Chapter 5 Further Analysis by Additive Functionals

CHAPTER 5

Further Analysis by Additive Functionals

Just as in the last three sections, we assume throughout this chaptei that we are given a symmetric Hunt process whose Dirichlet space is regular. It is shown in $ 5.1 that all equivalence classes of PCAF’s of the Hunt process are in one-to-one correspondence with all smooth measures introduced in $3.2. Such a correspondence is useful in that it enables us to reduce the calculus of functionals to the calculus of measures. Sections 5.2, 5.3, and 5.4 are concerned with additive functionals which are not necessarily non-negative. In particular, the AF A:] = tl(X,) -a(X,) generated by the function u in the Dirichlet space admits a unique decomposition A C U I = M I U I + “Ul, where McU1 is a martingale AF of finite energy and ““I is a continuous A F of zero energy (8 5.2). NP1 is not of bounded variation in t unless u is a potential of a signed smooth measure, in a certain sense (5 5.3). Thus, the decomposition in $5.2 may be regarded as a generalization (toward the Dirichlet space) of the Doob-Meyer decomposition of supermartingales and Ito’s formula on semi-martingales. A notion of stochastic integrals and a generalized Ito’s formula for the M A F Mrul are formulated in $5.4, which are then used to derive the Beurling-Deny second formula. A time changed process of the Hunt process by means of a PCAF gives rise to a Dirichlet space (the time changed Dirichlet space) living on the support of the AF. A rule relating the time changed Dirichlet space to the original one is established in $5.5. Each section contains examples concerning the Brownian motion or more general diffusions.

$5.1 Positive Continuous Additive Functionals and Smooth Measures Let ( X , m) be as in 4 1.1 and M be as in 34.3, namely, M = (Q, A’, X,, 123

124

FURTHER ANALYSIS BY ADDITIVE FUNCTIONALS

(z

P x ) is an m-symmetric Hunt process on X whose Dirichlet space a) on L2 ( X ; m) is regular. Potential theories relevent to (3; a)and M have been developed in the preceding two chapters. We now utilize these theories and study the structures of some important classes of additive functionals of the Hunt process M.') Let us call an extended real valued function A,(@),t 2 0, w E Q, an additive functional ( A F in abbreviation) if A,(o) is a perfect additive functional in the ordinary sense2) but with respect to the restricted Hunt process MI X - - N , N being a properly exceptional set which depends on A in general. More precisely, in order for A,(w)to be an AF the following conditions must hold: (A.l) A , ( * )is &-measurable, {F, being }the minimum completed admissible family (see 54.1); (A.2) there exist a set A E K- and an exceptional set N c X such that Px(A) = 1, vx E X - N , B,A c A , vt > 0, and moreover, for each w E A, A.(w) is right continuous and has the left limit on [0, w ] , A 0 ( o )= 0, ]A,(w)l < 00, vt < am), A,@) = AL(,, (U), vt 2 T(w) and A,+,(@) = A,(w) A,(Osw), vt, 3 2 0. The sets A and N referred to in the preceding paragraph are called a defining set and an exceptional set of the A F A,(w) respectively. Two AF's A ( ' ) and A ( 2 )are said to be equivalent if for each t > 0 Px(Ajl)= Aj2)) = 1 q.e. x E X . By virtue of Theorem 4.2.1, we can then find a common defining set A and a common properly exceptional set N of A ( ' ) and A ( 2 )such that All)(o) = Ai2)(w),vt > 0, vw E A . An AF which is non-negative and continuous on its defining set is called a positive continuous AF (PCAF in abbreviation). The set of all PCAF's is denoted by A:. The aim of this section is to characterize A: by the class S of smooth measures associated with (3; a)(see $3.2). TOthis end we first construct a PCAF for any ,u E &-the positive Radon measure of finite energy integral.

+

Lemma 5.1.1. For any u E 3; v E So, 0 < T < co and

E

> 0,

Proof. Taking a quasi-continuous Bore1 version II,we let E = { x E X : III(x) I > E } . Then the left hand side of (5.1.1) is dominated by eTJXp(x)v(dx) with p ( x ) = E,(e-"E). By virtue of Theorem 3.3.1 and Theorem 4.3.5, we have 1) The convention made in the last chapter is still in force in this chapter: any numerical functionfon X i s extended to X J by settingf(d) = 0. 2) Cf. R.M.Blumentha1-R.K.Getoor[14].

POSITIVE CONTINUOUS ADDITIVE FUNCTIONALS

125

Lemma 5.1.2. Let {u,,} be a sequence of quasi-continuous functions in 2T If {u,} is an 8 ,-Cauchy sequence, then there exists a subsequence {unk}satisfying the condition that for q.e. x E X . Px(u,,(Xr) converges uniformly in t on each compact interval of [0, w)) = 1. proof. Take {nk} such that J J ~ ( u , , ~+ u,,,, ~ unk+,- unk)< = 2-=. By virtue of the preceding lemma, Pu(n,) 5 eT2-k Ja,(Y) with A k = {sup Iu,,,(Xt) - U , , , + ~ ( X ~> ) ~2 - k } . By the Borel-Cantelli lemma, the 0515 T

P,-measure of the set A =0

= 5A, k

vanishes for every v

q.e. by virtue of Theorem 3.3.2.

E

So. Hence, Px(A)

Q.E.D.

Henceforth we use the notions ( v , u) = (u, Y ) = Jx u(x)v(dx) for a measure Y and a function u.

Theorem 5.1.1. For any p E So, there exists A E A: such that E,(J’; e-‘dA,) is a quasi-continuous version of U l p . Proof. Take a non-negative finite Bore1 and quasi-continuous version

u of U , p such that

(5.1.2)

I

nR,+lu(x) t u(x), U ( X ) = 0, v~ E N ,

+

a,v~ €

X - N,

where N is some properly exceptional set and {R,, a > 0} is the resolvent of the process M. We let =

n(u(x) - nR,+,u(x)), x I0

€

X -N

, X€N.

-

Then it is easy to see that g , *)yi p vaguely, R,g,(x) 1 u(x), vx E X N , and, moreover, R , g , is 8,-convergent to u. We define the approximating 1-order PCAF A”, of M by A”,,(t,w)= 16 e-”g,(X,(o))ds. Then, for any Y E So, (5.1.3) Eu((an(m> - Jdm))’) 5 2 M ,

Jm)IIRig, - RIgiIIg,’)

126

FURTHER ANALYSIS BY ADDITIVE FUNCTIONALS

with Mu = IIU2ullm.In fact, by setting g , , / = g , - g,, n side of (5.1.3) is equal to

> I, the left hand

Applying Schwarz inequality and noting B I(Rlgn,Rig,,) = (g,,,R,g,) d = = g l b ) , we obtain (5.1.3). k n ? Since Ev(A,(w) I Z) = A",,(?) e-'Ex,(J,,(co)) = A",(t) e-'Rlg,(Xr), we see that

+

+

(5.1.4) M,,(t) = A",(t)

+ e-'R,g,(X,),

is a martingale with respect to

0

tI w,

(z,Pw),u E So,.

By Doob's inequality')

which, combined with (5.1.3), means that there eixsts a subsequence Ink} such that

(5.1.5) Pv(Mnk(t)converges uniformly on [0, w)) = 1, '5~ E So0. In view of (5.1.4), (5.1.5), Theorem 3.3.2 and Lemma 5.1.2, we conclude that, by selecting a new subsequence if necessary, P,(A) = 1, x E X - fi, where

(5.1.6) A

=

{oE s'i: ~?,,(oc), w ) < w, Ank(t,o)converges uniformly in t on each finite interval of [O, w)}

and fiis some properly exceptional set containing N . Let us set A"(t,o)= lim A",,(t, o)for w E A and A"(t,o)= 0 for o @ "k'm

A . Furthermore, let A(t,o) = 1: esdA"(s,o).A is then a PCAF with A and fl being its defining set and exceptional set respectively. In order to complete the proof, it Suffices to show E,(A"(w))= (u, u), 1) Cf. P.A.Meyer [65].

