on a representation of additive functionals of zero quadratic variation

Potential Anal (2013) 38:1173–1186DOI 10.1007/s11118-012-9311-z

On a representation of additive functionals of zeroquadratic variation

Alexander Walsh

Received: 19 January 2012 / Accepted: 10 September 2012 / Published online: 25 September 2012© Springer Science+Business Media B.V. 2012

Abstract For a Markov process X associated to a Dirichlet form, we use continuousadditive functionals obtained by Fukushima decompositions in order to represent theclass of additive functionals of zero quadratic variation. We do not assume that X issymmetric.

Keywords Fukushima decomposition · Markov process · Dirichlet form ·Additive functional · Zero energy process · Quadratic variation

Mathematics Subject Classifications (2010) 60H05 · 60J55 · 60J25 · 31C25

1 Introduction and Main Result

For X, a symmetric Markov process, the set N of continuous additive functionals ofzero energy plays a very important role. In fact, for a function u in F , the domain ofthe Dirichlet form associated to X, Fukushima [4] has shown that u(X) admits thedecomposition

u(Xt) = u(X0) + Mut + Nu

t , (1.1)

where Mu is a martingale and Nu an element in N .Because of its importance, many authors have studied representations for ele-

ments into the class N , among them: Nakao [10], Oshima and Yamada [12], Wolf[14], Yamada [16, 17] and Yor [18].

When X is a symmetric diffusion, Eq. 1.1 is still valid for u in Floc, the set offunctions locally in F . Moreover, if X is irreducible, the set J = {Nu : u ∈ Floc}produces every element in N , in fact, it is shown by Oshima and Yamada [12] that

A. Walsh (B)Department of Industrial Engineering and Management, Technion, Haifa, Israele-mail: [email protected]

1174 A. Walsh

the class of elements locally in N coincides with the set of elements locally in J . (Seealso Theorem 5.5.6 in [4]).

If X is not continuous, there is not a decomposition (1.1) for the functions ulocally in F and therefore the set J is not well defined, nevertheless, another classof additive functionals plays its role. Define the set I by:

I ={

Nu −∫ .

0u(Xs)ds : u ∈ F

}.

It is shown by Nakao [10] that if X is a conservative symmetric Markov process,the class I coincides with the class N . But regardless of whether the process isconservative or not, the set I is very useful, if fact Nakao [9] has constructedstochastic integrals with the elements of I as integrators and then he has obtainedextensions of the Itô formula for the process u(X). In this sense, the set I replacethe set of bounded variation processes for which, Lebesgue–Stieltjes integrals arewell-defined.

Besides, in the stochastic calculus theory, many extensions of the Itô formulahave been established. In most of them, a new set of processes arises, the class ofzero quadratic variation process. This is the case of the Fukushima decomposition,because the elements in the class N are necessarily of zero quadratic variation. Manypositive continuous additive functionals are examples that the converse is not true.

In this direction, it is quite natural to ask if the zero quadratic variation processesare at least locally of finite energy and in affirmative case, if it is possible to representthem using elements in the class I . The aim of this paper is to give a positive answerto this question.

We do not assume that the Markov process X is symmetric. Nevertheless, thestochastic integral of Nakao can be extended to our case (see [15]) and thereforerepresentations of additive functionals as elements of the class I are useful in orderto develop related stochastic calculus.

Our main result is Theorem 1.1 below, it extends the work of Oshima and Yamada[12] and Nakao [10] because it removes the hypothesis of symmetry of X, but also thehypothesis of continuity, irreducibility and conservativeness. Moreover it relaxes theconditions of the processes to be represented. In order to introduce it, we recall thefollowing definition. An increasing sequence of nearly Borel finely open sets {Gn} issaid to be a nest if

Px( limn→∞ τGn = ζ ) = 1

for all x except in an exceptional set, where τGn denotes the exit time of Gn and ζ thelife time of X.

Theorem 1.1 Let A be a local continuous additive functional of zero quadraticvariation. Then there exists a nest of f inely open sets {Gn}n∈N and a sequence {un}n∈N ⊂F such that

At = Nunt −

∫ t

0un(Xs)ds for all t < τGn

Px—a.s. for every x except in a exceptional set.

On a representation of additive functionals of zero quadratic variation 1175

This paper is organized as follows: In Section 2 we introduce the notation,definitions and hypothesis used throughout this paper. In Section 3 we establish somepreliminary results that will be useful in the proof of Theorem 1.1. Finally in Section 4we will prove Theorem 1.1.

2 Notation and Basic Definitions

In this paper we use mostly notation and vocabulary from the book of Fukushimaet al. [4] still available for the non necessarily symmetric case (see Ma and Röckner[8] and Oshima [11]).

