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APPEARED IN BULLETIN OF THE

arXiv:math/9204224v1 [math.CV] 1 Apr 1992RESEARCH ANNOUNCEMENTAPPEARED IN BULLETIN OF THEAMERICAN MATHEMATICAL SOCIETYVolume 26, Number 2, April 1992, Pages 245-252A GENERAL CORRESPONDENCE BETWEENDIRICHLET FORMS AND RIGHT PROCESSESSergio Albeverio and Zhi-Ming Ma1. IntroductionThe theory of Dirichlet forms as originated by Beurling-Deny and developedparticularly by Fukushima and Silverstein, see e.g.

[Fu3, Si], is a natural functionalanalytic extension of classical (and axiomatic) potential theory. Although someparts of it have abstract measure theoretic versions, see e.g.

[BoH] and [ABrR],the basic general construction of a Hunt process properly associated with the form,obtained by Fukushima [Fu2] and Silverstein [Si] (see also [Fu3]), requires the formto be defined on a locally compact separable space with a Radon measure m andthe form to be regular (in the sense of the continuous functions of compact supportbeing dense in the domain of the form, both in the supremum norm and in thenatural norm given by the form and the L2(m)-space). This setting excludes infinitedimensional situations.In this letter we announce that there exists an extension of Fukushima-Silverstein’sconstruction of the associated process to the case where the space is only supposedto be metrizable and the form is not required to be regular.

We shall only sum-marize here results and techniques, for details we refer to [AM1, AM2]. Before westart describing our results let us mention that some work on associating strongMarkov processes to nonregular Dirichlet forms had been done before, by findinga suitable representation of the given nonregular form as a regular Dirichlet formon a suitable compactification of the original space.

In an abstract general settingthis was done by Fukushima in [Fu1]. The case of local Dirichlet forms in infinitedimensional spaces, leading to associated diffusion processes, was studied originallyby Albeverio and Høegh-Krohn in a rigged Hilbert space setting [AH1–AH3], un-der a quasi-invariance and smoothness assumption on m. This work was extendedby Kusuoka [Ku] who worked in a Banach space setting.

Albeverio and R¨ockner[AR¨O1–4] found a natural setting in a Souslin space, dropping the quasi-invarianceReceived by the editors May 23, 1990 and, in revised form, March 19, 19911980 Mathematics Subject Classification (1985 Revision). Primary 31C25; Secondary 60J45,60J25, 60J40, 60J35c⃝1992 American Mathematical Society0273-0979/92 $1.00 + $.25 per page1

2SERGIO ALBEVERIO AND ZHI-MING MAassumption. They also derived the stochastic equation satisfied by the process (forquasi–every initial condition) (the ultimate result in this direction is contained in[AR¨O5], where also a compactification procedure by Schmuland [Sch] and a tight-ness result of Lyons-R¨ockner [LR] are used).

For further results in infinite dimen-sional local Dirichlet forms see also [BoH, So1, So2, FaR, Tak, Fu4]. The converseprogram of starting from a “good” Markov process and associating to it a, nonnec-essarily regular, Dirichlet form has been pursued by Dynkin [D1, D2], Fitzsimmons[Fi1], [Fi2], Fitzsimmons and Getoor [FG], Fukushima [Fu5], and Bouleau-Hirsch[BoH].

For further work on general Dirichlet forms see also Dellacherie-Meyer [DM],Kunita-Watanabe [KuW], and Knight [Kn].Our approach differs from all the above treatments, in that we construct directlya strong Markov process starting from a given Dirichlet form without the assump-tion of regularity.In fact we manage to extend the construction used in [Fu3,Chapter 6], to the nonregular case. By so doing we obtain necessary and sufficientconditions for the existence of a certain right process, an m-perfect process as ex-plained below, properly associated with the given, regular or nonregular, Dirichletform.

Our construction relies on a technique we have developed in [AM1] to asso-ciate a quasi-continuous kernel, in a sense explained below, to a given semigroup(this is related to previous work by Getoor [Ge1] and Dellacherie-Meyer [DM]).Our method of construction of the process is related to some work by Kaneko [Ka]who constructed Hunt processes by using kernels which are quasi-continuous withrespect to a Cr,p-capacity. We also mention that our work provides an extension ofa result by Y. Lejan, who obtained in [Le1, Le2] a characterization of semigroupsassociated with Hunt processes: in fact Lejan’s work [Le1, Le2] provided an essen-tial idea for our proof of the necessity of the condition (ii) in the main theorembelow.2.