2)

d is the set defined by (4.1.6) for 9 = X - #.

127

POSITIVE CONTINUOUS ADDITIVE FUNCTIONALS

v~ E Soo,by virtue of Theorem 3.3.2. Since M , ( w ) = A",(w) is L2(Pv)convergent, so is the martingale M,(t). Hence, E,,(A"(t)) e-'(u, p,u) = lim E,,(M,(t)) = lim E,(A",(oo))= lim(v, Rig,) = (Y, u ) . By letting t

+

n-m

n-m

"4m

tend to infinity and noting the bound (v,p,u) = B , ( U l v , p , u )5 J ~ ' , ( Y ) J g !(a, u), we get the desired equality. Q.E.D.

Lemma 5.1.3 Consider p and A of the preceding theorem. Then for any a > 0 and bounded non-negative Bore1 functionf, Ex(/: e-arf(X,)dA,) is a quasi-continuous version of U , ( f - p ) . Proof. It is sufficient to consider the case where a = 1 and f = Z, G being an open set with p(dG) = 0. Let #(x) = Ex(I: e-fIG(Xr)dAf), y(x)= ~-'zX-G(Xf)dAt);then both 4 and y are 1-excessive and 4 4- y = U l p . By Theorem 3.2.1 and Lemma 3.3.2, 4 and y are quasi-continuous versions of 1-potentials of some measures 3. and Y E So respectively. We know from Lemma 4.4.1 and the equalities 4 = HY4, v/ = H X - G y that Supp[A] c G and Supp[v] c X - G. Since p = A v , we

&(I,"

+

have Iz = IGw,u.

Q.E.D.

Theorem 5.1.2. For p (up to equivalence).

E S,,

A E A: of Theorem 5.1.1 is unique

Proof. Suppose that A ( ' ) , A(') E A: are associated with p E So in the manner of Theorem 5.1.1. Then Ex(J:e-ldA(') I ) = e-'dAj2)) = u(x), x E: X - N , for some properly exceptional set N . In the same way as in the proof of Theorem 5.1.1,

Henceforth we use the notations (fA), = ~ ' f ( X S ) d A , ,U $ f(x) E x ( ~ m e - u r f ( X r ) d Afor , ) A E A t and J E B+.

=

128

FURTHER ANALYSIS BY ADDITIVE FUNCTIONALS

Lemma 5.1.4. For p E So and A E A,+, the following conditions are equivalent to each other: (i) Ull is a quasi-continuous version of U , p , > 0, f, h E s+, (ii) (h, U g f ) = ( F p , &A), (iii) Eh.m((fA)r) = ( f * p , p h h ) d s , t > 0, f, h E a+, (iv) lim ( l / t )I $ , . ~ ( ( ~ A=) ~(Fp, ) h ) for any y-excessive function h (y 2 rl0

0)and f E B+, (v> lim ( l / t ) Eh.m(Ar) = (p,6, h E nK fl0 vi) a(h,~ g + y f ) ( f i p , h), a 00, for any y-excessive function h (y 1 0 and f E B+, (vii) lim U ~ I= ) (p, A), h E a+n E a-,

*+

Proof. (i) is equivalent to (ii) by Lemma 5.1.3. We can also see the equivalence of (ii) and (iii) by the uniqueness of the Laplace transform. The implications (iii) j (iv) j(vi) =$ (ii), (iii) j (v) and (ii) 3 (vii) are clear. Suppose that (v) is satisfied and put c,(x) = E,(A,). Then

for any pexcessive function h in ST By taking the Laplace transform we get (ii) for h of this type and f = 1, which is enough to obtain (i). In the same way we can derive the implication (vii) 3 (i). Q.E.D. Lemma 5.1.5. Let p E So and A 5.1.4. Then for any closed set F c X

E Af

be related as in Lemma

where h is any y-excessive function (y 2 0), f E 9+ and F (O) is the fine interior of F, i.e., F(O) = {x E X: PX(ox-,,> 0 ) = 1) (cF). Proof. It suffices to give the proof when h is bounded excessive and belongs to L2(X;m) and f E Z9+ is bounded. By Lemma 5.1.3, Ugf is a quasi-continuous version of V,(fp). By virtue of Theorem 4.4.1, a&.,,,

(~~-pe-aff(Xf)dAr) = n(h, Ug f - H,X-”U$ f)

POSITIVE CONTINUOUS ADDITIVE FUNCTIONALS

which increases to

h(x)f(x)p(dx) as a

t

03.

129

Here R:h(x) =

Before proceeding to the main theorem of this section, we need one more lemma. Recall the definition of the smooth measure given in 53.2: p E S if p is a positive Bore1 measure charging no set of zero capacity and there exists an increasing sequenc- IF,,} of compact sets satisfying conditions (3.2.10), (3.2.11) and (3.2.12). Thus the class S contains all positive Radon measures charging no set of zero capacity.

Lemma 5.1.6. Let {Fn} be an increasing sequence of closed sets. IFn} satisfies (3.2.12) if and only if (5.1.7)

P,(lim ox+, n-==

< C) = 0 q.e.

x

E X.

Proof. Condition (3.2.12) is equivalent to (5.1.8)

P,(lim n-co

~ 7 G - p= ~

w) = 1 q.e.

for any relatively compact open set G on account of (4.3.5). (5.1.7) implies (5.1.8) because of the quasi-left continuity of the Hunt process M. To get (5.1.7) from (5.1.8), it suffices to choose a sequence {G,} of relatively compact open sets with GIc GI+,,GI t X , and observe the inequality

2

Aox-0,.

Q.E.D.

Theorem 5.1.3. The family of all equivalence classes of A: and the family S are in one to one correspondence specified by the following relation : (5.1.9)

lim+Eh.m((f*A),) rl0

=

( f - p , h ) , A E Af, p E S,

for any y-excessive function h(y 2 0) and f E S+. The proof of Theorem 5.1.3 is accomplished by the following two lemmas.

130

FURTHER ANALYSIS BY ADDITlVE FUNCTIONALS

Lemma 5.1.7. (5.1.9) holds.

Given A

there exists a unique p E S such that

E A:,

Proof. For a given A E A$ with a properly exceptional set N c X , we put &x) = e-rf(X,)e-Ardt),x E X - N , where f is a Bore1 function in L z ( X ;m)such thatf(x) > 0, vx E X . Then, $(x) > 0, vx E X - N . It is easy to see that the difference R , f - 4 is 1-excessive on X - N . In particular, the function 4 is, together with R , f , finely continuous on X - N , and consequently, right continuous along sample paths. This enables us to obtain the folrowing expression of the difference: (5.1.10)

U ~ # ( X=) RJ(x)

- &x),

x E X - N.

Lemma 3.3.2 and Theorem 4.3.2 now imply that Ui+is, together with R , f , a quasi-continuous function in F a n d so is 4. Accordingly there exists a sequence {En} of increasing closed sets such that Cap(X- En) 1 0,

n

-

m

00,

N c fl ( X - En)and "= 1

$1

is continuous for each n.