Throughout this paper, we assume that X = (�, {Ft}t≥0, {Xt}t≥0, {Pz}z∈E) is aHunt process on a locally compact separable metric space E, properly associatedto a regular Dirichlet form E with domain F in a Hilbert space L2(E; m). We do notassume that E is symmetric. More concretely, E is a non-symmetric Dirichlet formon L2(E; m). Set E1(u, v) := E(u, v) + (u, v), where (., .) denotes the inner productin L2(E; m). It is known that F is a Hilbert space with inner product E1(u, v) :=12 (E1(u, v) + E1(v, u)). Denote by ζ the life time of X and � the extra point such thatXt(ω) = � for all t ≥ ζ(ω) and ω ∈ �. A real function on E is extended to a functionon E ∪ � by setting f (�) = 0.

An (Ft)-adapted process A is an additive functional (AF in abbreviation) ifthere exists � in F∞ and a properly exceptional set N such that Px(�) = 1 forx ∈ E \ N, θt� ⊂ � for all t ≥ 0 and for all ω ∈ �, t �→ At(ω) is finite càdlág on[0, ζ(ω)), A0(ω) = 0, At(ω) = Aζ (ω) for t ≥ ζ(ω) and A(ω) has the additive property,At+s(ω) = At(ω) + As(θtω), s, t ≥ 0.

A local AF is a process that satisfies all requirements to be an AF except that theadditive property is required only for t, s ≥ 0 with t + s < ζ(ω).

In the sequel, we say that � is a defining set for A or a defining set admitting aproperly exceptional set N or that N is a properly exceptional set for A.

An AF A is said to be continuous (resp. càdlàg, resp. finite) if it has a defining set� such that A.(ω) is continuous (resp. càdlàg, resp. finite) in [0,∞) for any ω ∈ �.The abbreviations CAF and PCAF stand for, “continuous additive functional” and“positive continuous additive functional” respectively.

We say that a local AF A is continuous or a local CAF if there exists a definingset � such that A.(ω) is continuous in [0, ζ(ω)[ for any ω ∈ �.

Denote by M the set of MAF’s (martingale additive functionals), by◦

M the set ofMAF’s of finite energy and by N the set of CAF’s of zero energy.

For any u ∈ F , Mu and Nu denote the elements of◦

M and N respectively, that arepresent in Fukushima decomposition of u(Xt) − u(X0), t ≥ 0, i.e.,

u(Xt) − u(X0) = Mut + Nu

t for t ≥ 0, Px—a.s. for q.e x ∈ E.

Where following the usual notation, “q.e.” stands for quasi everywhere, that is,outside of a properly exceptional set. Moreover we use the convention that anyelement in F is represented by its quasi-continuous version.

For a nearly Borel set B ⊂ E, σB and τB represent the first hitting time to B andthe first exit time from B respectively, i.e,

σB := inf{t > 0 : Xt ∈ B} and τB := inf{t > 0 : Xt /∈ B}.

1176 A. Walsh

It is well known that for a nearly Borel set B, σB and τB are (Ft)-stopping times.Set O := {G ⊂ E : G is nearly Borel and finely open} and for A subset of E, set

OA := {G ∈ O : G ⊂ A}. For G ∈ O define

G :={{Gn} ⊂ OG : Gn ⊂ Gn+1 ∀n and Px

(lim

n→∞ τGn = τG

)= 1 for q.e. x ∈ E

}.

We denote E by . In this paper, we refer to the term “zero quadratic variation”used in Theorem 1.1 in the following sense.

Definition 2.1 An AF V is said to be strictly of zero quadratic variation if for anyT > 0, any sequence (tn)n∈N converging to zero and g ∈ L1(E; m) ∩ L∞(E; m),

limn→∞

�T/tn ∑�=0

Egm[(Vtn(�+1) − Vtn�)

2] = 0.

A local AF A is said to be of zero quadratic variation if there exists {Gn} in and asequence (Vn)n∈N of AF’s of strictly zero quadratic variation such that At = Vn

t fort < τGn Px—a.s. for q.e. x ∈ E.

Any AF of zero energy is of strictly of zero quadratic variation (see (5.2.14) in[4]). Therefore a local AF locally of zero energy is of zero quadratic variation.

3 Preliminary Results

In this section we will establish a series of lemmas and remarks that will help us inthe demonstration of Theorem 1.1. The following lemma is actually Lemma 4.6 in [7]that we recall for the reader’s convenience

Lemma 3.1 For an element G in O and an increasing sequence {Gn} of nearly Borelf inely open subsets of G, the following are equivalent:

(i) {Gn} ∈ G.(ii) Px(limn→∞ τGn = τG) = 1 for m—a.e. x ∈ E(iii)

⋃∞n=1 Gn = G q.e.

As consequence of Lemma 3.1 we have,

Lemma 3.2 Let {Gn}n∈N be an element of and for each n, let {Gn,k}k∈N be in Gn .Then there exists an increasing sequence {nk}k∈N ⊂ N and {Gk}k∈N ∈ such that foreach k ∈ N, Gk ⊂ Gnk, j for some j ∈ N.

Proof Let ϕ be an element of L1(E; m), bounded and m—a.e. strictly positive on E.In view of Lemma 3.1 we assume, by taking subsequences if necessary,

Pϕm

(ζ − τGn >

1

2n, ζ < ∞

)+ Pϕm

(τGn < n + 1, ζ = ∞)

<1

2n+1.