Main resultsWe shall now present briefly our main results. Let X be a metrizable topologicalspace with the σ-algebra X of Borel subsets.

A cemetery point ∆/∈X is adjoinedto X as an isolated point of X∆≡X∪{∆}. Let (Xt) = (Ω, M, Mt, Xt, Θt, Px) be astrong Markov process with state space (X∆, X∆) and life time ζ ≡inf{t ≥0|Xt =∆}, where, as in the usual notations of e.g.

[BG], (Ω, M) is a measurable space,(Mt, t ∈[0, ∞]) is an increasing family of sub σ-algebras of M, Θt being the shiftand Px being the “start measure” (i.e. the measure for the paths conditioned tostart at x) on (Ω, M), for each x ∈X∆.

We denote by (Pt) the transition functionof (Xt) and by (Rα), the resolvent of (Xt), i.e.Ptf(x) = Ex[f(Xt)]andRαF(x) = ExZ ∞0e−αtf(Xt) dt,where x ∈X, Ex being the expectation with respect to Px, for all functions f forwhich the right-hand sides make sense.We call Xt a perfect process if it satisfies the following properties:(i) Xt has the normal property: Px(X0 = x) = 1, ∀x ∈X∆;(ii) Xt is right continuous: t →Xt(ω) is a right continuous function from [0, ∞)to X∆, Px a.s., for all x ∈X∆;

DIRICHLET FORMS AND RIGHT PROCESSES3(iii) Xt has left limits up to ζ : lims↑t Xs(ω) := Xt−(ω) exists in X, for allt ∈(0, ζ(ω)), Px-a.s., ∀x ∈X;(iv) Xt has a regular resolvent in the sense that R1f(Xt−)I{t<ζ} is Px-indistin-guishable from (R1f(Xt))−I{t<ζ} for all x ∈X and for all f in the space bXof all bounded X-measurable functions. We have set (R1f(Xt))−I{t<ζ} :=lims↑t R1f(Xs)I{t<ζ} (where we always make the convention that Z0−= Z0for any process Zt, t ≥0).Remarks.

(i) A strong Markov process satisfying only (i), (ii) will be called a rightprocess with Borel transition semigroup.If X is a Radon space this definitioncoincides with the one in [Sh, Definition (8.1)] and [Ge2, (9.7)]. (ii) A special standard process in the sense of [Ge2, (9.10)] and, in particular,a Hunt process (cf.

e.g. [BG]), always satisfy (iii), (iv), hence they are particularperfect processes (see [Ge2 (7.2), (7.3)], [Sh (4.7.6) and (9.7.10)] for the proof).Thus we have the following inclusions, cf.

[Ge2, p. 55]:(Feller) ⊂(Hunt) ⊂(special standard) ⊂(perfect) ⊂(right) .In what follows we shall assume that m is a σ-finite Borel measure on X. Aprocess (Xt) is called m-tight if there exists an increasing sequence of compact sets{Kn}, n ∈N of X such thatPx{limn σX−Kn ≥ζ} = 1,m-a.e. x ∈X,where for any subset A of X∆, σA := inf{t > 0: Xt ∈A} is the hitting time of A.A process (Xt) is called an m-perfect process if it is a perfect process and ism-tight.

It follows from an idea of T. J. Lyons and M. R¨ockner [LR] that anystrong Markov process (Xt) in a metrizable Lusin space is m-tight if it satisfies (ii),(iii), see [AMR]. Thus in a metrizable Lusin the concepts of perfect process andm-perfect process coincide.

(In the special case of (Xt) being a standard processon a locally compact metrizable space the above conclusion that (Xt) is m-tightcan also be derived from [BG, (9.3)]. )Let us now give the correlates of above definitions for Dirichlet forms.

Let (E, F)be a Dirichlet form on L2(X, m), i.e. E is a positive, symmetric, closed bilinear formon L2(X, m) such that unit contractions operate on E (i.e.

E(f #, f #) ≤E(f, f),f # = (f ∨0) ∧1, all f in the definition domain F of E). We set as usual E1(f, g) ≡E(f, g) + (f, g), ∀f, g ∈F, where (f, g) is the L2(X, m)-scalar product of f and g.In the sequel we always regard F as a Hilbert space equipped with the innerproduct E1.

