On Semi-martingale Characterizations of Functionals of Symmetric Markov Processes

For a quasi-regular (symmetric) Dirichlet space (E, F) and an associated symmetric standard process (X t , P x), we show that, for u ∈ F, the additive functional u * (X t) − u * (X 0) is a semimartingale if and only if there exists an E-nest {F n } and positive constants C n such that |E(u, v)| ≤ C n v ∞ , v ∈ F Fn,b. In particular, a signed measure resulting from the inequality will be automatically smooth. One of the variants of this assertion is applied to the distorted Brownian motion on a closed subset of R d , giving stochastic characterizations of BV functions and Caccioppoli sets.


Introduction
We consider a Hausdorff topological space X and a σ-finite positive Borel measure m on X.Let (E, F) be a (symmetric) Dirichlet form on L 2 (X; m), namely, E is a Markovian closed symmetric form with domain F linear dense in L 2 (X; m).In this paper, we follow exclusively [MR 92] for the definition of E-quasi notions.For a closed set F ⊂ X, we set

An increasing family {F
for some E-nest {F n }. 'E-quasieverywhere' or 'E-q.e.' will mean 'except for an E-exceptional set'.A function defined E-q.e. on X is said to be E-quasicontinuous if there exists an E-nest {F n } such that the restriction of u to each set F n is continuous there.When the Dirichlet form is quasi-regular in the sense of [MR 92], any u ∈ F admits an E-quasicontinuous version which will be denoted by u * .
We consider a quasi-regular Dirichlet form (E, F) and an associated standard process M = (X t , P x ) on X.A purpose of the present paper is to prove that, for u ∈ F, the additive functional(AF in abbreviation) of the process M is a semimartingale ,namely, a sum of a martingale and a process of bounded variation, if and only if there exists an E-nest {F n } and positive constants where v ∞ denotes the m-essential sup norm of v ∈ L ∞ (X; m).
The existence of a special standard process associated with a quasi-regular Dirichlet form is well known [MR 92].In the present paper, we give another construction of an associated tight special standard process as an image by a quasi-homeomorphism of a Hunt modification of a Ray process.The importance of the notion of quasi-regularity of a Dirichlet form is in that it is not only sufficient but also necessary for the existence of an associated right process which is m-special standard and m-tight ([AM 91],[MR 92]).For instance, given simply an m-symmetric right process M on a Lusin topological space X, M automatically becomes m-tight and m-special standard, and consequently the associated Dirichlet form E on L 2 (X; m) becomes quasi-regular and M can be modified outside some E-exceptional set to be a tight special standard process ([MR 92], [Fi 97]).
We note here that, when the Dirichlet space is regular in the sense of [FOT 94], the E-quasi notions of [MR 92] defined above can be identified with those classical quasi notions introduced for instance in [FOT 94] in terms of the (E 1 -)capacity.Indeed, an increasing sequence of closed sets is an E-nest iff it is a generalized nest in the sense of [FOT 94], a set is E-exceptional iff it is of zero capacity, and a function is E-quasicontinuous iff it is quasi continuous in the sense of [FOT 94] (see Lemma 2.1).We shall take those identifications for granted for the moment.
When the Dirichlet space is regular and M is an associated Hunt process, the following facts are already known ([F 80], [FOT 94,Theorem 5.4.2]): the AF A [u] for u ∈ F can be uniquely decomposed as A [u] = M [u] + N [u] , (3) where M [u] is a martingale AF of finite energy and N [u] is a continuous AF of zero energy.N [u]  needs not be of bounded variation.It is of bounded variation if and only if there exists a set function ν on X and an E-nest {F n } such that ν| Fn is a finite signed measure for each n and ν charges no E-exceptional set (such a set function ν is called a smooth signed measure with an attached E-nest {F n }) and further In this case moreover, expressing ν as a difference ν 1 − ν 2 of two positive smooth measures and denoting by A k the positive continuous AF associated with ν k by the Revuz correspondence, k = 1, 2, it holds that The above mentioned facts for a regular Dirichlet form and an associated Hunt process will be systematically extended in §5 to a general quasi-regular Dirichlet space and an associated standard process.Here we will make use of a regular representation and an associated quasi-homeomorphism of the underlying spaces as will be formulated in §2 following [F 71a], [F 71b], [FOT 94], [CMR 94].Such a method was already adopted in [MR 92] in their specific context of a (local) compactification where a quasihomeomorphism is realized by an embedding, and they called it a transfer method.But we would like to use this term in the present more general context.A Dirichlet space is called strongly regular if the associated resolvent admits a version possessing a Ray property (see §2 for precise definition).In §4, we crucially need a refined transfer method involving a strongly regular representation.
Thus we basically need to show for u ∈ F the equivalence between the inequality (2) and the existence of a signed smooth measure ν satisfying equation ( 4).This will be done in §3 and §4.While the inequality (2) follows readily from the equation ( 4), the proof of the converse implication, especially the derivation of the smoothness of a signed measure is a very delicate matter.It has been known however that a kind of strong Feller property of the resolvent of the associated Markov process M yields the desired smoothness( [CFW 93], [FOT 94,Theorem 5.4.3],[F 97a]).In §3, we shall work with a strongly regular Dirichlet space to show that this requirement can be weakened to a Ray property of the resolvent of the associated process.We shall then employ in §4 a transfer method involving a strongly regular representation of the Dirichlet space prepared in §2 to complete the proof of the desired equivalence.