Let us put

(5.1.11) Fn = lx E E , : #(x) 2

+\

and prove that {Fn}satisfies condition (5.1.7). To this end, set B,

X - N : pl(x) 2

=

{x E

nI 1 ,on = a," and a = lim on.Since $ is finely continuous n-m

on X - N , we have for x E X - N , E,(jb", e-rf(Xt)e-A*d t ) = E,(e-"n 1 e-Au$(XuJ) 2 F. By letting n tend to infinity, we can see P,(a < C) = 0, x E X - N , in view of the strict positivity off. Hence {Fn}satisfies (5.1.7) because of the inclusion X - F,, c (X- En) U B, and (4.3.5). Now let A , = I,*A. By virtue of the inequality Uinl 5 nUA#(E 3) and Lemma 3.3.2 again, there exists a unique p,,E Sosuch that U i n l is a quasi-continuous version of the potential Ul,un.But then

because Ufa,l = U)4,1Fnis a version of UIIP,.plby Lemma 5.1.3. We can now define a measure p by IFn*p= ZFn-pn, n = 1,2, . . , p ( X - U F,) = 0. p is smooth in view of Theorem 3.2.3 and Lemma 5.1.6. It remains to show that A and p are related by (5.1.9). By Lemma 5.1.4, we see for a n y f E 93+ and y-excessive function h, (fp, h ) =

131

POSITIVE CONTINUOUS ADDITIVE FUNCTIONALS

lim lim a(h, U:fy f)= lim lim a(h, U4+y(ZFn*f)).By (5.1.7), we get

n-m

(I-co

(fp,h)

= lim u--

(I-m

"-m

a(h, U s ; f ) , which is obviously equivalent to (5.1.9).

Lemma 5.1.8. Given ,ti E S, there exists A alence such that (5.1.9) holds.

E Af

Q.E.D.

uniquely up to equiv-

Proof. By Theorem 3.2.3 and Lemma 5.1.6, there exists a sequence IF.} of increasing closed sets satisfying (3.2.11) and (5.1.7) and ZFn-pE So for each n. By Theorem 5.1.1, Theorem 5.1.2 and Lemma 5.1.4, there exists A(")E A$ uniquely up to the equivalence such that A(")is related to ZFn-,u by (5.1.9). But then

because ZFn*A(I)is related by (5.1.9) to ZPn.ZF,.p= Z.,*,t.i Choose a properly exceptional set N and a defining set A which are common to all A ( " ) such that, for any o E A , Aj") (0) = ( I , , * A '"+'))t(o), vt > 0, n = 1, 2, , and a(oj) (= lirn ax-p,(o))2

--

n-m

c(o).For w E A let

Obviously A E Af. Since A , = A,(") for t Lemma 5.1.5, aEh., a

t

00,

(

< a,,

e - ( ( ~ + y ) r . f ( ~ , ) d ~t r ]JF

Fie) being the fine interior. Note that the set

(0)

we see from

h(x)f(x)~dx),

u F, - U FAo)is excepn

tional because of (5.1.7). By letting i z tend to infinity, we get lim a(h, U;+yf) = ( f - p , h ) , proving that A is related to ,u by (5.1.9). u-m

The uniqueness of A is then clear because ',,*A is related to ZF;p in the manner of Theorem 5.1.2 for each n. Q.E.D.

The proof of Theorem 5.1.3 is now completed. For A E A:, the associated smooth measure (according to Theorem 5.1.3) is denoted by p,+ The next lemma plays an important role in the subsequent sections.

Lemma 5.1.9. For any A

(5.1.14) E,(A,) 5 (1

E A:,

v E So, and

t

> 0,

+ t > l ! U , ~ l l ~ * p A ( ~ )( 5 .I

Proof. Assume first that ,u = pA is of finite energy integral: p E So.

132

FURTHER ANALYSIS BY ADDITiVE FUNCTIONALS

By setting c,(x) (5.1.15) c,

E

= &(A,),

x

E

X , we claim that

F a n d ZY(c,, v) = ( p , fl - p , f l ) , vv E X

By Lemma 5.1.4, the potential u i ( x ) of A is a quasi-continuous version of the potential U , p of p. Hence, ( p , c,) S e' ( p , u:) = e' 2$'l(p) < co. On the other hand, (l/s) (c,, c, -pscr) = (l/s)(c,, c, - p f c S ) = (l/s)(cr prcr,c,) which is, by Lemma 5.1.4, equal to (l/s) J: ( p , p.(c, - p,c,)) du = (1/s) (p, 2c,+, - c, - czr+Jdu. Therefore, we get lim (l/s)(c,, C, - pScJ SlO

= @, 2 ~ '- c~,) = (p, C, - p , ~ , < ) 00, proving that c, E 3'- and 2$'(cf,c,) = (p, c, - p f c t ) .Similarly, we have the relation in (5.1.15). For p E So and v E Soo, (5.1.15) gives us ,!?,(A,) = ( v , c,) =

+

+

al(uly, c,) = < p , F - Pf(W))( C f , U 1 v ) 5 II u14I,(P(m j x c,(x)rn(dx)), which proves the inequality (5.1.14) by noting that

sup (1/t) J,c,(x)rn(dx) = lim (lit) E,(A,) f10

t>O

= p(X).

When p = p A is a general smooth measure, we may consider the increasing sequence {F"} of closed sets of Theorem 3.2.3. Since ZFn*Ais 5 (1 t ) llulvllm*p(~n). BY related to ZFn-pE So, we have EV((ZFn*,4),) letting n tend to infinity, we arrive at (5.1.14). Q.E.D.

+

Finally, we briefly mention the sweeping out of PCAF. Consider p E

Soand the associated A E A,+. Ufil is a quasi-continuous version of

U,p.

Take any Bore1 set D c X.By virtue of Lemma 4.4.1, the function HyU,!,l is then a quasi-continuous version of the potential U l p D p, D being the 1sweeping out o f p on D (see the end of $3.3.). Since p D E So, we can consider a PCAF B associated with pn. B is called the 1-sweeping out of A on the set D.Thus, (5.1.16) Uil(x) = HfUj,l(x)

q.e.

B, does not increase on the complementary set of D in the following sense:

(5.1.17) PJB,

= 0,

'jt

< oD)= 1

q.e. x

E

X.

Indeed, we see from Theorem 4.2.3 that H f H f J ( x ) = Hff(x) q.e. x E Xfor a n y f E S + ( X ) (cf. 114; I, (11,9)]). Therefore,(5.1.16) implies E,(lT e-'dB,) = Uil(x) - H f U i l ( x ) = 0 q.e. x E .'A Example 5.1.1. Let M = (8,4 X,, P,) be the Brownian motion on Rw. The associated Dirichlet form B on L2(Rn)is given by (l/2D, H'(R"))(see Example 4.3.1). The absolutely continuous measure

POSITIVE CONTINUOUS ADDITIVE FUNCTIONALS

(5.1.18) p(dx) =f(x)dx,

133

f z 0,

is smooth with respect to the form 8 whenfis locally integrable. In fact, pis then a positive Radon measure charging no set of zero capacity since p vanishes on any set of Lebesgue measure zero. The PCAF of M associated with p by Theorem 5.1.3 is

Sb

(5.1.19) A,(o) = f(X,(w))ds. The same assertions hold even when p diverges around a set of zero capacity. Suppose that n 2 2. Then, for example, (5.1.20) f ( x ) = \xIu, x

E

R",

gives a smooth measure for any real a. Indeed, F,, = {x E R": l/n 1X I 5 n} satisfies not only (3.2.10), (3.2.11), but also (3.2.12) because the one point set {0} is of zero Newtonian capacity (see Example 3.3.1). The associated PCAF A , is still given by (5.1.19). A , satisfies the finiteness condition (5.1.21) P,

(r

IX,lads < m)

=

1, t

> 0,

whenever x # 0, since {0}is polar (see Example 4.3.1). Because of the law of the iterated logarithm, however, (5.1.21) is violated for x = 0 when a 5 -2. Thus A , is not necessarily a PCAF in the ordinary sense. But it is always a PCAF in our sense because we can ignore the polar set {0} as an exceptional set of A,. When n = 1, the Brownian motion on R' admits no non-empty exceptional set (Example 4.5.1). Hence, the present definition of AF reduces to the ordinary one. By virtue of the remark at the end of Example 3.3.3, we easily see that a measure p on R1 is smooth if and only if p is a positive Radon measure. In particular, the &measure a{,,] concentrated at one-point y admits a PCAF 2l(t,y,w) according to Theorem 5.1.3. I(t,y) is called the Brownian local time. It is known that there exists a set A c Q with Px(A)= 1, vx E R', such that for each o E A , I(t,y,w) is continuous in two variables t and y." Notice that the formula of Lemma 5.1.4(iii) serves as a characterization of the correspondence of p E S and A E A&. By integrating the corresponding and A , = 2I(t,y), we get formula for p = (5.1.22)

J' IE(X,)ds= 2 s

Furthermore, we get

E

l(t,y)dy, E c R'.