For any closed subset F ⊂X, we setFF ≡{f ∈F : f = 0 m-a.e. on X −F},FF is then a closed subset of F.The following definitions are extensions of corresponding definitions in [Fu3].

Anincreasing sequence of closed sets {Fk}k≥1 is called an E-nest if Sk FFk is E1-densein F. A subset B ⊂X is said to be E-polar if there exists an E-nest {Fk} such thatB ⊂\k(X −Fk) .

4SERGIO ALBEVERIO AND ZHI-MING MAA function f on X is said to be E-quasi-continuous if there exists an E-nest {Fk}such that f|Fk, the restriction of f to Fk, is continuous on Fk for each k.It is not difficult to show that every E-polar set is m-negligible. We denote byTt resp.

Gα the semigroup resp. resolvent on L2(X, m) associated with (E, F); i.e.,if A is the generator of Tt,√−Af,√−Ag= E(f, g)∀f, g ∈F = D√−A.(cf.

[Fu3]). We setH = {h: h = G1f with f ∈L2(X; m), 0 < f ≤1m-a.e.

}We remark that H is nonempty because we assumed m to be σ-finite. For h ∈Hwe define the h-weighted capacity Caph as followsCaph(G) := inf{E1(f, f): f ∈F, f ≥hm-a.e.

on G},for any open subset G of X, andCaph(B) := inf{Caph(G): G ⊃B, G open}for any arbitrary subset B ⊂X. It is possible to show, see [AM2], that Caph is aChoquet capacity enjoying the important property of countable subadditivity.

Therelation between this notion of capacity and the notion of E-nest is expressed bythe following proposition:Proposition. An increasing sequence of closed subsets {Fk} of X is an E-nest ifand only if for some h ∈H (hence for all h ∈H) one hasCaph(X \ Fk) ↓0as k →∞.For the proof we refer to Proposition 2.5 of [AM2].Denote by Cap the usual 1-capacity given by E, cf.

[Fu3]. One has obviouslyCaph(B) ≤Cap(B) for every B ⊂X.

Hence the following corollary holds:Corollary. Every set B ⊂X with Cap B = 0 is an E-polar set.Every nest{Fk} resp.

every quasi-continuous function in the sense of [Fu3] is an E-nest resp.an E-quasi-continuous function in our sense (we remark that [Fu3] the space X issupposed to be locally compact separable and m to be supported by X and Radon).Now let (Xt) be a Markov process with transition function Pt. We say that(Xt) is associated with E if Ptf = Ttf m-a.e.

for all f ∈L2(X, m), t > 0, and itis properly associated with E if Ptf is an E-quasi-continuous version of Ttf for allf ∈L2(X, m), t > 0. The main result we obtain is the followingTheorem.

Let (E, F) be a Dirichlet form on L2(X; m). Then the following familyof conditions (i)–(iii) is necessary and sufficient for the existence of an m-perfectprocess (Xt) associated with E:(i) there exists an E-nest {Xk} consisting of compact subsets of X;(ii) thereexistsanE1-densesubsetF0ofFconsistingofE-quasi-continuous functions;(iii) there exists a countable subset B0 of F0 and an E-polar subset N such thatσ{u: u ∈B0} ⊃B(X) ∩(X −N) .

Dirichlet forms and right processes5Moreover, if an m-perfect process (Xt) is associated with E, then it is alwaysproperly associated with E.Remarks. (i) If E is a regular Dirichlet form in the sense of [Fu3] then all conditionsare satisfied.

But the regularity assumption on the Dirichlet form E usually assumedin the literature, cf. [Fu3, Si], is not necessary for the existence of an m-perfectprocess (it is even not necessary for the existence of a diffusion process).

Assumein fact each single point set of X is a set of zero capacity (e.g. X = Rd, d ≥2,E the classical Dirichlet form associated with the Laplacian on Rd).

Let µ be asmooth measure (in the sense of [Fu3]), which is nowhere Radon (i.e. µ(G) = ∞forall nonempty open subsets G ⊂X): the existence of such nowhere Radon smoothmeasures has been proven in [AM4] (see also [AM3, AM5]).We consider the perturbed form (Eµ, Fµ) defined as follows:Fµ := F ∩L2(X; m);Eµ(f, g) := E(f, g) +ZXfgµ(dx)∀f, g ∈Fµ .It has been proven in [AM5, Proposition 3.1] that (Eµ, Fµ) is again a Dirichlet form.One can check that (Eµ, Fµ) satisfies all conditions in the theorem, see [AM7],hence the theorem is applicable and there exists an m-perfect process associatedwith (Eµ, Fµ).