For each n ∈ N, there exists an �n ∈ N such that �n ≥ n for all n and

Pϕm

(τGn − τGn,�n

>1

2n, τGn < ∞

)+ Pϕm

(τGn,�n

< n + 1, τGn = ∞)<

1

2n+1.

Set Gn := ⋂k≥n Gk,�k and let Gn be the fine interior of Gn. Then Gn ⊂ Gn,�n .

We will prove that {Gn}n∈N ∈ . Evidently each Gn is finely open and Gn ⊂ Gn+1.By Proposition 10.6 of [13], each Gn is nearly Borel measurable and τGn = τGn

=inf{τGk,�k

: k ≥ n} Px—a.s. for all x ∈ E. Then for any n ∈ N we have

Pϕm(τGn < n, ζ = ∞) ≤∞∑

k=n

Pϕm(τGk,�k< n, ζ = ∞)

≤∞∑

k=n

(ak + b k),

where,

ak := Pϕm(τGk,�k< k, τGk = ∞, ζ = ∞) and b k := Pϕm(τGk,�k

< k, τGk < ∞, ζ = ∞).

For any k ∈ N, ak ≤ Pϕm(τGk,�k< k, τGk = ∞) < 1

2k+1 and for b k we have

b k ≤ Pϕm(τGk,�k< k, τGk − τGk,�k

> k−1, τGk < ∞, ζ = ∞)

+ Pϕm(τGk,�k< k, τGk − τGk,�k

≤ k−1, τGk < ∞, ζ = ∞)

≤ Pϕm(τGk − τGk,�k> k−1, τGk < ∞) + Pϕm(τGk < k + 1, ζ = ∞)

≤ 1

2k.

Therefore ak + b k ≤ 3/2k+1 and Pϕm(τGn < n, ζ = ∞) ≤ 3/2n. It follows by Borel–Cantelli that Px(limn→∞ τGn = ∞, ζ = ∞) = Px(ζ = ∞) for m—a.e. x ∈ E. In asimpler way, we can prove that Pϕm(ζ − τGn > n−1, ζ < ∞) ≤ 21−n and thenPx(limn→∞ τGn = ζ, ζ < ∞) = Px(ζ < ∞) for m—a.e. x ∈ E. This finishes the proofthanks to Lemma 3.1. ��

We recall that an AF M is called an MAF (M ∈ M) if it is finite, càdlàg and forq.e x in E, Ex[M2

t ] < ∞ and Ex[Mt] = 0 for any t ≥ 0. The hypothesis that it be finiteand càdlàg is not necessary, in fact,

Lemma 3.3 Let M be an AF admitting a properly exceptional set N such that for x ∈E \ N and t ≥ 0, Ex[M2

t ] < ∞ and Ex[Mt] = 0. Then M is an MAF.

Proof The only point that we have to show is that there exists a defining set on whichM is finitely càdlàg. Evidently there exists a defining set � of M such that M.(ω) isfinite on [0,∞) for any ω ∈ �. We assume without loss of generality that N is aproperly exceptional set for �. For ω ∈ � such that ζ(ω) > 0 define

Msζ−(ω) = inf

s<ζ(ω),s∈Q

sup{Mr(ω) : s ≤ r < ζ(ω), r ∈ Q}

Miζ−(ω) = sup

s<ζ(ω),s∈Q

inf{Mr(ω) : s ≤ r < ζ(ω), r ∈ Q}

1178 A. Walsh

and Msζ−(ω) = Mi

ζ−(ω) = 0 if ζ(ω) = 0. Besides, define

� = {ω ∈ � : ζ(ω) ∈ (0,∞) and Msζ−(ω) = Mi

ζ−(ω) ∈ R} ∪ {ω ∈ � : ζ(ω) ∈ {0,∞}}.Clearly � ∈ F∞. The fact that for ω ∈ �, M(ω) is càdlàg on [0, ζ(ω)), leads to

Msζ−(ω) = lim supt↑ζ(ω) Mt(ω) and Mi

ζ−(ω) = lim inft↑ζ(ω) Mt(ω). Therefore, M(ω) isfinite càdlàg on [0, ∞) for ω ∈ �.

We shall prove that � is a defining set for M. We can check easily that θt� ⊂ � forall t ≥ 0. We must prove that Px(�) = 1 for all x ∈ E\N. For any x ∈ E\N, M is a(Px, (Ft))-martingale. Thus for x ∈ E\N, M has a Px-modification càdlàg denotedby Mx. For x in E\N, set �x := � ∩ {w ∈ � : Mx

. is càdlàg} ∩ {w ∈ � : Mt(ω) =Mx

t (ω)∀t ∈ Q+}. We have then Px(�x) = 1. For ω ∈ �x such that 0 < ζ(ω) < ∞, one

obtains Msζ−(ω) = lim supt↑ζ(ω) Mx

t (ω) = lim inft↑ζ(ω) Mxt (ω) = Mi

ζ−(ω). Consequently�x ⊂ � and Px(�) = 1. ��

For f in L2(E; m) and t > 0 define pt f (x) = Ex[ f (Xt)] and R1 f (x) =∫ ∞0 e−t pt f (x)dt, where X is the dual process of X.