The condition (2) is accordingly more easily verifiable than the existence of a signed smooth measure satisfying equation (4).Actually it is enough to require inequality (2) holding for v in a more tractable dense subspace of F b,Fn .For instance, consider the simple case that the nest is trivial: 2) is reduced to the condition that, for u ∈ F, there exists a positive constant C such that the inequality holds for any v in the space As will be seen in §6, the above condition is equivalent to the one obtained by replacing the space F b with its subspace L satisfying (L) L is an E 1 -dense linear subspace of F b , and, for any > 0, there exists a real function φ (t) such that and φ (L) ⊂ L.
We shall further see in §6 that inequality (6) holding for v ∈ L is not only sufficient but also necessary for the AF N [u] t to be of bounded variation with an additional property that where the integral inside the braces denotes the total variation of N [u] on the interval [0, t].The property (7) says that this PCAF has a finite Revuz measure.
When X is an infinite dimensional vector space and E is obtained by closing a pre-Dirichlet form defined for smooth cylindrical functions on X, we may take as L the set of all smooth cylindrical functions to check inequality (6).
When the Dirichlet space is regular, a natural choice of L is a dense subspace of C 0 (X) satifying the condition (L).In this case, inequality (6) for L is evidently equivalent to the existence of a unique finite signed measure ν satisfying the equation ( 4) for all v ∈ L. Our general theorem in §4 assures that this ν is automatically smooth, namely, it charges no set of zero E 1 -capacity.
When the Dirichlet space is not only regular but also strongly local, we shall extend in §6 the last statement to a function u ∈ F loc satisfying equation (4) for a signed Radon measure ν and for all v belonging to a more specific subspace L of F ∩ C 0 (X).We will see that this property of u is equivalent to the condition that N [u] t is of bounded variation and satisfies, for any compact set K, In §7 we will apply the last theorem of §6 to the energy forms E and the associated distorted Brownian motions M living on closed subsets of R d .In particular, we shall improve those results obtained in [F 97a],[F 97b] to complete stochastic characterizations of BV functions and Caccioppoli sets.
Finally we mention a celebrated paper [CJPS 80], in which it was already proved that, given a general Markov process M = (X t , P x ) on a general state space X, a function u on X produces a semimartingale u(X t ) under P x for every x ∈ X if and only if there exist finely open sets E n with such that u is a difference of two excessive functions on each set E n .In the present paper, we restrict ourselves to the case that M is symmetric and the semimartingale property of u(X t ) is required to hold only for E-q.e. starting point x ∈ X.As is clear from the above explanations, our necessary and sufficient conditions of the type (2) are more easily verifiable in many cases where X are of higher dimensions.

Representation and quasi-homeomorphism
We say that a quadruplet (X, m, E, F) is a Dirichlet space if X is a Hausdorff topological space with a countable base, m is a σ-finite positive Borel measure on X and E with domain F is a Markovian closed symmetric form on L 2 (X; m).The inner product in L 2 (X; m) is denoted by (•, •) X and we let We note that the space Given two Dirichlet spaces (X, m, E, F), ( X, m, Ẽ, F ), we call them equivalent if there is an algebraic isomorphism Φ from F b onto Fb preserving three kinds of metrics: One of the two equivalent Dirichlet spaces is called a representation of the other.
The underlying spaces X, X are said to be quasi-homeomorphic if there exist E-nest {F n }, Ẽ-nest { Fn } and a one to one mapping Fn such that its restricition to each F n is homeomorphic to Fn .
We say that the equivalence as above is induced by a quasi-homeomorphism if there exists a mapping q as above such that Φu(x) = u(q −1 (x)) x ∈ X 0 .
Then m is the image measure of m by q and ( Ẽ, F) is the image of (E, F) by q.Furthermore q is quasi-notion preserving ([CMR 94,Cor.3.6]): 1. Let {E n } be an increasing sequence of closed subsets of X.It is an E-nest if and only if Let us now recall three kinds of regularity of Dirichlet spaces.We call a Dirichlet space (X, m, E, F) quasi-regular if there exists an E-nest consisting of compact sets, each element in a certain E 1 dense subspace of F admits its E-quasicontinuous version and there exists an E-nest {F n } such that the points of ∪ ∞ n=1 F n are separated by a certain coutable familiy of Equasicontinuous functions belonging to F. Every element of F then admits a quasicontinuous version.
When X is locally compact, we denote by C 0 (X) (resp.C ∞ (X)) the space of continuous functions on X with compact support (resp.vanishing at infinity).We call a Dirichlet space (X, m, E, F) regular if X is a locally compact separable metric space, m is a positive Radon measure on X with full support and the space F ∩ C 0 (X) is E 1 -dense in F and uniformly dense in C 0 (X).
Such a family C 1 is said to be attached to the Ray resolvent R α (x, B).
A Dirichlet space (X, m, E, F) is called a strongly regular if X is a locally compact separable metric space, m is a positive Radon measure on X with full support and the associated L 2resolvent is generated by a Ray resolvent and a set C 1 attached to this Ray resolvent is contained in the space F ∩ C ∞ (X).Any strongly regular Dirichlet space is regular (cf.[F 71b, Remark 2.2] and [FOT 94,Lemma 1.4.2]).
We shall prove the following two theorems by combining those results in [F 71a We prepare a lemma about identifications of quasi-notions in the regular case.Suppose (X, m, E, F) is a regular Dirichlet space.The associated capacity Cap is defined for any open set A ⊂ X by and for any set B ⊂ X by In [FOT 94], 'q.e.' means 'except for a set of zero capacity'.A family {F n } of increasing closed subsets of X is then said in [FOT 94] to be a nest if and to be a generalized nest if for any compact set K ⊂ X A function u defined 'q.e.' on X is said in [FOT 94] to be quasicontinuous if for any > 0 there exists an open set A with Cap(A) < such that u| X−A is continuous.