134

FURTHER ANALYSIS BY ADDITIVE FUNCTIONALS

for the PCAF A associated with any positive Radon measure p on R'.

$5.2 Decomposition of Additive Functionals of Finite Energy Let X , m and M be as in the preceding section. In the following three sections we deal with those AF's of the process M which are not necessarily non-negative. For any AF A , of M, we set (5.2.1) e(A) = lim 1 E,(A;) 110

whenever the limit exists. e(A) is called the energy of A . First of all, we shall exhibit three important classes of AF's of finite energy.

x

(I) AF'sgenerated byfunctions. Suppose that a function u on possesses a version ii(ii = u m-a.e.) such that ii is finely continuous q.e. and finite q.e. Then

(5.2.2)

Apl = a(X,) - a(&),

t

>0

defines an AF in our sense and, indeed, a unique equivalence class independent of the choice of the version zi by virtue of Lemma 4.2.5. A["]is well defined for any u E X because we may take as ii a quasicontinuous version of u (Theorem 4.3.2). Moreover, APl is of finite energy and (5.2.3) e(Acul)= gres(u,u), u

E9

according to the formula (4.5.22). Here gres is the resurrected form deequals 8 if and only if there is no killing inside fined by (4.5.20). gres A.

Suppose that the Dirichlet space (F 8) ,of the process M is transient in the sense of 4 1.5. Let (E, 27)be the extended Dirichlet space. Then the Indeed, as we saw at the end of53.1, relation (5.2.3) extendes to all u E Fc. each u E Xeadmits a quasi-continuous version zi, by which the AF A["]is still well defined. On the other hand, we can see in the same way as the proof of Lemma 1.5.4 that zi satisfies the relation (4.5.7). Hence, (5.2.3) together with (4.5.5) extends to u E Fe.

DECOMPOSITION OF ADDITIVE FUNCTIONALS OF FINITE ENERGY

Consider the family

@I) Martingale AF’s of finite energy.

(5.2.4)

135

A= { M : M is an A F such that for each

1

> 0 E,(M:) < 03

and E,(M,) = 0 q.e. x E X } . Since E,,,(M;7) is subadditive in t , e ( M ) is well defined and (5.2.5)

1

e(M)= ! :s

Em(M,2) ( 5

for any M E A. We set

(5.2.6)

A= { M E A:e ( M ) < m}.

M E A i f and only if M is a square integrable ordinary perfect A F with mean zero of the Hunt process M I X - - N , N being some properly exceptional set (depending on M in general). (M,, K ,P,),z,, is then a martingale for each x E X - N . Hence, we can utilize a method of P.A.Meyer [67;111, ThCortme 31 to conclude that there exists for each M E A’a unique PCAF ( M ) E A t such that for any t > 0 (5.2.7)

EX((M),)= E,(M,2) q.e. x

E X.l)

We call M E A a martingale AF(MAF). ( M ) is called its quadratic variation. Denote byp<,, the smooth measure associated with ( M ) according to Theorem 5.1.3. puo, is called the energy measure of the MAF M. From (5.2.1), (5.2.7), and (5.1.9), we see that the energy of an MAF is just half of the total mass of its energy measure: (5.2.8)

1

e ( M ) = Tp<”>(X), M

(110 CAF’s of zero energy.

E

A.

Consider the family

(5.2.9) JY’, = { N : N is a continuous AF, e ( N ) = 0, E,(IN,I)< 00 q.e. for each r

> 0).

The quadratic variation of N E Ncvanishes in the following sense: (5.2.10)

flT

C (N(k+,,

k-1

/,

- Nk/n)2

-

0,

-

03,

in L’(P,,,)

1) Meyer’s construction gives an ordinary “non-perfect” PCAF ( M ) of M I x - N . Then ( M ) can be modified to be an element of A: by “perfection” (see P.A. Meyer 1681 and references therein).

136

FURTHER ANALYSIS BY ADDITIVE FUNCTIONALS

because the expectation of the left hand side equals

Since the right hand side of (5.2.12) is dominated by

N of (5.2.11) is of zero energy, i.e., N

E NC.

t

sds.V; f ) ,

We are particularly interested in the sum of the classes (11) and (119: (5.2.13)

S '

=

A@ Nc

namely, &consists of AF's A such that (5.2.14) A ,

=M

,+ N,,

M

E

.A?, N

E Nc.

Clearly &is a linear space of AF's of finite energy. Moreover, the expression (5.2.14) of A E &is unique because d Mc= {0}where 0 denotes the additive functional identically zero. In fact, if A E d i s of zero energy, then p(A)vanishes by (5.2.8) and so does ( A ) . Hence, A = 0 by (5.2.7). We define the mutual energy of A , B E d by (5.2.15) e ( A , B ) = lim 1 E,(A,B,). I10

We know by Schwarz inequality that e(A, B) = 0 when either A or B is in M,. Therefore, that

DECOMPOSITION OF ADDITIVE FUNCTIONALS 07 FINITE ENERGY

(5.2.16)

e(A) = e(M) if A

+N,

=M

137

M E .,&,N c M C .

The main purpose of this section is to show that any AF ALu3 (u E .T) in the first class (I) actually belongs to d,i.e., A["]is expressed as a sum of elements of .,&and Ncuniquely. To this end we first prove that the space -,&is a real Hilbert space with inner product (5.2.15). Theorem 5.2.1. .,&is a real Hilbert space with inner product e. More specifically, for any e-Cauchy sequence M ( " )E .,&, there exist a unique M E .,& and a subsequence n k such that lim e(M(")- M ) = 0 and for n-c=

q.e. x E X , P,(lim MPk) = M , uniformly on any finite interval o f t ) = 1. "k-m

Proof. By virtue of Lemma 5.1.9 and identities (5.2.7), (5.2.8), we have (5.2.17) E,(M,2) 5 2(1

+ 0 J I U L 4 1 d 4 ) ,u

E So,,.

(M,, PJao is then a square integrable martingale and Doob's inequality implies (5.2.18)

P,(SUP lMsl 09sT

> a> 5 ~2

(

+1T)llul~ll~e(M).

Assume that M(")E .,&constitutes an e-Cauchy sequence and select

a subsequence nk

-

1

00

such that e(M("k+')- M("k) < Fk.