Lemma 3.4 There exists {Gn}n∈N in such that for any AF A of strictly zero quadraticvariation and n ∈ N

limt→0

1

tEm[A2

t : t < τGn ] = 0

Proof Let ϕ be a function in L1(E; m) such that 0 < ϕ(x) ≤ 1 for any x ∈ E. Setg(x) := R1ϕ(x). Then g ∈ L1(E; m) and g(x) > 0 for any x ∈ E. For any n ∈ N setGn := {x ∈ E : e−1/n p 1

ng(x) > 1

n }. Since g is 1-excessive with respect to X,⋃Gn =

E, Gn ⊂ Gn+1 and each Gn is nearly Borel and finely open with respect to X. ThenGn is quasi-open ([7], Proposition 4.1) and therefore finely open q.e., that is, thereexists Gn ∈ O such that Gn = Gn q.e. Without loss of generality we assume that Gn ⊂Gn+1. By Lemma 3.1, {Gn}n∈N ∈ . Besides, let A be an AF of strictly zero quadraticvariation, (tk) a sequence converging to zero and n ∈ N.

0 = limk→∞

�1/ntk ∑�=0

Egm[(Atk(�+1) − Atk�)

2]

= limk→∞

�1/ntk ∑�=0

∫E

ptk�g(x)Ex[A2tk ]m(dx)

≥ limk→∞

�1/ntk ∑�=0

∫E

e−1/n p 1ng(x)Ex[A2

tk : tk < τGn ]m(dx)

≥ 1

n2lim

k→∞1

tkEm[A2

tk : tk < τGn ]

This shows that limt→01t Em[A2

t : t < τGn ] = 0. ��

The following result is a little modification of II.4.14 of [2] and will be used tobuild sequences of .


Lemma 3.5 Let Y be in F∞ and N be a properly exceptional set such that for anyx ∈ E\N, Y ◦ θt → Y as t → 0 Px—a.s. and there exists δ > 0 such that

supx∈E

Ex[supt<δ

|Y ◦ θt|] < ∞.

Then for any open set I ⊂ R, {x : Ex[Y] ∈ I}\N ∈ O.

Proof We have E\N ∈ O and Px(σN = ∞) = 1 for all x ∈ E\N. The functionf (x) := Ex[Y] is universally measurable (see Theorem I.5.8 in [2]) hence 1E\N(x) f (x)

is nearly Borel measurable, indeed

1E\N(x) f (x) = limα→∞ 1E\N(x)αRα f (x).

For a real a, set A := {x : Ex[Y] < a}\N, then A is nearly Borel. We shall prove thatA is finely open. For x in A and ε > 0 such that Ex[Y] < a − ε, set B(x) = Bε(x) ∪ Nwhere Bε(x) := {y : Ey[Y] ≥ a − ε/2}. Then B(x) ∈ Bn, and E\A ⊂ B(x). Besides itis not difficult to prove that Px(σB(x) = 0) = 0. This shows that A belongs to O. Withthe same arguments we can show that for any a ∈ R, {x : Ex[Y] > a}\N ∈ O. ��

The following fact will be used in the proof of the next lemma,

F∞ = Fζ := {A ∈ F∞ : A ∩ {ζ ≤ t} ∈ Ft ∀t ≥ 0}.Indeed, obviously Fζ ⊂ F∞. For any s ≥ 0 and A ∈ B(E), {Xs ∈ A} = {Xs ∈ A} ∩{s < ζ } ∈ Fζ , thus F0∞ ⊂ Fζ . Therefore, F∞ = ⋂

μ∈P(E) F0,μ∞ ⊂ ⋂

μ∈P(E) Fμζ = Fζ

(see (6.20) in [13]).

Lemma 3.6 Let A be a local AF with def ining set �. Then A can be extended to an AFA with def ining set � such that for ω ∈ � satisfying ζ(ω) < ∞, the function t �→ At(ω)

is continuous at t = ζ(ω), where At = sups≤t |As|.

Proof The proof is based in a similar argument used in [3], Remark 2.2. For w ∈� and s < t let A∗

s,t(ω) := sup{Ar(ω) : s ≤ r < t, r ∈ Q} and A∗ζ (ω) := inf{A∗

s,ζ(ω)(ω) :s < ζ(ω), s ∈ Q} if 0 < ζ(ω) and A∗

ζ (ω) = 0 if ζ(ω) = 0. For any t ≥ 0, set

At(ω) :={

At(ω) if t < ζ(ω)

A∗ζ (ω) if t ≥ ζ(ω).