Proof.Let M = (X t , P x ) be a Hunt process on X which is associated with the form E in the sense that the transition semigroup p t f of the process M is a version of the L 2 semigroup T t f associated with E for any non-negative Borel function f ∈ L 2 (X; m).
(i) Denote by σ E the hitting time of a set E: In view of Lemma 5.1.6 of [FOT 94], we know that an increasing sequence {F n } of closed sets is a generalized nest iff where ζ denotes the life time of M.
If ( 9) is true, then, for any bounded Borel ϕ ∈ L 2 (X; m), the function belongs to the space F Fn and converges as n → ∞ to the 1-resolvent R 1 ϕ of M m-a.e. and in E 1 -metric as well.Here we set the value of ϕ at the cemetery ∆ to be zero.Since the family Since u is quasicontinuous, it vanishes q.e.Choosing ϕ in ( 11) to be strictly positive on X, we arrive at the property (9).
for some generalized nest {F n } by virtue of (i).Then Cap(K ∩ N ) = 0 for any compact set K and hence Cap(N ) = 0. Conversely, any set N of zero capacity satisfies the inclusion (12) for a certain nest {F n } in the sense of [FOT 94] ,which is an E-nest by (i).Hence N is E-exceptional.
To see the equivalence (iii), let u be E-quasicontinuous with associated E-nest {F n }.Since {F n } is a generalized nest by (i), u is quasicontinuous in the sense of [FOT 94] on each relatively compact open subset of X, which in turn readily implies that it is quasicontinuous on X in the sense of [FOT 94].The converse implication is trivial as in the proof of (ii). 2 For a regular Dirichlet space, the notion 'q.e.' now becomes a synomym for 'E-q.e.' Further the condition ( 9) with 'q.e.' being replaced by 'E-q.e.' becomes a stochastic characterization of an E-nest.
Let L be a closed subalgebra of L ∞ (X; m) satisfying condition (L) and X be its character space.By virtue of [FOT 94,Th.A.4.1], there exists then a regular Dirichlet space with underlying space X which is equivalent to the given Dirichlet space.Such a regular Dirichlet space is called a regular representation with respect to L. [F 71a, Th.3] went further asserting that there exists a subalgebra L satisfying not only (L) but also (R), and the regular representation with respect to this L becomes strongly regular.[FOT 94,Th.A.4.1] is a reformulation of [F 71a,Th.2]just by removing the irrelevent assumption that X is locally compact and m is Radon.In the same way, [F 71a,Th.3]can be reformulated.2 Proof of Theorem 2.2 In view of [F 71b,Th.2.1] (c.f. [FOT 94,Th.A.4.2]), we see that, whenever two regular Dirichlet spaces are equivalent, then the equivalence is induced by a quasi-homeomorphism.Here, the quasi-homeomorphism was formulated by the nest defined by the associated capacities, but it is a quasi-homeomorphism in the present sense because of Lemma 2.1 (i).
On the other hand, [CMR 94,Th.3.7]shows that a Dirichlet space is quasi-regular if and only if it is equivalent to some regular Dirichlet space by means of a quasi-homeomorphism.We arrive at Theorem 2.2 by combining those two facts. 2 Corollary 2.1 If two quasi-regular Dirichlet spaces are equivalent, then the equivalence is induced by a quasi-homeomorphism.
Corollary 2.2 Any quasi-regular Dirichlet space is equivalent to a strongly regular Dirichlet space by means of a quasi-homeomorphsim.

Smoothness of signed measures for a strongly regular Dirichlet spaces
In this section, we work with a fixed strongly regular Dirichlet space (X, m, E, F).For the associated Ray resolvent R α (x, B), there exists a substochastic kernel µ(x, B) such that The set X b of all branching points is called the branch set.One can then construct a Markov process M = (X t , P x ) on X called a Ray process with resolvent R α (x, B), which is known to enjoy the following specific properties ([R 59], [KW 67]).We denote by X ∆ the one point compactification of X and put 2) The sample path X t = X t (ω) is cadlag; X t (ω) ∈ X ∆ is right continuous for all t ≥ 0, and have the left limit and to be 1-excessive if For a 1-supermedian function f , f denotes its 1-excessive regularization : Actually f is then 1-excessive and f (x) ≤ f (x) x ∈ X.We shall use the following description of the branch set(cf.[KW 67]).For the family C 1 (⊂ F ∩ C ∞ (X)) of 1-supermedian functions attached to the Ray resolvent, we set For a Borel set B ⊂ X, we let with the convention that inf φ = ∞ and we further let When B is open, p B = ṗB and it is a 1-excessive Borel measurable function.
The facts stated in the next lemma are taken from [F 71b] but we shall present alternative elementary proof of them.
Proof.(i) Take any g ∈ C 1 .Since αR α+1 g is E 1 -convergent to g ∈ F, ĝ is an E-quasicontinuous version of g.But g is continuous and hence g = ĝ E-q.e., namely, Cap(g > ĝ) = 0.By (13) and the countable subadditivity of the capacity, we arrive at (i).
(ii) Since p A (x) = 1 for all x ∈ A − X b and consequently m-a.e. on A by (M.6), the proof of [FOT 94, Lemma 4.2.1](where p A (x) = 1 ∀x ∈ A) works without any change in proving that p A is a version of e A . 2 We now proceed to the proof of a proposition which is an intermidiate but crucial step in establishing the equivalence of the inequality (2) and the exisitence of a smooth signed measure satisfying (4).