Applying the Borel-CanteIli lemma to this inequality, we see (5.2.19) P,(A,)

=

1, Y E So,

where A , = {w E SZ; Ms("k)converges uniformly in s on each finite interval}. In view of Theorem 3.3.2, (5.2.19) implies Px(Ao)= 1 q.e. x E X. Denote by r k a defining set of M("k)and put A , = nr,. Furthermore, we set M,(w) = lim M,("*)(w),s 2 0, for w ")-'=

n A , . M , is then an AF k

E A.

with defining set A , n A , . Since MP' is L2(Pu)convergent to M , by virtue of (5.2.17), we see Eu(M:) < 00 and E,(M,) = 0 for any v E So,. It is easy to conclude from this with the help of Theorem 3.3.2 that E,(M,2) < 00 and Ex(Mt)= 0 q.e. x E X , that is, M E 4. For any E > 0, choose N

138

FURTHER ANALYSIS BY ADDITIVE FUNCTIONALS

such that e(M(")- M("1) < E , n, m > N . We have (1/2t) E,((M,'") - M,)') - E by Fatou's lemma and (5.2.5). Then e ( M ( " )- M ) 5 E by (5.2.5) I again. Hence, M E .&and M ( " )is e-convergent to M . Q.E.D.

Theorem 5.2.2. uniquely as

For any u

the AF A["] can be expressed

E

Moreover, it holds that

u). (5.2.21) e(Mcul)= grres(u, In the transient case, the above statements extend to any function u in the extended Dirichlet space E.

Proof. We already know the uniqueness of the decomposition (5.2.20) and it suffices to show the existence of such Mruland ""I. We start with the special case where u is in the range of the resolvent: u = R,f,fbeing a Borel function in L2(X;m).In this case (5.2.20) is reduced to the usual semi-martingale decomposition, namely, we may set

As we have seen, ""1 of (5.2.22) belongs to Nc.Since u is an element of A?] = u(X,) - u(XJ is of finite energy and Mpl of (5.2.22) satisfies the relation (5.2.21) in view of (5.2.3). We easily see that Mi"' E JZ') and hence MrulE A. Next, take any Borel function u E r a n d define the approximating function u,, by (5.2.23) u, = nR,+,u

= R,f,,

with

.fn

= n(u - nR,+,u).

By the uniqueness of the decomposition (5.2.20) for u,'s, we have then M [ G - M[%I = M[u*-uml and (5.2.24) e(MC"J- M"+J)= gres(un- u,, 1 ) Cf. footnote of pp. 132.

U,

- urn).

DECOMPOSITIONOF ADDITIVEFUNCTIONALS OF FINITE ENERGY

139

Since un E F a r e 8,-convergent to u and a,(v,v) dominates gres(v,V ) for any v E T,we conclude that M C u n l is an e-Cauchy sequence in the space By virtue of Theorem 5.2.1, the formula

A.

makes sense and

(5.2.26) MculE

2, e(MCu3) = grres(u, u),

uE

X

of (5.2.25) belongs to the space It only remains to show that Mc.Note that a subsequence nk exists and (5.2.27) P,(Ntc".kl converges to NfUl uniformly on any finite interval of t)=1

q.e.

x

E X,

because the same statements for ACUnland M["nl hold on account of Lemma 5.1.2 and Theorem 5.2.1 respectively. From this and (5.2.22), we know that NrU1is a CAF. On the other hand, we have from (5.2.25) N,CUI = &-u,I - ( M y - MCYnl) Nf"nl and, consequently, - 1 ) 3 e(MLU1 - MCUnl). lim Em ((NP')*) 2 3 e(ACU-"nl

+ +

110

By virtue of (5.2.3), (5.2.24) and (5.2.25)) the right hand side equals 6 gres (u - u,, u - u,), which can be made arbitrarily small with large n. Therefore, e(NLU1) = 0 and NLulE xc. When the process is transient, we can extend the decomposition (5.2.20) and the relation (5.2.21) to any u E re in the same way as above by making use of an approximating sequence u, E 3which is 8-convergent t o u. Q.E.D. Corollary 1. The following linearity and continuity hold: M [ a u + b o I = aMLu1 + ~MCVI NCuu+bvl = + b"v1, a, b E R',u, v E F ( u , v E Zein transient case). (ii) u,, u E T ( u , , u E Fein transient case), a(u, - u, u, - u) 0, n 00 a subsequence {un,},lim M,Lunkl= MPI, lim Nf"W = NPl

(i)

-

9

-.

"k-"

"k-."

uniformly on each finite interval of t , P,-as. for q.e. x

E.'A

In fact, the relation (i) fol!ows from the uniqueness of the decomposi-

140

FURTHER ANALYSIS BY ADDITIVE FUNCTIONALS

tion (5.2.20). The equality (5.2.21) means that the linear transform (5.2.28)

@ :u

-+

Mc":

(z

is continuous from the Dirichlet space Z ) (the extended Dirichlet space (Fe, Z ) in transient case) into the space ( 2 , e ) . Combining this with Theorem 5.2.1 and Lemma 5.1.2, we conclude the continuity statement (ii). (5.2.21) also implies the following embedding statement telling us that the structure of the Dirichlet space may be studied entirely within the framework of the space A o f martingales of finite energy. Corollary 2. Suppose that the process M is transient and that there is no killing inside X. Then the transformation Q, of (5.2.28) is an a)can be identified by CD isometry and the extended Dirichlet space (Fe, with a closed subspace of (2, e).

Having established decomposition (5.2.20), we are now in a position 1 W" in (5.2.20). In this to study some basic properties of AF M C Uand section we give a theorem by which we can compute the energy measure of the martingale part Mcul.Denote by Fb the family of all m-essentially bounded functions in K Theorem 5.2.3. (5.2.29)

s

X

When

z1

E

Kb,

f ( x ) , ~ ~ ~ [ , , ~=, (28res(u*f, dx) u) - 2Yres(uz,f), f~

&.

Proof. By (5.1.9) and (5.2.7), (5.2.30)

s

X

f ( ~ ) , u < ~ ~ ~= ! ) (lim d x ) Et.,((MF"1)2) I10

w h e n f i s a bounded y-excessive function ( y 2 0) in K Since MP1 = a(XJ - a(X,) - Npl and Ef.,,,((NP1)')5 llfIlm E,,,((N,c"3)z),we can see that the right hand side of (5.2.30) equals

-

which is equal to the right hand side of (5.2.29) in view of (4.5.20). (5.2.29) is now valid f o r f = aR,g, g E Fb. After letting a 00, we get (5.2.29) for general f E Fb.Q.E.D.

CONTINUOUS ADDITIVE FUNCTIONALS OF ZERO ENERGY

141

Example 5.2.1. Consider the case where X i s an Euclidean domain D and M is an m-symmetric diffusion process on D whose Dirichlet form 8 on L2(D;m) possesses CZ(D) as its core. We assume for simplicity that M admits no killing inside D. From Theorem 2.2.2 and Theorem 4.5.3, we know then that 8 = gre3 and 8 should have the expression

with some measures Y,, satisfying (1.2.3). In 45.4, we give still another derivation of this formula using the stochastic calculus. Now the right hand side of (5.2.29) equals

and thus (5.2.29) implies

Suppose that every i',, is absolutely continuous with respect to the Lebesgue n

measure. Take Bore1 density functions a,,(x) such that C aif(x)&tj2 0, Vx

E D,"E E

(5.2.32)

,.]=I

R". In view of Theorem 5.1.3, the identity (5.2.31) then implies

, ,k

(iW"}, = 2 r (

au au - - a,,)(X,)ds,

ax, ax,

ii E

C,"(D).

Furthermore, if a,, satisfies

for some positive constants 1, and A,, and if m(dx) = dx, then (5.2.32) holds for any u E Fb(namely, for any bounded function in HA(D)) because so does (5.3.31).