First, we shall prove the (Ft)-adaptedness of A. Let I ⊂ B(R) and t ≥ 0. Itis clear that A∗

ζ ∈ F∞ = Fζ then {At ∈ I} ∩ {ζ ≤ t} = {A∗ζ ∈ I} ∩ {ζ ≤ t} ∈ Ft. Since

{At ∈ I} ∩ {t < ζ } = {At ∈ I} ∩ {t < ζ } ∈ Ft we obtain that {At ∈ I} ∈ Ft which givesthe (Ft)-adaptedness of A.

Now we shall prove the additivity of A(ω) for ω in �. We will prove onlythe case, t < ζ(ω) ≤ t + s, for the other cases, the additivity is evident. Thanksto the right continuity of As(ω) for s < ζ(ω) we have that A∗

ζ (ω) =lim sups↑ζ(ω) As(ω). Since ζ(θtω) = ζ(ω) − t > 0, (Aζ ◦ θt)(ω) = lim sups↑(ζ(ω)−t) As(θtω) =lim sups↑ζ(ω) As(ω) − At(ω) = Aζ (ω) − At(ω). Finally, At+s(ω) = Aζ (ω) = At(ω)+(Aζ ◦ θt)(ω) = At(ω) + As(θtω).

1180 A. Walsh

Finally, if ω ∈ � and ζ(ω) < ∞, Aζ (ω) = lim sups↑ζ(ω) As(ω). Then, the continuityof t �→ At(ω) at t = ζ(ω) is evident. ��

From now on, for any local AF A, A denotes the process defined in Lemma 3.6.For any G ∈ O we define XG by

XGt =

{Xt if t < τG

� if t ≥ τG.

Remark 3.7 Let A be an AF admitting a properly exceptional set N and G bean element of O. Suppose that N contains (E\G)\(E\G)r . (Here, for a set B, Br

denotes the set of regular points for B.) With the arguments used in the proof ofLemma 2.1 in [12], we show that (As∧τG , s ≥ 0) is an AF of XG admitting N asproperly exceptional set.

Lemma 3.8 Let A be a local CAF. There exists {Gn} in such that for any t ≥ 0 andn ∈ N, supx∈Gn

Ex[At∧τn ] < ∞, where τn := τGn .

Proof For n ∈ N set fn(x) := Px(n−1 < ζ) and Gn := {x ∈ E : fn(x) > 0}. For eachn, fn is 0-excessive. Indeed, Ex[ fn(Xt)] = Px(n−1 + t < ζ) ↑ fn(x) as t ↓ 0. Then fn

is nearly Borel measurable and finely continuous. Consequently Gn belongs to O.Moreover

⋃n Gn = E and thanks to Lemma 3.1, {Gn} is in .

For n in N, set ψn(x) := Ex[exp(−An−1)]. Let N be a properly exceptional setfor A. It is easy to see exp(−An−1 ◦ θt)1{t<ζ } → exp(−An−1) and hence exp(−An−1 ◦θt) → exp(−An−1) as t → 0, Px—a.s. for x ∈ E \ N. Then by Lemma 3.5, for anyn ∈ N, Gn := {x ∈ Gn : ψn(x) > n−1} \ N ∈ O. It is clear that for any n, Gn ⊂ Gn+1.Moreover, if x ∈ E \ N, there exists k, n ∈ N such that fn(x) > 0 and ψn(x) > k−1,then x ∈ Gk∨n. Hence

⋃Gn = E q.e., then, by Lemma 3.1, {Gn} belongs to . In

order to finish the proof we have to prove that

supx∈Gn

Ex[At∧τn ] < ∞ ∀t ≥ 0.

The following argument is used in the proof of Theorem 5.5.6 of [4]. Set t := n−1

and take λ > 0 such that β := 1 − t + e−λ < 1. For any x in Gn, we have t < ψn(x) ≤1 − Px(At ≥ λ) + e−λ, thus Px(At ≥ λ) ≤ β. Set ηk := inf{s > 0 : As = kλ}. We claimthat

ηk+1 ≥ ηk + η1 ◦ θηk when ηk < ∞. (3.1)

Indeed, since A is a local CAF, A is continuous in [0, ζ ) and thanks to Lemma 3.6,A is continuous in R+. Then ηk = inf{s > 0 : As ≥ kλ} and Aηk = kλ if ηk < ∞.Moreover, in this case we have As ◦ θηk < λ for all s ∈ [0, η1 ◦ θηk). Then As+ηk ≤


Aηk + As ◦ θηk < (k + 1)λ for all s ∈ [0, η1 ◦ θηk). This shows that As < (k + 1)λ forall s ∈ [0, ηk + η1 ◦ θηk) and Eq. 3.1 follows immediately. Using Eq. 3.1 we obtain

Px(At∧τn ≥ (k + 1)λ) = Px(ηk+1 ≤ t ∧ τn)

≤ Px(η1 ◦ θηk ≤ t, ηk ≤ t ∧ τn)

= Ex(PXηk(At ≥ λ) : ηk ≤ t ∧ τn)

≤ βPx(ηk ≤ t ∧ τn) ≤ βk+1,

which leads to

Ex(At∧τn) =∞∑

k=0

∫ (k+1)λ

kλ

Px(At∧τn ≥ y)dy ≤ λ

∞∑k=0

Px(At∧τn ≥ kλ) ≤ λ

1 − β.