Proposition 3.1 Let u ∈ F and w ∈ F b .Suppose there exists a finite signed measure ν = ν u,w on X such that Then ν is smooth, namely, ν charges no set of zero capacity.Moreover it holds that where v * is any E-quasicontinuous version of v.
Lemma 3.2 Assume the condition of Proposition 3.1.
is uniformly bounded and v n converges to a function v ∈ F pointwise and in L 2 (X; m).Then v satisfies (14).
(ii) Fix an arbitrary non-negative v 1 ∈ F ∩ C ∞ (X) and α > 0 and let Next take any bounded Borel 1-excessive function Lemma 3.3 Assume the condition of Proposition 3.1 and denote by ν the total variation of the finite signed measure ν.
Proof.(i) We use the description (13) of the branch set.Take g On the other hand, ĝ is a bounded Borel 1-excessive function and defines the same element of F as g because αR α+1 g is E 1 -convergent to g ∈ F. Therefore, by Lemma 3.2 (i), and consequently (ii) Because of lemma 3.1 (i), there exists an decresing open sets A n including X b such that lim n→∞ Cap(A n ) = 0. Due to Lemma 3.1 (ii) and Lemma 3.2 (ii), we then have In view of p An is E-bounded, uniformly bounded and L 2 -convergent to zero.Therefore, from ( 17) and Lemma 3.2 (i), we have for where Since the ω-set in the braces of the left hand side of ( 16) is contained in the measurable ω-set Λ, we get to (16). 2 Proof of Porposition 3.1 Take any compact set K with Cap(K) = 0.For the first assertion of the proposition, it suffices to show that Choose a sequence {A n } of relatively compact open sets such that On the other hand, M is quasi-left continuous under P ν by virtue of (M.5) and Lemma 3.3 (ii): for any stopping times σ n increasing to σ, We now have the equation ( 17) for p An by Lemma 3.1 (i) and Lemma 3.2 (ii).In the same way as in the proof of the preceding lemma, we see that the left hand side of this equation tends to zero and consequently which means that ν( ṗK > 0) = 0.
For the second assertion, take any v ∈ F b and let v ∞ = M.We can then find a sequence of functions to an E-quasicontinuous version v * of v.As lemma 3.2 (i), the desired identity (15) now follows from ( 14) for v n . 2 We are now in a position to prove the main theorem of this section.
Theorem 3.1 For u ∈ F, the next two conditions are equaivalent: (i) There exists an E-nest {F n } for which the inequality (2) is valid for some positive constants C n .
(ii) There exists a signed smooth measure ν with some attached E-nest {F n } for which the equation (4) is valid.
Proof of the implication (ii) ⇒ (i).Suppose (ii) is fulfilled.Take any v ∈ F b,Fn and let M = v ∞ .Then v * ≤ M E − q.e. and further v * = 0 E − q.e. on X \ F n .Therefore the absolute value of the right hand side of ( 4) is domonated by C n • M with C n being the total variation of ν on the set F n .

2
The converse implication (i) ⇒ (ii) will be proved in the following more specific form.
Proposition 3.2 Suppose, for u ∈ F, the inequality (1) is valid for some E-nest {F n }.Then there exists a smooth signed measure ν with an attached In the rest of this section, we assume that u ∈ F satisfies (1) for an E-nest {F n }.We shall construct ν and {F n } of Proposition 3.2 by a series of lemmas.
First fix an arbitrary w belonging to the space F Fn,b for some n.
Since this inequality holds for any v in the space F ∩C ∞ (X) which is uniformly dense in C ∞ (X), there exists a unique finite signed measure ν = ν w for which the equation ( 14) is valid.In virtue of Proposition 3.1, ν w charges no set of zero capacity and Lemma 3.4 2 This lemma says that, roughly speaking, (w * ) −1 dν w is independent of w and a candidate of the measure ν we want to construct.To make a rigorous construction, we need to consider a Hunt process associated with the given strongly regular Dirichlet space in order to use a general theory in [FOT 94].Such a process can be immediately constructed from the Ray process M = (X t , P x ) already being considered .In fact, Lemma 3.1 (ii) readily implies the following ([F 71b, Th.3.9]): for any set B ⊂ X with Cap(B) = 0, there exists a Borel set N ⊃ B with Cap(N ) = 0 such that X − N is M-invariant in the sense that Since Cap(X b ) = 0 by Lemma 3.1 (i), the branch set is included in a set N of the above type.On account of the properties (M.1) ∼ (M.5) of the Ray process M, we can get a Hunt process on X still associated with the form E first by restricting the state space of M to X ∆ − N and then by making each point of N to be a trap( see [FOT 94,Th.A.2.8,A.2.9] for those procedures).
We may call the resulting Hunt process a Hunt modification of the Ray process M.
In what follows in this section, M = (X t , P x ) denotes a Hunt process on X associated with the form E. For a strictly positive bounded Borel function ϕ on X with ϕ ∈ L 2 (X; m), we put where the right hand side is defined by (10).Here, we set ϕ(∆) = 0.By [FOT 94,Th.4.4.1], we know that ρ n is an E-quasicontinuous Borel function in F Fn,b .We then introduce the sets E n is a quasi-closed Borel set, increasing in n and owing to the stochastic charcterization (9) of the E-nest {F n }.
We define ν by For m > n, we have from Lemma 3.4 ν ρn (dx) on E n which means that the above definition makes sense.ν| En is then a finite signed measure for each n and ν charges no set of zero capacity.Moreover In fact, the above definition of ν and Lemma 3.4 imply that and we are led to (23) from ( 19).
In order to construct an appropriate E-nest from {E n }, we prepare a lemma.