55.3 Continuous Additive Functionals of Zero Energy We saw in 55.2 (111) that any AF in t h e class Mcis of zero quadratic variation in a certain sense. Nevertheless, Mccontains many CAF's which are not of bounded variation. I n this section we concentrate o u r attention o n the second term NculE Mcin the decomposition (5.2.20) of Theorem 5.2.2 a n d give some answers to the next basic questions:

142

FURTHER ANALYSIS BY ADDITIVE FUNCTIONALS

(I) When is NfUl of bounded variation in t? (11) Where is the support of NP1 located? Let us start with the following characterization of NCU1. Theorem 5.3.1. The following conditions are equivalent for an AF A in transient case): and for a function u E F ( u E (i) A = NCUI, (ii) A E Mc and, for each t > 0,

(5.3.1) &(A,)

= p @ ( x ) - zi(x),

(iii) A E Mcylim E,(A,) f10

= 0,

q.e. x E X , q.e. and

Proof. The implication (i) a (ii) 3 (iii) is clear. Assume that an AF A satisfies the condition (iii). Let us put c,(x) = E,(A,); then we see from A E Mcand the relation

(5.3.3)

Cr+s(x)

= c,(x) +p,cs(x)

q.e. for each t, s > 0,

that c, E L2(X;m), t > 0. By (5.3.2), (5.3.3) and (1.5.5), we have that for any v E E t > 0, and T > 0, (v --pTv, c,) = lirn l/s(S,v - psSTv,c,) = lim l/s(STv- p,STv, si 0

s10

- ,zT (S,v - ptSTv,u) = - 8(S,v, u - p,u) = -(v - p Tv ? 21 --p r u). Therefore, I, = (v, c, 3- u - p,u) satisfies the equation I, = I f + T - I , of the linearity. Since lim (lit) 1, = 0 by (5.3.2), I , = 0 and, consequently,

c,) =

I10

(5.3.4) c, =p,u - u m-a.e. In particular, this means that c, E F ( c , E 3,in transient case) and lim c, = 0 in 5Moreover, c,(x) is a q.e. limit of a quasi-continuous func110

tion pSc,-,(x) as s 1 0 in view of (5.3.3) and a condition in (iii). Since the convergence also takes place in the topology of 6 c, is quasi-continuous and (5.3.4) is strengthened as (5.3.1) (see $3.1). The implication (iii) 3 (ii) is proven. Finally, we assume condition (ii) for A . In order to derive (i), we set B, = NP1 - A, = a(X,) - ii(X,) - MP1 - A,, t > 0, q t ( X ) = E,(B:) and claim that

CONTINUOUS ADDITIVE FUNCTIONALSOF ZERO ENERGY

(5.3.5) q x x ) = 0 q.e.

for each t

143

> 0.

Clearly B, is a CAF of zero energy and E,(B,) = 0 on account of the assumption (ii). But then q,(x)nz(dx)= 0 because the left hand side is subadditive in t. We thus have q,(x) = 0 m-a.e., from which we can conclude psqr = 0 q.e. for each s > 0. By using Fatou’s lemma, qr(x) = p,,,q,(x) = 0 q.e., getting (5.3.5). Q.E.D. E,(lim (By+,,,- B,,,)’) 5

Ix

”-”

*-m

An A F A is said to be of bounded variation if A,(o) is of bounded variation in t on each compact subinterval of [0, c(o))for every fixed o in a defining set of A . A CAF A is of bounded variation if and only if A can be expressed as a difference of two PCAF’s: (5.3.6)

A,(w) = Ai”(w) - A ~ ” ( w ) ,A “ ) , A”’

E A:.

If an A F A is continuous and of bounded variation, then its total variation {A},(= IdA,I) is a PCAF’) and an expression (5.3.6) is provided by A:’) = { A },, A,(’) = { A }, - A, for instance. For the sake of definiteness, we say that each measure u in Sois offinite 1-order energy integral. In transient case, the same role is played by a measure Y offinite 0-order energy integral which is defined as a positive Radon measure satisfying

J

X

IvwIVw

s c JW,

v

E

F n c,(x).

Let us say that a signed Radon measure v on X is of finite 1-order (resp. 0-order) energy integral if so is the total variation I v I . This happens if and only if u can be expressed as u = u ( I ) - v (’), u (‘1 and v (’) being positive Radon measures of the same type. We can then define the a-potential of u by U,v = Uav(‘)- U , Y ( ~(resp. ) 0-potential of v by Uv = Uv(l)U v @ )which ) does not depend on the choice of v ( ’ ) and u(’). Moreover, the is integrable with quasi-continuous version G of any v E F(resp. v E Fe) respect to the total variation J u l and (5.3.7)

Z?,(U,v, v) = ( v , a)

(5.3.7)’

8(UU, v) = ( v , fl)

2,

E

2q

v €2 K.

Given a signed Radon measure u of finite I-order (resp. 0-order in 1) The measurability of ( A ] follows from Darboux’s theorem (Cf,[76; 0141).

144

FURTHER ANALYSIS BY ADDITIVE FUNCTIONALS

transient case) energy integral, we express it as v way and set

=v(l)- Y

in the above

( ~ )

(5.3.8) A = A‘” - A‘2’, where A ( ’ )and A ( 2 ) are PCAF corresponding to v ( ’ ) and v ( * ) respectively according to 9 5.1. A of (5.3.8) is a CAF of bounded variation and does not depend on the choice of v ( l )and Y ( ~ ) .

Lemma 5.3.1. For any signed Radon measure v as above, (5.3.9) NFUl” =

L-

U,V(X,)ds - A,, t 2 0,

and in the transient case. (5.3.10) NCU”= -A, where A is the CAF of bounded variation associated with v by (5.3.8). Proof. In view of the linearity (Corollary 1 to Theorem 5.2.2), we may assume that v is non-negative. Take v E So and a quasi-continuous version u of U,v. By virtue of (5.2.25) and (5.2.27), we have then Npl = lim f0 (nR,+,u(X,) -fn(X,))ds with f n = n(u - r ~ R , + ~ uthe ) , convergence n-”

being uniform in t on each finite interval P,-as. for q.e. x E X , by taking a subsequence if necessary. Hence the identity (5.3.9) follows from Lemma 5.1.2 and the method of the construction of A E A$ carried out in the proof of Theorem 5.1.1. In order to prove (5.3.10), observe that any positive Radon measure u of finite O-order energy integral is also an element of So and (5.3.11) CJu

=

U,V

+R(V),

where R denotes the O-order resolvent of the process M. Let f = F ;then R, f ( E 5) converges as CY 1 0 to Rf in (Z, a).Therefore, by the continuity (Corollary 1 to Theorem 5.2.2) and by an obvious modification of Lemma 5.1.2, we see

(5.3.10) follows from this and (5.3.1 1).

Q.E.D.

145

CONTINUOUS ADDITIVE FUNCTIONALS OF ZERO ENERGY

Before formulating an answer to the question (I) raised in the beginning of this section, we remind the readers of Theorem 3.2.3 and Lemma 5.1.6 (resp. their obvious modifications in transient case) :if a non-negative Bore1 measure Y is smooth, then there exists an increasing sequence { F k } of closed set satisfying

(5.3.12) PJim

k-==

< C) = 0 9.e.

x E X,

(5.3.13) Zpk*v E S,(resp. IPk*v is of finite 0-order energy integral) for each k. Let us call such a sequence {Fk} a nest associated with Y E S. Given two smooth measures Y ( ' ) and ~ ( ' 1 , we can always choose a common nest { F k } associated with Y ( ' ) and Y('). We then let

and call uk the restriction to Fk of the difference Y ( ' ) - Y ( ' ) . Each v k is a signed Radon measure of finite 1-order energy integral (resp. of finite 0order energy integral in the transient case). If we let

(5.3.15) Tk = {u E E ii

=0

q.e. on X - Fk},

then by Theorem 4.4.1 and (5.3.7),

(5.3.16)

a,(u,Vk

-Ha.k(Uz),

V ) = (Yk,

a),

v'8 E r k ,

The same statement for a = 0 holds in the transient case. Keeping these in mind we proceed to the proof of the next theorem. Theorem 5.3.2. The following two conditions are equivalent to each other for u E F ( u E 3,in the transient case): (i) NLfdl is a CAF of bounded variation. (ii) there exist smooth measures ~ ( ' 1and Y(') such that

