ,t1 dµζ dν(ζ)= fζ 1 dµζ dν(ζ) ZX ZX ZZ ZX ZZ = f dµ , ZX a contradiction. The reverse implication follows from (3.21) in a similar way.

(ii) Assume (E , D(E )) is transient for every ζ N c for some ν-negligible N Z. That is, D(E ) ζ ζ ∈ ⊂ ζ e is a Hilbert space with inner product E for every ζ N c. By (the proof of) [15, Lem. 1.5.5, p. 42], the ζ ∈ space D(E ) is E1/2-dense in D(E ) for every ζ N c. Thus, the space of ν-measurable vector ﬁelds S ζ ζ ζ e ∈ C is underlying to each of the direct integrals

SC ⊕ D(E)1 = D(Eζ )1 dν(ζ) , ZZ SC ⊕ Fe := D(Eζ )e dν(ζ) , ZZ SC ⊕ 2 2 L (µ)= L (µζ ) dν(ζ) . ZZ In particular, there exists a sequence (u ) simultaneously D(E) -, F - and L2(µ)-fundamental in n n ⊂ C 1 e 2 the sense of Deﬁnition 2.3. Denote by ιζ,e the identity of L (µζ ), regarded as the continuous embedding ι : D(E ) D(E ) and note that ζ,e ζ 1 → ζ e

ζ ιζ,eun um D = Eζ (un,um)= un um D un um 2 7→ h | i (Eζ)e h | i (Eζ)1 − h | iL (µζ ) is ν-measurable for every n,m. By [12, §II.1.4, Prop. 2, p. 166] this implies that ζ ι is a ν-measurable 7→ ζ,e ﬁeld of bounded operators. Writing

⊕ ιe : D(E)1 Fe , u ιζ,euζ dν(ζ) , → 7→ZZ and arguing as in the proof of Proposition 2.13, the map ιe is injective, and thus it is a continu- D D ous embedding of (E)1 into the Hilbert space (Fe, E). As a consequence, K := clFe ιe (E)1 is a

Hilbert space with scalar product E. By deﬁnition of D(E)e, the identity map ιe is a continuous embedding (D(E) , E) (K, E) (F , E). In particular, (D(E) , E) is a Hilbert space, and the form (E, D(E)) e ⊂ ⊂ e e is transient by [15, Thm. 1.6.2, p. 58].

Claim: D(E)e = K = Fe and (3.20) holds. By (the proof of) [15, Lem. 1.5.5, p. 42], the space D(E) is E1/2-dense in D(E) , thus the same holds for . It suﬃces to show that is E1/2-dense in F as well. e C C e 34 L. DELLO SCHIAVO

We denote by ⊥ the E-orthogonal complement of in F , resp. by ⊥ζ the E -orthogonal complement C C e C ζ of in D(E ). By assumption, ⊥ζ = 0 for every ζ N c. Finally, by the direct-integral representation C ζ C { } ∈ of Fe,

⊕ ⊥ = ⊥ζ dν(ζ)= 0 , C ZZ C { } similarly to the proof of the Claim in Step 4 of Theorem 3.4.

We say that u, v are E-equivalent if E(u v)=0, and we write u v. Let the analogous deﬁnition ∈C − ∼ for u v be given. By the direct-integral representation (2.24) of (E, D(E)), it is readily seen that ∼ζ (3.22) u v u v for ν-a.e. ζ Z. ∼ ⇐⇒ ∼ζ ∈

Assume now that (E, D(E)) is transient. That is, D(E)e is a Hilbert space with inner product E. It 1/2 suﬃces to show (3.20). Since E is non-degenerate on D(E)e, it is non-degenerate on , thus u v 1/2 C 1/2 ∼ if and only if u = v µ-a.e. By (3.22), Eζ is non-degenerate on for ν-a.e. ζ Z, thus ( , Eζ ) is a C ∈ C 1/2 pre-Hilbert space for ν-a.e. ζ Z. For each ζ Z denote by Kζ the abstract completion of ( , Eζ ), ∈ ∈1/2 C endowed with the non-relabeled extension of Eζ . It is a straightforward veriﬁcation that there holds the direct-integral representation

SC ⊕ (3.23) D(E)e = Kζ dν(ζ) . ZZ By definition of D(E ) , the completion embedding ι : K extends to a setwise injection ¯ι : D(E ) ζ e ζ C → ζ ζ ζ e → K . Indeed, let u D(E ) and (u ) be its approximating sequence. Since (u ) is, by definition, ζ ζ ∈ ζ e n n ⊂C n n E1/2-Cauchy, it converges to some h K by completeness of K . Set ¯ι (u ) := h , and note that the ζ ζ ∈ ζ ζ ζ ζ ζ 1/2 D definition is well-posed since Eζ is a norm in Kζ . Thus, (Eζ )e, identified with a subset of Kζ via ¯ιζ , D 1/2 is a pre-Hilbert space with scalar product Eζ , and in fact it holds that (Eζ )e = Kζ by Eζ -density of in K . We note that equality D(E ) = K is not a mere isomorphism of Hilbert spaces, but rather C ζ ζ e ζ an extension of the completion embedding ι , thus preserving the lattice property of regarded as a ζ C subspace of both D(Eζ )e and Kζ. Together with (3.23), this shows (3.20).