Proof.In view of the definition of the quasi-closed set ([FOT 94, pp68]), we can find, for any > 0, an open set ω with Cap(ω) < such that Γ n − ω are closed for all n.For any compact K, (Γ n − ω) ∩ K are decreasing compact sets.Since Cap is a Choquet capacity, We then get lim n→∞ Cap( Let us define increasing closed sets F n by and prove that lim Since we have We let Γ n are then quasi-closed and On account of the preceding Lemma and (21), we conclude that (25) is proven and {F n } is an E-nest by Lemma 2.1.Moreover For the measure ν defined by ( 22), ν| F n is therefore a finite signed measure for each n.Since ν charges no set of zero capacity, it becomes a smooth measure for which the present E-nest {F n } is attached.
Finally take any w belonging to the space Thus the equation ( 23) leads us to completing the proof of Proposition 3.2. 2 Theorem 3.1 is proved.Here we add a theorem corresponding to a special case of Theorem 3.1 where the E-nests are trivial.
Theorem 3.2 For u ∈ F, the next two conditions are equivalent: (i) The inequality (6) holds for some constant C > 0.
(ii) There exists a finite signed measure ν charging no E-exceptional set such that the equation (4) holds for any v ∈ F b .
Proof.The proof of the implication (i) ⇒ (ii) is the same as the corresponding proof of Theorem 3.1.The converse can be viewed as a special case of Proposition 3.1 (the case where w = 1). 2

Transfers of analytic theorems to a quasi-regular Dirichlet space
First, we transfer Theorem 3.1 from a strongly regular Dirichlet space to a quasi-regular Dirichlet space.
Theorem 4.1 Let (X, m, E, F) be a quasi-regular Dirichlet space.For u ∈ F, the next two conditions are equivalent: (i) There exists an E-nest {E n } for which the inequality (2) is valid for some positive constants C n .
(ii) There exists a signed smooth measure ν with some attached E-nest {E n } for which the equation (4) is valid.
Proof.By Corollary 2.2, the quasi-regular Dirichlet space (X, m, E, F) is equaivalent to a certain strongly regular Dirichlet space ( X, m, Ẽ, F ) by a quasi-homeomprphism q; there exist E-nest {F n }, Ẽ-nest {F n }, q is a one to one mapping from X 0 = ∪ ∞ n=1 F n onto X0 = ∪ ∞ n=1 Fn , its restriction to each F n is homeomprphic to Fn , and the map from F b to Fb defined by q is E-quasi notions preserving as is explained in §2.

Suppose that the condition
By virtue of Theorem 3.1, there exists a smooth signed measure ν on X with an attached Ẽ-nest { Ẽ n } for which the equation Then we can easily see that {E n } is an E-nest, ν is a smooth signed measure on X with the attached E-nest {E n }.By rewiting the above equation and noting that v * (x) = ṽ * (qx) gives an E-quasicontinuous version of v = Φ −1 (ṽ), we arrive at the equation ( 4) holding for u, ν, { Ẽ n }, getting the conclusion (ii).
The converse implication (ii) ⇒ (i) can be directly shown as the corresponding proof of Theorem 3.1. 2 In exactly the same way, Theorem 3.2 can be transfered from a strongly regular Dirichlet space to a quasi-regular Dirichlet space.
Theorem 4.2 Let (X, m, E, F) be a quasi-regular Dirichlet space.For u ∈ F, the next two conditions are equivalent: (i) The inequality (6) holds for some constant C > 0.
(ii) There exists a finite signed measure ν charging no E-exceptional set such that the equation (4) holds for any v ∈ F b .
For later convenience, we also transfer Proposition 3.1 in the following manner: Proposition 4.1 Let (X, m, E, F) be a quasi-regular Dirichlet space.Let u ∈ F and w ∈ F b .Suppose there exists a positive constant C = C u,w such that Then there exists uniquely a finite signed smooth measure ν for which the equation ( 15) is valid.
Proof.In the same way as in the proof of Theorem 4.1, the above inequality is honestly inherited to a strongly regular representation, where we can use Proposition 3.1 to obtain the conclusion as above which can be transfer back to the quasi-regular Dirichlet space also in the same way as in the proof of Theorem 4.1.2 To this end, let us first consider a regular Dirichlet space (X, m, E, F) and an associated Hunt process M = (X t , P x ) on X.In this context, we start with mentioning some probabilistic characterizations of E-exceptional sets in a convenient way for later use.
properly exceptional if it is a nearly Borel measurable m-negligible set and its complement X − N is M-invariant.We call N (M-)properly exceptional in the standard sense if, in the above definition of the proper exceptionality, the M-invarance of X − N is weakened to the M-invariance up to the life time in the following sense: Remark 5.1 The last notion of the exceptionality makes sense not only for the present Hunt process but also for a standard process, and we shall later utilize it for a standard process associated with a quasi-regular Dirichlet space.
Lemma 5.1 The following conditions for a set N ⊂ X are equivalent each other: (iii) N is contained in a properly exceptional set.
(iv) N is contained in a properly exceptional set in the standard sense.
Proof.The equivalence of the first three conditions are proven in [FOT 94].The implication (iii)⇒(iv)⇒(ii) is also obvious. 2 Remark 5.2 (i) In studying a Hunt process M associated with a regular Dirichlet space (X, m, E, F), the state space of M needs not to be the entire space X.Indeed, we may well consider a Borel subset N 0 of X with m(N 0 ) = 0 and a Hunt process M with state space X − N 0 such that it is associated with a Dirichlet form E on L 2 (X; m) in the sense that its transition function on X − N 0 generates the L 2 -semigroup corresponding to E. Then the set N 0 becomes automatically E-exceptional, because it is properly exceptional with respect to the trivial extension M of M to X (M is obtained from M by joining every point of N 0 as a trap, see [FOT 94,Th.A.2.9]).M is still associated with E and hence Lemma 5.1 applies.