We assume that N contains (E\Gn)\(E\Gn)r for all n ∈ N (if it is not the case one can

always expand N). Hence (As∧τn , s ≥ 0) is an AF of XGn admitting N as a properlyexceptional set (see Remark 3.7). Using the Markov property of XGn and the additiveproperty of (As∧τn , s ≥ 0) it is easy to show by induction that supx∈Gn

Ex[A(kt)∧τn ] < ∞for any k ∈ N. This finishes the proof of the lemma. ��

Let G be an element of O and denote by EG the restriction of E to FG × FG. ThenEG is also a Dirichlet form and the process XG

t is associated to EG (see Theorem 4.3in [7]). Moreover, the form EG is quasi-regular (Lemma 3.4 of [6]), nevertheless,thanks to a regularization method (see Chapter V of [8]), all results of regularDirichlet forms used in this paper are valid for EG. When we introduce a class ofAF’s associated to EG, we add the symbol (EG) in order to differentiate it from thesame class associated with E . For example for u element of FG, Nu(EG) denotesthe CAF of zero energy associated to XG obtained from Fukushima decomposition

for u(XGt ) − u(XG

0 ).◦

M (EG) denotes for example the set of MAF’s of XG of finiteenergy. For f ∈ L2(E; m) set pt f (x) = Ex[ f (Xt)], R1 f (x) = ∫ ∞

0 e−t pt f (x)dt. ForG ∈ O, pG

t and RG1 are defined in similar way but with XG replacing X.

Lemma 3.9 Let G be an element of O and u in FG. Then

Nut∧τG

= Nut (EG) for t ≥ 0 Px—a.s. for q.e x ∈ G.

Proof First we shall prove the lemma for u = RG1 f with f ∈ L2(G; m). In this case,

Nut (EG) = ∫ t∧τG

0 (u(Xs) − f (Xs))ds. On the other hand, for any w ∈ FG, E(u, w) =EG(u, w) = ( f − u, w)m. Then it follows by Lemma 5.4.4. of [4] (which is valid alsofor non symmetric forms. See [15].) that for Px—a.s. for q.e x ∈ G,

Nut∧τG

=∫ t∧τG

0(u(Xs) − f (Xs))ds = Nu

t (EG).

For the general case, let u ∈ FG and ( fn) ⊂ L2(G; m) such that un := RG1 fn

converges to u with respect to EG1 and hence, with respect to E1. Then Px—a.s. for

q.e x ∈ G, Nunt (EG) and Nun

t converge uniformly on any compact to Nut (EG) and Nu

trespectively. ��

1182 A. Walsh

The following lemma can be found in Nakao [10] under the assumption that E issymmetric. We relax this assumption. For f ∈ L2(E; m) denote its L2(E; m)-normby ‖ f‖.

Lemma 3.10 Let (ct)t>0 be a function such that for every t > 0, ct belongs to L2(E; m),ct+s = ct + ptcs for t, s > 0 and limt→0 ‖ct‖ = 0. Then there exists a unique u inL2(E; m) such that ct = ptu − u − ∫ t

0 psuds.

Proof Since ‖ct+s‖ ≤ ‖ct‖ + ‖cs‖, limt→∞ ‖ct‖/t exists in R. Set u = − ∫ ∞0 e−tctdt and

Cα = ∫ ∞0 e−αtctdt, α > 0. Then u and Cα are in L2(E; m). Straightforward computa-

tions show that for any α > 0

αCα = (α − 1)Rαu − u. (3.2)

One also has α∫ ∞

0 e−αt(ptu − u − Stu)dt = (α − 1)Rαu − u, where Stu :=∫ t0 psu ds. Hence by the right continuity of (ct) and (ptu − u − Stu) and the

uniqueness of the Laplace transform we have that ct = ptu − u − Stu. Let v beanother function satisfying ct = ptv − v − Stu. Thanks to Eq. 3.2 we have for anyα > 0, u − v = (α − 1)Rα(u − v). In particular, for α = 1 we obtain u − v = 0. ��

4 Proof of Theorem 1.1

Proof of Theorem 1.1 In view of the commentary following the proof of Lemma 3.8,when G belongs to O, we can apply to EG all the results so far used and establishedfor the form E .