(5.3.17)

a(u,

V ) = (Yk,

@),

vfl E F k ,

for every k. Here u k is the restriction to Fk of the difference Y (') - u ('I IFk} being a common nest associated with Y ( ' ) and u('). F k is the space defined by (5.3.15). 9

146

FURTHER ANALYSIS BY ADDITIVE FUNCTIONALS

is Proof. We only give the proof for u E 37 Suppose NculE Jc of bounded variation; then Nc”l = - A ( ’ ) A @ )for some A ( ’ )and A(’) E A:. Denote by Y(’) and Y ( ~ )the smooth measures associated with A ( ’ ) and A @ )respectively. Then the relation lim (lit) E,.,(NPI) = -(vk, v}

+

110

holds for a function ZI = h - Hl,kh( E x k ) , h being any l-excessive function in F i n accordance with Theorem 5.1.3. Hence, the relation (5.3.17) is valid for such type of functions in view of Theorem 5.3.1. For any v E Fk, we can choose a sequence {v,} of the above type which is 8’convergent to v. Since Y k is of finite 1-energy integral, v, is L’(X;1 ~ ~ 1 ) convergent to 8 as well. Thus, (5.3.17) is established for any v E r k for each k. ConverseIy, assume that the relation (ii) holds. Let A “ ) and A ( 2 )be the PCAF associated with the smooth measures Y (’1 and Y (2) respectively. We set A = - A ( ] ) A @ )and claim that

+

(5.3.18)

NC”’= A .

Fix k and rewrite (5.3.17) as

+

where ,uk = u.m v k . Since the left hand side of (5.3.19) equals g,(u - H,,ka,v) for any v E r k , (5.3.16) and (5.3.19) imply that (5.3.20)

U

- Hi,#

=

u#c, - H i , k ( G k ) -

+

Now the CAF IFk-A= --IFk-A(’) Z,,-A(2) is associated with the measure -Yk. Hence, B, = u(X,>ds- (IFk-& is associated with ,uk and, by Lemma 5.3.1, we arrive at the equality

On the other hand, Hl,k(i55k)is also a 1-potential of a measure nk,uk (a difference of swept-out measures in So).Denote by B ( k )the difference of PCAF‘s associated with akpk(we may ~ a l l B (the ~ )sweeping out of B on the set X - Fk;See the last part of 9 5.1). Then, once again using Lemma 5.3.1, we have

CONTINUOUS ADDITIVE FUNCTIONALS OF ZERO ENERGY

s:

147

(5.3.20), (5.3.21), (5.3.22), and Corollary 1 to Theorem 5.2.2 imply Np-Ri,rPI = -

Hl,kfi(Xa)ds

+ (IFk*A)t+ B,",

< (rk, by virtue of (5.1.17),

Since B," = 0, 'jf

f

> 0.

we conclude

It is easy to see that H1,kfi E Fconstitutes an Z? ,-Cauchy sequence. Since we may take as fi a quasi-continuous version of u in the restricted sense (see §3.1), the property (5.3.12) of the nest {Fk}, Theorem 4.3.2, and the quasi-left continuity of the process M imply Iim ffl,kfi(x)= Ex(e-G?(Xc))= 0 q.e. x

E X,from

k-0

which we get lim g l ( H l , k f iH1,ko) , = 0. k-or

We can now derive the desired equality (5.3.18) from (5.3.23) by letting k tend to infinity and by using (5.3.21), Corollary to Theorem 5.2.2, and Lemma 5.1.2. Q.E.D. Corollary. The following conditions are equivalent to each other for fe in the transient case). (i) u is a difference of two I-excessive functions in Firesp. 0-excessive functions in Ye). (ii) u = U1v(resp. u = Uv) with a signed Radon measure v of finite 1order (resp. 0-order) energy integral. (iii) N["' is a CAF of bounded variation and the associated measure Y - Y (') in Theorem 5.3.2 has the additional property that lim Yk I is a u E SY-(resp. u E

k-m

positive Radon measure of finite 1-order (resp. 0-order) energy integral. In fact, relation (i) e (ii) and (ii) 3 (iii) are implied in Theorem 3.2.1 and Lemma 5.2.1 respectively. (iii) j(ii) is trivial. Denote by SY-* the dual space of the Dirichlet space F w i t h metric &: Each element u E f d e f i n e s a unique T["'E T*by (5.3.24)

a(u,V)

=

(F"', v), v

CEST

Sometimes it is convenient to express the relation (5.3.2) as (5.3.25) lim+E,.,(A,) t10

=

- (T'"', v},

v

E s':

In the transient case. the dual mace SY-,* of the extended Dirichlet mace

148

FURTHER ANALYSIS BY ADDITIVE FUNCTIONALS

(Ye, a)plays a more specific role in characterizing the CAF's W": Theorem 5.3.3. Suppose the process M is transient. Then the family {NIU1, u E Xe}of CAF's stands in one-to-one correspondence with the :of the extended Dirichlet space, the correspondence being dual space F lim E,(A,) = 0 q.e. and characterized by theconditions that A E NC, 110

= - ( T, w), v E K. lim~E,.,(A,) 1 I10 Moreover, A is of bounded variation if and only if T E F,* is expressible as T = v ( l ) - Y Q ) , I J ( ~ ) , Y ( ~ )E S , in the sense of (5.3.17).

In fact, the relation (5.3.24) defines a one-to-one correspondence between 9 - z and Yein this case. Hence, Theorem 5.3.3 follows from the preceding two theorems. In order to formulate an answer to the question (ii) raised in the beginning of this section, recall the definition of the a-spectrum a,(u) of u E 3'given in $3.3. In particular, a set a(u) is called the (0-) spectrum of u E Y if O(U) is the compIement of the largest open set G such that 8(u, v) vanishes for any w E F n C,(m with Supp[v] c G.

Theorem 5.3.4. For any u E F ( u E Fe in transient case), the AF NPl vanishes on the complement of the spectrum F = a(u) of u in the following sense: (5.3.26) P,(NfU1 = 0, vt

< oF)= 1

q.e. x E X .

Proof. First of all, consider a measure p E Soand an AF A E A: which are related to each other by Theorem 5.1.3. SinceZ,*p andI,*A are related to each other for any Bore1 set E, we can say that the support of p is just the complement of the largest open set G on which A vanishes in the sense that

(5.3.27) P,((Z,.A),

= 0, vt

> 0) = 1,

q.e. x E X .