(ii) Given a Hunt process M on X − N 0 as above, we call a set N M-properly exceptional (resp.M-properly exceptional in the standard sense M-invariant up to the life time).With this slight modification of the notion of proper exceptionality, not only Lemma 5.1 but also what will be stated below about additive functionals remain valid for a Hunt process M on X − N 0 as above.
(iii) We can and we shall allow an analogous freedom of choice of the state space about a standard process associated with a quasi-regular Dirichlet space, accompanyed by an analogous modification of the proper exceptionality in the standard sense.
We quote from [FOT 94] those basic notions and theorems concerning additive functionals of the Hunt process M on X.By convention, any numerical function f on X is extended to the one-point compactification X ∆ by setting f (∆) = 0. Let Ω, {F t } t∈[0,∞] , ζ, θ t be the sample space, the minimum completed admissible filtration, the life time and the shift operator respectively attached to the Hunt process M.An extended real valued function A t (ω) of t ≥ 0 and ω ∈ Ω is called an additive functional (AF in abbreviation) if it is {F t }-adapted and there exist Λ ∈ F ∞ with θ t Λ ⊂ Λ, ∀t > 0 and a properly exceptional set N ⊂ X with and Λ (resp.N ) in the above definition is called a defining (resp.exceptional) set for the AF A. We regard two AF's to be equivalent if (1) Then we can find a common defining set Λ and a common properly exceptional set N of A (1)  and A (2) such that A (1) Here we have to use Lemma 4.1 together with the fact that the ω-set An AF A t (ω) is said to be finite, cadlag and continuous respectively if it satisfies the respective property at every t ∈ [0, ∞) for each ω in its defining set.A [0, ∞]-valued continuous AF is called a posiitve continuous AF (PCAF in abbreviation).We denote by A + c the set of all PCAF's.We shall call an AF A t (ω) of bounded variation if it is of bounded variation in t on each compact subinterval of [0, ζ(ω)) for every fixed ω in a defining set of A.
A positive Borel measure µ is called a smooth measure if µ charges no E-exceptional set and there is an E-nest {F n } such that µ(F n ) is finite for each n.The totality of smooth measures is denoted by S.There is a one to one correspondence between (the equivalence classes of) A + c and S by the Revuz correspondence: holding for any non-negative Borel function f and γ-excessive function h, γ ≥ 0. The smooth measure corresponding to a PCAF A in the above way is said to be the Revuz measure of A.
For any u ∈ F, there exists a unique finite smooth measure µ u on X, which satisfies the following equation in the case that u ∈ F b : µ u is called the energy measure of u ∈ F.
The energy e(A) of an AF A is defined by For u ∈ F, A [u] t defined by ( 1) is (up to the equivalence specified above) a finite cadlag AF of finite energy.The families of martingale AF's of finite energy and CAF's of zero energy are defined respectively by . and e(M ) < ∞}, . and e(N ) = 0}.
For any u ∈ F, the AF A [u] admits a decomposition (3) uniquely up to the equivalence specified above, where FOT 94,Th.5.2.2]).The energy measure µ u of u coincides with the Revuz measure of the quadratic variation FOT 94,Th. 5.2]).Furthermore [FOT 94,Th.5.4.2] asserts the following: the CAF N [u] is of bounded variation if and only if the condition (ii) of Theorem 4.1 is valid for some signed smooth measure ν.In this case moreover, N [u] admits an expression (5) for some PCAF's A k corresponding to smooth measures ν k , k = 1, 2, by the Revuz correspondence and it holds that ν = ν 1 − ν 2 .Now we turn to a general quasi-regular Dirichlet space (X, m, E, F).Let ( X, m, Ẽ, F ) be its regular representation and be a Hunt process on X associated with the latter (one may take for instance a strongly regular representation and a Hunt modification of the associated Ray process).By Theorem 2.2, the two Dirichlet spaces are related each other just as in the first paragraph of the proof of Theorem 4.1.We shall use the notations {F n }, X 0 , { Fn }, X0 , q, Φ, appearing in that paragraph without repeating the explanation.We are ready to construct a standard process on X associated with E as an image of M by q −1 in a similar way to [F 71b].
Applying the stochastic characterization (9) to the Ẽ-nest { Fn }, we can find an M-properly exceptional Borel set Ñ1 including X − X0 such that Px ( lim In other words, we have Adjoin a point ∆ to X as an extra point (as a point at infinity if X is locally compact) and extend q to a one to one mapping from X 0 ∪ ∆ onto X0 ∪ ∆ by setting q(∆) = ∆.We define an E-exceptional Borel set N 1 ⊂ X by We let The element of Ω (resp.{F t }) is denoted by ω (resp.Λ) instead of ω (resp.Λ).Finally let us define X t , ζ, P x by With these definitions of elements, we put M1 is a Hunt process on X − Ñ1 which is associated with the regular Dirichlet form Ẽ. We shall call M 1 the image of the Hunt process M1 by the quasi-homeomorphism q −1 .
Theorem 5.1 M 1 defined by (30)∼( 33) is a standard process on X − N 1 associated with the quasi-regular Dirichlet form E. Further M 1 is special and tight. Proof.