Moreover, if (E, F) satisfies the local property, then so does (Eµ, Fµ)and hence there exists a diffusion process associated with (Eµ, Fµ) (see (ii) below).But clearly (Eµ, Fµ) is not regular, in fact there is even no nontrivial continuousfunction belonging to Fµ, since µ is nowhere Radon. See [AM7] for details.

(ii) An m-perfect process is a diffusion (i.e. Px{Xt is continuous in t ∈[0, ζ)} = 1,for q.e.

x ∈X}) if and only if the associated Dirichlet form (E, F) satisfies the localproperty (in the sense that E(f, g) = 0 if supp f ∩supp g = ∅), see [AM6]. (iii) By requiring F0 in (ii) to consist of strictly E-quasi-continuous functionswe obtain a necessary and sufficient condition for the existence of a Hunt processassociated with (E, F), see [AM8].

(iv) By introducing a dual h-weighted capacity and employing a Ray-Knightcompactification method it is possible to extend the above theorem to nonsymmetricDirichlet forms satisfying the sector condition. (v) Applications of the above theorem to infinite dimensional spaces X are inpreparation.

They allow in particular to construct infinite dimensional processeswith discontinuous sample paths, with applications to certain systems with infin-itely many degrees of freedom.AcknowledgmentsWe are very indebted to Masatoshi Fukushima, who greatly encouraged our work,and carefully read the first version of the manuscripts of [AM1–2] suggesting manyimprovements. We are also very grateful to H. Airault, J. Brasche, P. J. Fitzsim-mons, R. K. Getoor, W. Hansen, M. L. Silverstein, S. Watanabe, R. Williams, J. A.Yan, Zhang, and especially Michael R¨ockner for very interesting and stimulatingdiscussions.

We also profited from meetings in Braga and Oberwolfach and aregrateful to M. De Faria, L. Streit resp., H. Bauer, and M. Fukushima for kind invi-tations. Hospitality and/or financial support by BiBoS, A. von Humboldt Stiftung,DFG, and Chinese National Foundation is also gratefully acknowledged.

6SERGIO ALBEVERIO AND ZHI-MING MANote added in Proof. After we finished this work we learned from P. J. Fitzsimmonsthat every (nearly) m-symmetric right process is an m-special standard process (see[Fi1]), and our notion of perfect process is equivalent to a special standard process.Thus our theorem gives in fact a general correspondence between Dirichlet formsand right processes.

We are most grateful to P. J. Fitzsimmons for his remark.References[AH1]S. Albeverio and R. Høegh-Krohn, Quasi-invariant measures, symmetric diffusionprocesses and quantum fields, Les m´ethodes math´ematiques de la th´eorie Quantiquede Champs, Colloques Internationaux du C.N.R.S., no. 248, Marseille, 23–27 juin 1975,C.N.R.S., 1976.

[AH2], Dirichlet forms and diffusion processes on rigged Hilbert spaces, Z. Wahrsch.verw. Gebiete 40 (1977), 1–57.

[AH3], Hunt processes and analytic potential theory on rigged Hilbert spaces, Ann.Inst. Henri Poincar´e, 13 (3), (1977), 269–291.[ABrR]S.

Albeverio, J. Brasche, and M. Røckner, Dirichlet forms and generalized Schr¨odingeroperators, Proc. Sønderborg Conf., Schr¨odinger operators (H. Holden, A. Jensen, eds.

),Lecture Notes Phys., vol. 345, Springer-Verlag, Berlin, 1989, pp.

1–42. [AHPRS1] S. Albeverio, T. Hida, J. Potthoff, M. R¨ockner, and L. Streit, Dirichlet forms interms of white noise analysis.

I, Construction and QFT examples, Rev. Math.

Phys.1 (1990), 291–312. [AHPRS2], Dirichlet forms in terms of white noise analysis II, Closability and diffusionprocesses, Rev.

Math. Phys.

1 (1990), 313–323.[AK]S. Albeverio and S. Kusuoka, Maximality of infinite dimensional Dirichlet forms andHøegh-Krohn’s model of quantum fields, in Memorial Volume for Raphael Høegh-Krohn, Ideas and Methods in Mathematical Analysis, Stochastics, and Applications,vol.

II (S. Albeverio, J.-E.Fenstad, H. Holden, T. Lindstrom, Cambridge Univ. Press(to appear).[AKR]S.