(iii) Assume (E, D(E)) is recurrent. By [15, Thm. 1.6.3, p. 58] there exists a sequence (u ) D(E), n n ⊂ so that lim u (x)=1 for µ-a.e. x X, and lim E(u )=0. By the Markov property for (E, D(E)) n n ∈ n n we may assume that u [0, 1]. By regularity of (E, D(E)), we may assume that (u ) + (τ). n ∈ n n ⊂ C ⊂ C0 By e.g. [19, Prop. 3.1(iii), p. 690], E(u v) E(u)+ E(v), thus, up to passing to a suitable non- ∨ ≤ relabeled subsequence, we may assume that (u ) is monotone non-decreasing. Then, lim u 1 τ- n n n n ≡ locally uniformly on supp[µ] = X by Dini’s Theorem, and therefore lim u (x)=1 for µ -a.e. x X n n ζ ∈ for every ζ Z. By the direct integral representation (2.24), it is readily seen arguing by contradiction ∈ that lim E (u )=0 for ν-a.e. ζ Z. As a consequence, (E , D(E )) is recurrent for ν-a.e. ζ Z, again n ζ n ∈ ζ ζ ∈ by [15, Thm. 1.6.3, p. 58].

Suppose now that (E, D(E)) is given as the direct integral of Dirichlet forms in Theorem 3.4, and assume that (E , D(E )) is recurrent for ν-a.e. ζ Z. We show that (E, D(E)) is recurrent. A proof in ζ ζ ∈ the setting of Theorem 3.11 is nearly identical, and therefore it is omitted. Recall the notation in §3.1 and argue by contradiction that (E, D(E)) is not recurrent. By Corol- lary 3.15, there exists an E-invariant subset Xd, with µXd > 0, so that (Ed, D(Ed)) is transient. ERGODIC DECOMPOSITION OF DIRICHLET FORMS 35

Since is µ-essentially countably generated by ∗, we may and shall assume without loss of gener- X0 X ality that X ∗, so that B := s(X ) . Since µX > 0, we have νB > 0. It is not diﬃcult to show d ∈ X d ∈ Z d that the direct-integral decomposition of L2(µ) splits as a direct sum of Hilbert spaces

⊕ ⊕ 2 2 2 2 2 L (µ) = L (µX ) L (µXc ) = L (µζ ) dν(ζ) L (µζ ) dν(ζ) . ∼ d d ∼ ⊕ ZB ⊕ZBc

Since Xd is E-invariant, a corresponding direct-integral decomposition of D(E) is induced by Corol- lary 3.15

⊕ ⊕ D D D D D (E)1 ∼= (Ed)1 (Ec)1 ∼= (Eζ )1 dν(ζ) (Eζ )1 dν(ζ) . ⊕ ZB ⊕ZBc

Applying the reverse implication in (ii), it follows from the transience of (Ed, D(Ed)) that D(Eζ , D(Eζ )) is transient for ν-a.e. ζ B. Since νB > 0, this contradicts the assumption. ∈

Ergodic decomposition of measures. Let (X, τ, ,µ) be a locally compact Polish probability space. X Since (X, ) is a standard Borel space, the space M of all σ-ﬁnite measures on (X, ) is a standard X X Borel space as well when endowed with the σ-algebra generated by the family of sets

η M : a <ηA

M := Ω, , (Mt) , (Px) , ξ F t≥0 x∈X∂ be the properly associated right process. We set

p (x, A) := P ω Ω: M (ω) A , x X , t 0 , A . t x { ∈ t ∈ } ∈ ∂ ≥ ∈ X∂

The semigroup T• of (E, D(E)) is thus well-deﬁned on bounded Borel measurable functions, by letting

T : (R) (R) t Xb −→ Xb , t 0 . f f(y) pt( , dy) ≥ 7−→ ZX ·

Definition 3.17. We say that a σ-ﬁnite measure η on (X, ) is T -invariant if X •

Ttf dη = f dη, f b(R) , t 0 . ZX ZX ∈ X ≥

An invariant measure η is T•-ergodic if every E-invariant set is either η-negligible or η-conegligible. We denote by Minv, resp. Merg, the set of all σ-ﬁnite T•-invariant, resp. T•-ergodic, measures.

The formulation of the following result is adapted from [6, Thm. 1]. In light of Corollary 3.15, we may restrict to the case of recurrent Dirichlet forms.

Corollary 3.18. Let (X, τ, ,µ) be a probability space satisfying Assumption 2.18, and let (E, D(E)) X be a recurrent quasi-regular Dirichlet form on L2(µ). Then, there exists a properly E-coexceptional subset X of X, and a surjective map π : X M so that inv inv → erg (i) for every λ M the set π−1(λ) is λ-conegligible; ∈ erg 36 L. DELLO SCHIAVO

(ii) for every η M , ∈ inv

η = λ dη(λ) , η := π♯η ; ZMerg