Let (An)n∈N be a sequence of AF’s of strictly zero quadratic variation and {Gn} in such that A = An on [[0, τn[[ Px—a.s. for q.e. x ∈ E, where τn := τGn . Thanks toLemma 3.8 we can assume that for any t ≥ 0, supx∈Gn

Ex[At∧τn ] < ∞ and thanks toLemma 3.4 we assume that for any n

limt→0

1

tEm[(An

t )2 : t < τn] = 0 (4.1)

For any n ∈ N, we use the following notation: Xn := XGn , En := EGn andpn

t f (x) := Ex[ f (Xnt )] for any f ∈ L2(Gn; m). Moreover (Nn, Hn) denotes a Lévy

System for Xn.We assume without loss of generality that each Gn is relatively compact and then

m(Gn) < ∞. For any t set cnt (x) := Ex[At∧τn ]. Then (cn

t )t>0 satisfies the condition ofLemma 3.10 (with Gn and pn replacing E and p respectively), thus if we define vn by

vn(x) = −∫ ∞

0e−tEx[At∧τn ]dt,

it follows by Lemma 3.10 that vn belongs to L2(Gn; m) and

Ex[At∧τn ] = pnt vn(x) − vn(x) −

∫ t

0pn

s vn(x)ds, m—a.e. for x ∈ E, t ≥ 0.


Since At∧τn satisfies the condition of Lemma 3.5 (with θs∧τn instead of θs), for anyt ∈ R+, the function cn

t is quasi continuous. We can hence assume that vn is quasicontinuous and therefore, for q.e. x ∈ E (Proposition IV.5.30 in [8])

Px(t �→ vn(Xt) is càdlàg on [0, ζ )) = 1.

i.e., vn is q.e. finely continuous. Thanks to Lemma 4.1.6 in [4], we can also assumethat vn is Borel measurable. For n ∈ N and t ≥ 0 set

Cnt := At∧τn +

∫ t

0vn(Xn

s )ds and Mnt := vn(Xn

t ) − vn(Xn0 ) − Cn

t . (4.2)

Thanks to Lemma 3.3 and the fact that vn is bounded, Mn belongs to M(En).

There exists {Gn,k}k∈N ∈ Gn such that 1Gn,k ∗ Mn ∈ ◦M (En) for any k. Therefore, by

Lemma 3.2 (and by taking a subsequence of (Gn) if necessary), there exists {H0n}n∈N

in such that, for any n, 1H0n∗ Mn ∈ ◦

M (En) and H0n ⊂ Gn. Besides, following the

same arguments used in the proof of Lemma 3.8 in [7], we can construct {H1n}n∈N and

{H2n}n∈N in and sequences {gn}n∈N, {hn}n∈N in F such that for all n ∈ N

1. H2n ⊂ H1

n ⊂ H0n.

2. ‖gn‖∞ = 1, gn = 1 q.e. on H1n and gn = 0 q.e. on E \ H0

n.3. ‖hn‖∞ = 1 hn = 1 q.e. on H2

n and hn = 0 q.e. on E \ H1n.

Now we shall prove that wn := vngn belongs to FH0n. For fixed n ∈ N and any f ∈

L2(H0n; m) set p0

t f := Ex( f (X H0n

t )) and p0t f := Ex( f (X H0

nt )), t ≥ 0, where X is the

dual process of X. Then,

(wn, wn − p0t wn) = 1

2Em[(wn(X0) − wn(Xt))

2, t < τH0n) + (w2

n, 1 − p0t 1) = It + Jt.

It ≤ Em[v2n(X0)(gn(X0) − gn(Xt))

2 : t < τH0n]

+ Em[g2n(Xt)(vn(X0) − vn(Xt))

2 : t < τH0n]

≤ 2‖v2n‖(gn, gn − ptgn) + 4Em[(1H0

n∗ Mn)2]

+ 4Em

[(An

t +∫ t

0vn(Xs)ds

)2

: t < τn

].

Since gn ∈ F , it follows from [1] that supt>01t (gn, gn − ptgn) < ∞. Therefore in view

of Eq. 4.1 and since 1H0n∗ Mn

supt>0

1

tIt < ∞. (4.3)

Besides, (w2n, 1 − p0

t 1) ≤ ‖vn‖∞(g2n, 1 − p0

t 1) and since gn ∈ FH0n

we have thanks toLemma 3.3 of [5]

supt>0

1

tJt < ∞. (4.4)

Eqs. 4.3 and 4.4 lead to supt>0(wn, wn − p0t wn) < ∞ then by [1], wn ∈ FH0

n.

1184 A. Walsh

From Eq. 4.2 and Fukushima decomposition with respect to En, for any t < τH1n

wn(Xnt ) − wn(Xn

0 ) = Mnt + Cn

t

wn(Xnt ) − wn(Xn

0 ) = Mwnt (En) + Nwn

t (En).