Albeverio, S. Kusuoka, and M. R¨ockner, On partial integration in infinite dimen-sional space and applications to Dirichlet forms, J. London Math. Soc.

42 (1990),122–136.[AM1]S. Albeverio and Z. M. Ma, A note on quasicontinuous kernels representing quasi-linear positive maps, Forum Math.

3 (1991), 389–400. [AM2], Necessary and sufficient conditions for the existence of m-perfect processesassociated with Dirichlet forms, S´em.

Probabilities, Lect. Notes in Math., Springer,Berlin, (1991) (to appear).

[AM3], Nowhere Radon smooth measures, perturbations of Dirichlet forms and sin-gular quadratic forms, Proc. Bad Honnef Conf.

1988, (N. Christopeit, K. Helmes,and M. Kohlmann eds), Lecture Notes Control and Information Sciences, vol. 126,Springer-Verlag, Berlin, 1989, pp.

3–45. [AM4], Additive functionals, nowhere Radon and Kato class smooth measure associ-ated with Dirichlet forms, SFB 237, preprint, to appear in Osaka J.

Math. [AM5], Perturbation of Dirichlet forms—lower semiboundedness, closability and formcores, J. Funct.

Anal. 99 (1991), 332–356.

[AM6], Local property for Dirichlet forms on general metrizable spaces (in prepara-tion). [AM7], Diffusion process associated with singular Dirichlet forms, SFB237 Preprint,Bochum; Proc.

Lisboa Conf. (A.

B. Cruzeiro, ed.

), Birkha¨user, New York, 1991 (toappear). [AM8], Characterization of Dirichlet spaces associated with symmetric Hunt pro-cesses, Bochum Preprint (1991), to appear in Proc.

Swansea Conf. Stochastic Analysis(A. Truman, ed.) 1991.[AMR]S.

Albeverio, Z. M. Ma, and M. R¨ockner, A remark on the support of Cadlag processes,,Bochum Reprint (1991). [AR ¨O1]S. Albeverio and M. R¨ockner, Classical Dirichlet forms on topological vector spaces—closability and a Cameron-Martin formula, J. Funct.

Anal. 88 (1990), 395–436.

DIRICHLET FORMS AND RIGHT PROCESSES7[AR ¨O2], Classical Dirichlet forms on topological vector spaces—the construction of theassociated diffusion process, Probab. Theory and Related Fields 83 (1989), 405–434.

[AR ¨O3], New developments in theory and applications of Dirichlet forms (with M.R¨ockner), Stochastic Processes, Physics and Geometry, Proc. 2nd Int.

Conf. Ascona-Locarno-Como 1988 (S. Albeverio, G. Casati, U. Cattaneo, D. Merlini, and R.

Moresi,eds. ), World Scient., 1989, pp.

27–76. [AR ¨O4], Stochastic differential equations in infinite dimension: solutions via Dirichletforms, Probab.

Theory and Related Fields 89 (1991), 347–386. [AR ¨O5], Infinite dimensional diffusions connected with positive generalized white noisefunctionals, Edinburgh preprint, 1990, Proc.

Bielefeld Conference, White Noise Anal-ysis (T. Hida, H. H. Kuo, J. Potthoff, and L. Streit, eds. ), pp.

1–21.[BG]R. M. Blumenthal and R. K. Getoor, Markov processes and potential theory, AcademicPress, NY, 1968.[BoH]N.

Bouleau and F. Hirsch, Propri´et´es d’absolue continuit´e dans les espaces de Dirich-let et applications aux ´equations diff´erentielles stochastiques, S´em. de Probabilit´es,Lecture Notes in Math., vol.

20, Springer-Verlag, Berlin and New York, 1986.[D1]E. B. Dynkin, Green’s and Dirichlet spaces associated with fine Markov processes, J.Funct.

Anal. 47 (1982), 381–418.

[D2], Green’s and Dirichlet spaces for a symmetric Markov transition function,Lecture Notes of the London Math. Society vol.

79, Cambridge Univ. Press, 1983, pp.79–98, Springer-Verlag, Berlin and New York, 1982.[DM]C.

Dellacherie and P. A. Meyer, Probabilit´es et potentiel, Chapters 9–16, Hermann,Paris, 1987.[Fi1]P. J. Fitzsimmons, Markov processes and nonsymmetric Dirichlet forms without reg-ularity, J. Funct.