The first assertion can be proved in the same way as in [F 71b, §4] where (X, m, E, F), ( X, m, Ẽ, F ) and M were a regular Dirichlet space, its strongly regular representation, and a Hunt modification of the Ray process associated with the latter respectively, and the process M 1 defined by (30)∼(33) was shown to be a Hunt process on X − N 1 .The only defference from the present situation was in that {F n }, { Fn } were nests in the sense of [FOT 94] rather than E-nest and Ẽ-nest, and accordingly we had the following stronger property than (29): In the present case, we can also see as in [F 71b] that {F t } t∈[0,∞] defined by ( 31) is the minimum completed admissible filtration for X t defined by (32).Thus M 1 is special, namely, {F t } is quasileft continuous because so is { Ft }.Let Kn be compact sets increasing to X such that Kn is included in the interior of Kn+1 and put K n = q −1 ( Kn ∩ Fn ).Then {K n } are increasing compact subsets of X − N 1 and we get obviously the tightness of M 1 : 2 The state space of M 1 can be enlarged to X if necessary by an trivial extension (namely, by making each point of N 1 to be a trap.)[MR 92,Th. 3.5] has given another construction of an m-tight special standard process associated with a quasi-regular Dirichlet space.
M 1 is defined in the same way as for an Hunt process.On account of Lemma 5.2 and Remark 5.4, we can then find a common defining set Λ ∈ F ∞ and a common exceptional set N (⊃ N 1 ) such that

A
(1) We can easily observe that, if A t (ω) is an AF of M 1 with a defining set Λ and an exceptional set N (⊃ N 1 ), then A t (ω) is an AF of M1 with some defining set contained in Λ and some exceptional set containing q(N )(⊃ Ñ1 ).Conversely any AF of M1 with a defining set Λ and an exceptional set Ñ (⊃ Ñ1 ) can be viewed as an AF of M 1 with the same defining set and the exceptional set q −1 ( (N )).Two AF's are equivalent with respect to M 1 iff so they are with respect to M1 .
Various classes of AF's of M 1 are defined in the same way as for an Hunt process.In particlur, we have the classes A + c of PCAF's, • M of margingale AF's of finite energy and N c of continuous AF's of zero energy for the process M 1 .For u ∈ F and its E-quasicontinuous version u * , we put ũ * (x) = u * (q −1 (x)) x ∈ X − Ñ1 .
Then ũ * is an Ẽ-quasicontinuous version of ũ = Φu and Hence A [u] (the left hand side) is (up to the equivalence) a finite cadlag AF and uniquely expressible as (3) for M 1 because so is A [ũ] (the right hand side) for M1 .
For the present quasi-regular Dirichlet form E, the class S of smooth measures and the notion of the energy measure µ [u] of u ∈ F b are defined also in the same way as for a regular Dirichlet form.q preserves the notion of the smoothness of positive measures.µ is the Revuz measure of a PCAF A of M 1 if qµ is the Revuz measure of A as a PCAF of M1 .The energy measure of u ∈ F b characterized by the equation ( 28) can be constructed as the image by q −1 of the energy measure of Φu ∈ Fb with respect to the regular Dirichlet form Ẽ.
Summing up what has been mentioned, we get (ii) Any u ∈ F admits a unique finite smooth measure µ u satisfying the equation (28) in the case that u ∈ F b .
(iii) For any u ∈ F, the AF (1) admits the decomposition (3) uniquely up to the equivalence where M [u] ∈ • M, N [u] ∈ N .M [u] admits its quadratic variation in A + c whose Revuz measure is the energy measure of u.
Finally we transfer Theorem 5.4.2 of [FOT 94].Recall that an AF A t (ω) is said to be of bounded variation if it is of bounded variation in t on each compact interval of [0, ζ(ω)) for every fixed ω in its defining set.and ψ From the validity of (6) for functions in L, we get It suffices to let n → ∞ and then ↓ 0. 2 Theorem 6.2For u ∈ F, the following conditions are equivalent: (i) The inequality (6) holds for any v in the space L satisfying condition (L).
(ii) There exists a unique finite signed smooth measure ν for which the equation ( 4) is valid for any v ∈ F b .
(iii) The continuous AF N [u] t in the decomposition (3) of the AF (1) is of bounded variation and satisfies the property (7).
Proof.The first two conditions are equivalent by virtue of Theorem 4.2 and Lemma 6.1.Assume (ii) and let (35) be the Jordan decomposition of ν.Then ν k , k = 1, 2, are finite smooth measures and, on account of Thorem 5.3, N [u] is expressible by the PCAF's A k with Revuz measure ν k , k = 1, 2. Denote by {N [u] } t the total variation of N [u] on [0, t].It is known to be an element of A + c .Since it is dominated by A 1 t + A 2 t , we get from the Revuz correspondence ( 27) arriving at (iii).
Conversely, assume (iii) and let On account of Theorem 5.3, condition (ii) holds for the signed smooth measure (35) where ν k is defined to be the Revuz measure of A k , k = 1, 2. Then ν is finite, because by condition (7) 2 Remark 6.1 Property (7) says that the Revuz measure of the PCAF {N [u] } t has a finite total mass.By [FOT 94,Th.5.3.1],any PCAF A and its Revuz measure µ are related by Hence, property (7) implies the P m -integrability If the process is conservative in the sense that p s 1 = 1, s > 0, then the integrability (36) implies property (7).
In the rest of this section, we assume that (X, m, E, F) is a regular Dirichlet space and M = (X t , P x ) is any associated Hunt process.Of course, the preceding two theorems remain valid under the present assumption.Let us further assume that E is strongly local: if u, v ∈ F are of compact support and v is constant on a neighbourhood of the support of u, then E(u, v) = 0.