Denote by Mn,d and Mwn,d(En) the discontinuous part of Mn and Mwn respectively.For any t < τH1

nwe have

Mn,dt = lim

ε→0

(∑s≤t

(wn(Xns ) − wn(Xn

s−))1{ε<|wn(Xns )−wn(Xn

s−)|}

−∫ t

0

∫(vn(y) − wn(Xn

s ))1{ε<|vn(y)−wn(Xns )|}Nn(Xn

s , dy)dHns

)

+∫ t

0wn(Xn

s )N(Xns , {�})dHn

s and,

Mwn,dt (En) = lim

ε→0

(∑s≤t

(wn(Xns ) − wn(Xn

s−))1{ε<|wn(Xns )−wn(Xn

s−)|}

−∫ t

0

∫(wn(y) − wn(Xn

s ))1{ε<|wn(y)−wn(Xns )|}Nn(Xn

s , dy)dHns

)

+∫ t

0wn(Xn

s )N(Xns , {�})dHn

s ,

where the above limits are in Px-probability for q.e. x ∈ E. Comparing the aboveequations we have for q.e. x and t < τH1

n,

Mn,dt − Mwn,d

t (En)

= limε→0

(∫ t

0

∫(wn(y) − wn(Xn

s ))1{ε<|wn(y)−wn(Xns )|} Nn(Xn

s , dy)dHns

−∫ t

0

∫(vn(y) − wn(Xn

s ))1{ε<|vn(y)−wn(Xns )|} Nn(Xn

s , dy)dHns

).

Since hn(y) = 0 for q.e. x ∈ E \ H1n and hn(x) = 1 for q.e. x ∈ H2

n, the last term ofthe above equation coincides on {t < τH2

n} with

limε→0

(∫ t

0

∫(hn(Xn

s ) − hn(y))(wn(y) − wn(Xns ))1{ε<|wn(y)−wn(Xn

s )|} Nn(Xns , dy)dHn

s

−∫ t

0

∫(hn(Xn

s ) − hn(y))(vn(y) − wn(Xns ))1{ε<|vn(y)−wn(Xn

s )|}Nn(Xns , dy)dHn

s

)

=∫ t

0

∫(hn(Xn

s ) − hn(y))(wn(y) − wn(Xns ))Nn(Xn

s , dy)dHns

−∫ t

0

∫(hn(Xn

s ) − hn(y))(vn(y) − vn(Xns ))Nn(Xn

s , dy)dHns

= 〈Mhn, j(En), Mn, j〉 − 〈Mhn, j(En), Mwn, j(En)〉.


where for two elements L, J of M(En), 〈L, J〉 denotes the compensator of thequadratic covariation [L, J] and L j denotes the jumping part of L.

Therefore we have Px—a.s. for q.e. x ∈ E on t < τH2n,

Mn,dt − Mwn,d

t (En) = Knt := 〈Mhn, j(En), Mn, j〉t − 〈Mhn, j(En), Mwn, j(En)〉t.

Thus, for q.e. x, Px—a.s. on t < τH2n, Cn

t = Mwn,ct − Mn,c

t − Knt + Nwn

t (En).Since the quadratic variation of Cn

t is equal to zero on {t < τH2n}, Pm—a.s., the

continuous martingale Mn,c − Mwn,c is equal to zero on {t < τH2n}, Pm—a.s. Then we

have Px—a.s. for m—a.e. on t < τH2n, Cn

t = −Knt + Nwn

t (En). In fact the precedentequation hold on {t < τH2

n} Px—a.s. for q.e. x ∈ E thanks to an argument of the

refinement used in the proof of Lemma 4.6 of [3].Kn is the difference of two positive CAF’s Kn,1 and Kn,2. Denote its Revuz

measures by μn,1 and μn,2. For any n there exists {Hn,k} ∈ Gn such that 1Hn,k ∗ μn,1

and 1Hn,2 belong to S0(En), the set of measures of finite energy integrals. (SeeSection 2.2.2 of [4]). It follows from Lemma 3.2 (and by taking a subsequence ofGn if necessary) that there exists a nest Gn such that Gn ⊂ Gn, 1Gn

∗ μn1 and 1Gn∗ μn2

belong to S0(En) for any n. Let γn,1 and γn,2 be the 1-potentials of 1Gn∗ μn,1 and

1Gn∗ μn,2 respectively. Set γn := γn,2 − γn,1. Then γn ∈ FGn and Px—a.s. for q.e.

x ∈ E, for any t < τGn, (see Lemma 5.4.1 in [4]) Kn

t = Nγnt (En) − ∫ t

0 γn(Xns )ds.

Set Gn := Gn ∩ H2n. Then (Gn)n∈N ∈ and Px—a.s. for q.e. x ∈ E on {t < τGn}

At = −∫ t

0vn(Xs)ds − Nγn

t (En) +∫ t

0γn(Xn

s )ds + Nwnt (En)

Then thanks to Lemma 3.9, Px—a.s. for q.e. x ∈ E on {t < τGn} At = Nunt −∫ t

0un(Xs)ds, where un = R1vn − γn + wn − R1wn. ��

Acknowledgements This work was part of my PhD thesis realized in the university of Paris VI.I would therefore like to thank sincerely my PhD advisor Nathalie Eisenbaum for every helpfuldiscussion that led to improvement of the results in this paper. In addition, I would also like to thankthe anonymous referee for a detailed and insightful report that greatly improved the exposition ofthe paper.

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http://tel.archives-ouvertes.fr/tel-00627558_v1/=

http://tel.archives-ouvertes.fr/tel-00627558_v1/=

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