Anal. 85 (1989), 287–306.

[Fi2], Time changes of symmetric Markov processes and a Feynman-Kac formula,J. Theoret.

Probab. 2 (1989), 487–502.[FG]P.

J. Fitzsimmons and R. K. Getoor, On the potential theory of symmetric Markovprocesses, Math. Ann.

281 (1988), 495–512.[Fu1]M. Fukushima, Regular representations of Dirichlet forms, Trans.

Amer. Math.

Soc.155 (1971), 455–473. [Fu2], Dirichlet spaces and strong Markov processes, Trans.

Amer. Math.

Soc. 162(1971), 185–224.

[Fu3], Dirichlet forms and Markov processes, North-Holland, Amsterdam, Oxfordand NY, 1980. [Fu4], Basic properties of Brownian motion and a capacity on the Wiener space, J.Math.

Soc. Japan 36 (1984), 161–175.

[Fu5], Potentials for symmetric Markov processes and its applications, Lecture Notesin Math., vol. 550, Springer-Verlag, Berlin and New York, pp.

119–133. [FaR]Fan Ruzong, On absolute continuity of symmetric diffusion processes on Banachspaces, Beijing preprint, 1989, Acta Mat.

Appl. Sinica (to appear).[Ge1]R.

K. Getoor, On the construction of kernels, Lecture Notes in Math., vol. 465,Springer-Verlag, Berlin and New York, 1986, pp.

443–463. [Ge2], Markov processes: Ray processes and right processes, Lecture Notes in Math.,vol.

440, Springer-Verlag, Berlin and New York, 1975.[Ka]H. Kaneko, On (r, p)-capacities for Markov processes, Osaka J.

Math. 23 (1986), 325–336.[Kn]F.

Knight, Note on regularization of Markov processes, Illinois J. Math.

9 (1965),548–552.[Ku]S. Kusuoka, Dirichlet forms and diffusion processes on Banach space, J. Fac.

ScienceUniv. Tokyo, Sec.

1A 29 (1982), 79–95.[KuW]H. Kunita and T. Watanabe, Some theorems concerning resolvents over locally com-pact spaces, Proc.

5th Berkeley Sympos. Math.

Stat. and Prob.

vol. II, Berkeley, 1967.[Le1]Y.

Le Jan, Quasi-continuous functions and Hunt processes, J. Math.

Soc. Japan 35(1983), 37–42.

[Le2], Dual Markovian semigroups and processes, Functional Analysis in MarkovProcesses (M. Fukushima, ed. ), Lecture Notes in Math., vol.

923, Springer-Verlag,Berlin and New York, 1982, pp. 47–75.

8SERGIO ALBEVERIO AND ZHI-MING MA[LR]T. Lyons and M. R¨ockner, A note on tightness of capacities associated with Dirichletforms, Edinburgh preprint, 1990, to appear in Bull. London Math.

Soc.[P]K. R. Parthasarathy, Probability measures on metric spaces, Academic Press, NY andLondon, 1967.[R]M.

R¨ockner, Generalized Markov fields and Dirichlet forms, Acta Appl. Math.

3(1985), 285–311.[Sch]B. Schmuland, An alternative compactification for classical Dirichlet forms on topo-logical vector spaces, Stochastics and Stoch.

Rep. 33 (1990), 75–90.[Sh]M. J. Sharpe, General theory of Markov processes, Academic Press, NY, 1988.[Si]M.

L. Silverstein, Symmetric Markov processes, Lecture Notes in Math., vol. 426,Springer-Verlag, Berlin and New York, 1974.

[So1]Song Shiqui, The closability of classical Dirichlet forms on infinite dimensional spacesand the Wiener measure, Acad. Sinica preprint, Beijing.

[So2], Admissible vectors and their associated Dirichlet forms, Acad. Sinica preprint,Beijing, 1990.[Tak]M.

Takeda, On the uniqueness of Markovian self-adjoint extensions of diffusion op-erators on infinite dimensional spaces, Osaka J. Math.

22 (1985), 733–742.CERFIM, Cas. Post.

741, CH 6900 Locarno, SwitzerlandCurrent address: Faculty of Mathematics and SFB 237, Ruhr University, D 4630 Bochum,Federal Republic of GermanyBiBos, University of Bielefeld, D 4800 Bielefeld, Federal Republic of GermanyCurrent address: Institute of Applied Mathematics, Academia Sinica, Beijing 100080, China

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