Then the associated Hunt process M can be taken to be a diffusion (namely, of continuous sample paths on [0, ζ)) with no killing inside.A function u is said to be locally in F (u ∈ F loc in notation) if for any relatively compact open set G there is a function w ∈ F such that u = w m − a.e. on G. Let u ∈ F loc .u admits an E-quasicontinuous version u * .The energy measure µ u of u is still well defined.The AF A [u] defined by (1) admits a decomposition (3) where M [u] ∈ • Mloc and N [u] ∈ N c,loc .The decomposition is unique up to the equivalence of local AF's.The quadratic variation M [u] ∈ A + c of M [u] has as its Revuz measure the energy measure µ u of u.See [FOT 94,§5.5]for details of the above notions, notations and statements.
As for the semimartingale characterization of N [u] ∈ N c,loc for u ∈ F loc , all statements in Theorem 5.3 remain true except that the E-nest {E n } appearing in the condition (ii) there is now required to consist of compact subsets of X ([FOT 94,th.5.5.4]).
The next condition for a subset C of C 0 (X) is taken from [FOT 94] (condition (C.2) of [FOT 94]).
(C) C is a dense subalgebra of C 0 (X).For any compact set K and relatively compact open set G with K ⊂ G, C admits an element u such that u ≥ 0, u = 1 on K and u = 0 on X − G.
Let C be a subset of F ∩ C 0 (X) satisfying both conditions L and C. Such a subset is similar to a special standard core in the sense of [FOT 94], and the only difference lies in the definition of a real function φ (t) appearing in condition L. For any open set G ⊂ X, we put ([FOT 94,Lem.2.3.4]).Theorem 6.3 The next conditions are equivalent for u ∈ F loc : (i) For any relatively compact open set G ⊂ X, there is a positive constant C G such that (ii) There exists a signed Radon measure ν on X charging no set of zero capacity such that is of bounded variation and satisfies property (8).

Stochastic characterizations of BV functions and Caccioppoli sets
Let R d be the d-dimensional Euclidean space and m 0 be the Lebesgue measure on it.We consider a non-negative locally integrable function ρ on R d and the associated energy form defined by The associated diffusion M ρ = (X ρ t , P ρ x ) on R(ρ) is called a distorted Brownian motion.The reason of this naming is in that, if we apply the decomposition (3) to coordinate functions then we get the expression of the sample path where B t is a d-dimensional Brownian motion and (ii) The distorted Brownian path X ρ t is a semimartingale in the sense that each component N i t , (1 ≤ i ≤ d), in the decomposition (42) is of bounded variation and additionally it satisfies that for every compact set K ⊂ R(ρ).
Proof.We apply Theorem 6.3 to the strongly local, regular Dirichlet form (E ρ , F ρ ) on L 2 (X; m) and an associated diffusion M ρ = (X ρ t , P ρ x ) on X, where We take which obviously has the properties (L) and (C).Taking as u the coodinate function ψ i ∈ F loc , the condition (i) of Theorem 6.It is easy to see that ρ ∈ BV loc if and only if the above condition is satisfied for each i = 1, 2,  (ii) The modified reflecting Brownian path (X t , P x ) = (X I D t , P I D x ) on D is a semimartingale in the sense that each component N i t of the second term N I D t in its decomposition (42) is of bounded variation and satisfies the additional property that In this case, the modified reflecting Brownian motion admits an expession of Skorohod type: for a PCAF L t with Revuz measure σ.

2
Proof of Proposition 3.2 Take a sequence ↓ 0. Since the sets E n defined by (20) are quasiclosed, we can find decresing open sets ω with Cap(ω ) < and E n − ω are closed for all n and .
Theorem 5.2 (i) The equivalence classes of PCAF's A + c of M 1 and the smooth measures S of E are in one to one (Revuz) correspondence by (27).
Condition (H) is fulfilled in the following two important cases:(I) ρ is non-negative continuous m 0 -a.e. on R d .(II) ρ(x) = I D (x), x ∈ R d , for an open set D ⊂ R d .In the second case, R(ρ) = D and the distorted Brownian motion M I D reduces to the modified reflecting Brownian motion on D associated with the strongly local regular Dirichlet formE I D (u, v) = 1 2 D ∇u(x) • ∇v(x) m 0 (dx), F I D = Ĥ1 (D) on L 2 (D; I D • m 0 )(= L 2 (D; m 0 )) studied in [F 97a].Here Ĥ1 (D) denotes the closure of the spaceC 1 0 (R d )| D in the Sobolev space H 1 (D).The term 'modified' is added because Ĥ1 (D) could be a proper subset of H 1 (D).A function ρ ∈ L 1 loc (R d ) is called BV (denoted by ρ ∈ BV loc ) if for any bounded open set V ⊂ R d , there exists a positive constant C V such that | V (div v)ρm 0 (dx)| ≤ C V v ∞ ∀v ∈ C 1 0 (V ; R d ).Suppose a non-negative function ρ ∈ L 1 loc (R d ) satisfies the condition (H).Then the following conditions are equivalent:(i) ρ ∈ BV loc .
with |n E | = 1 σ-a.e.such thatE div v m 0 (dx) = − ∂E v • n E dσ ∀v ∈ C 1 0 (R d , R d ).(45) By specifying Theorem 7.1 to ρ = I D , we get Theorem 7.2 The following conditions are equivalent for an open set D ⊂ R d : (i) D is Caccioppoli.

Theorem 2.1 Any Dirichlet space admits its strongly regular representation.
• • • , d.Thus, we get Theorem 7.1 from Theorem 6.3. 2 A measurable set E ⊂ R d is called Caccioppoli if I E ∈ BV loc .We know ([EG 92]) that E is Caccioppoli if and only if there exists a positive Radon measure σ on ∂E and a σ-measurable vector n E : ∂E → R d