Take u E F a n d let F = ~ ( u )Then . al(u - G,u) c F. By invoking the theorem of spectral synthesis (Theorm 3.3.4), we can find a sequence { p n ) of signed Radon measures of finite (1-) energy integrals such that Supp [pn]c F and that the potentials u, = U , p n are S,-convergent to u - G,u. By subtracting a subsequence if necessary, we then conclude from Corollary 1 to Theorem 5.2.2 that

CONTINUOUS ADDITIVE FUNCTIONALSOF ZERO ENERGY

149

uniformly on each finite interval of r P,-as. for q.e. x E X . Denote by A,(")the CAF of bounded variation associated with pn in

s'

the manner of (5.3.8). Lemma 5.3.1 then implies N,CYnl=

A,("),while Lemma 5.1.2 implies that

- J: E ( X , ) &

J; an(xJds -

a,(X,)ds converges to

s'

tl(Xs)ds

= --NfGlul in the same sense as above. Therefore, we get

from (5.3.28) that

(5.3.29) lim A,(")= - N p 3 , n-m

the convergence being in the same sense as (5.3.28). Since A(")vanishes on the complement of F in the sense of (5.3.27), we are led from (5.3.29) to (5.3.26). Q.E.D. The support Supp[A] of an AF A is defined by

(5.3.30) Supp[A] =

{X

E

X : P,(R = 0)= l } ,

where R(o) = inf {t > 0: A,(w)

(5.3.31) Supp [N"9 c a(u), vu

+ 0}. Theorem 5.3.4 means E

K

Example 5.3.1. Just as in Example 5.1.1, we consider the Brownian motion M on R". Let us examine the properties of CAF NP1for u E H1(Rn). If the distribution derivative du belongs to L2(Rn),then

which follows from Lemma 5.3.1 by the help of the expression u = G,(u - 1/2du) and Example 5.1.1. In general, the following expression is valid : (5.3.33) N p

= lim e"l0

's' 2

0

du,,(X,)ds, u

E H'(

V),

where u, = pe*u is the convolution with themollifier p , (see Problem 1.2.1), and IF"} is a sequence decreasing to zero and depending only on the function u. The convergence in (5.3.33) is uniform on each finite interval o f t P,-as. for q.e. x E Rn. Indeed, a, being convergent to u in the space H'(R*)as E 1 0, Corollary 1 to Theorem 5.2.2 and identity (5.3.32) for I( = u,lead us to (5.3.33).

150

FURTHER ANALYSIS BY ADDITIVE FUNCTIONALS

In view of Example 5.2.1, MLul for u E H1(Rn)is an MAF with the quadratic variation being given by

by means of a stochastic integral. (5.3.32), In the next section we express MLU1 (5.3.33), and (5.3.34) tell us that our decomposition a(X,) - zi(Xo) = Mpl NPI, u E H1(Rn),is an extension of Ito's formula. when n 2 3), the A F N[lr]vanishes (in the For any u E H1(Rn)(u E re sense of Theorem 5.3.4) on the complement of the support of the distribution derivative Au on account of Theorem 5.3.4 and (3.3.20). There are many u E H1(Rn),however, for which NP1are not of bounded variation as is shown in the discussion of the one-dimensional case below. given by Suppose that n 2 3 and consider the extended Dirichlet space Fe (1.5.13). By virtue of Theorem 5.3.3, the family {Ncul,u E Fs) of AF's and the family {1/2Au, u E 3-Jof distributions are in one-to-one correspondence. The latter family exhausts the dual space described explicitly as follows: .I) 9-: = {T:tempered distribution, f E L2(15 I ""1 is of bounded variation if and only if 1/2Au is represented by a difference of smooth measures in the sense of (5.3.17). This condition is certainly satisfied if 1/2Auisa signed Radon measure of finite energy integral. But the converse is not necessarily true. For instance, consider a signed measuref(x)dx on R3 with

+

Fz

(5.3.35) f ( x )

=

1

Ixj"sinm1

Lo

1x1 < 1

1x1 2 1.

f(x)dx can always be expressed as a difference f ' (x)du--f-(x)dx

measures and

of smooth

(5.3.36) A , = s r f ( X s ) d s defines a CAF of bounded variation of the 3-dimensional Brownian motion with a possible exceptional set being the origin 0 (see Example 5.1.1). The 0-order resolvent is now given by the Newtonian kernel g(x,y) = -limit

1) Cf. Example 1.5.1. and J. Deny [25].

4n Ix - YI ~

and the

STOCHASTIC CALCULUS RELATED TO THE DIKICHLET FORM

151

converges for every x E R3whenever (Y > -3. The Dirichlet integral of u equals (5.3.38) lim el0

J

xl>s.lyl>e

g(x, v)fW

f(u) dxdy.

When -3 < a, (5.3.38) is convergent; consequently, the u defined in (5.3.37) belongs to Fe. Moreover, -Ncul is given by (5.3.36) due to Corollary 1 of Theorem 5.2.2. When -3 < a 5 -512 however, (5.3.38) is not absolutely convergent, which means that -1/2Au =fdx can not be expressed as a difference of positive Radon measures of finite energy integrals. Nor can u be expressed as a difference of excessive functions belonging to F. (Corollary to Theorem 5.3.2). In the case of one-dimensional Brownian motion, a non-empty exceptional set is absent and the smooth measure reduces to the positive Radon measure (Example 5.1.1). This considerably simplifies the situation. Thus NCulfor u E H 1 ( R 1 )is of bounded variation if and only if the distribution 1/2u” is a signed Radon measure or equivalently u‘ is of bounded variation on each finite interval. When this is the case, N p Jis expressed as the local time integral:

In fact, from (5.1.22) we see that

which combined with (5.3.33) lead us to (5.3.39). Here Z,(t,-) = pc* I&.) As an example consider a function g E C,(R1) satisfying JRt g(x)dx = 0 and set u(x) = J”g(y)dy. Then ““1 is of bounded variation if and only if the function g is of bounded variation.

$5.4 Stochastic Calculus Related to

the Dirichlet Form

A real-valued stochastic process 2, is said to be a semi-martingale if 2,is expressible as a sum of a (local) martingale and a process of bounded variation. In the last section we saw that our process a(XJ - a(Xo)= MP1 N p ] for u E Y i s not necessarily a semi-martingale. Accordingly the transformation rule of semi-martingales due to H. Kunita and S . Wata-

+

nabe (a generalized Ito’s formula) does not apply directly. Nevertheless, we can extend a due rule of transformation for the martingale part MCU1. Our strategy is first t o establish a transformation rule of the energy measures of 12.1cu1’s in a purely analytical manner and then to translate it in terms of a new notion of stochastic integrals.

152

FURTHER ANALYSIS BY ADDITIVE FUNCTIONALS

Our assumptions for X , m and M are the same as in the preceding three of MCu1 Ed 'for sections. For simplicity the energy measure p(MLul> u E F is denoted by ,qU) and called the energy measure of u. When u E Fb (an essentially bounded function in F)(5.2.29) , reads as follows:

Actually the energy measure of u E Fb can be introduced directly by this formula because the right hand side of (5.4.1), being equal to

as we saw in the proof of Theorem 5.2.3, defines a positive linear functional on 9-n C,,(X). Since the total mass of p(=, equals 28IeS(u, u), we can define the bounded signed measure p(u.v)for u, v E 3by

Then we have for u, v,,f E Fb,

Lemma 5.4.1.

For u, v ,f

E

9-n

Co ( X ) ,

J being the jumping measure appearing in the Beurling-Deny first formula (Theorem 2.2.1 and Theorem 4.5.2). Proof. By virtue of (5.4.3), the left hand side of (5.4.4) equals

STOCHASTIC CALCULUS RELATED TO THE DIRICHLET FORM

153

- lim II(t) where 110

.,.

J

I I ( ~ )= 1 X X X - d u(x)zv(x)f(x>(1 - +(v)M~, d y ~ d x ) , 4 being any non-negative function in 3-n Co(X)such that neighbourhood of Supp [u] U Supp [v]U Supp If]. It then holds that

#, = 1 on a

because the function (v(y) - v(x))f(x)4(y)belongs to C,(X x X - d ) and the measure (l/t)(u(y) - u ( ~ ) ) ~ p , (dy)m(dx) x, is uniformly bounded by 28(u, u) (see the formula (4.5.7)) and vaguely converges to 2(u(y) - U ( X ) ) ~J(dx, dy) on X x X - d as t 4 0. On the other hand, we see from Lemma 4.5.2 and Theorem 4.5.2 that

+J

II(t) = 1( u 2 v ~4 - p14) u(x)zv(x)f(x) (I - ptl(x)) m(dx) cont X u(x)~v(x)f(x)k(dx) = g r e s ( u 2 v ~4) verges as t I o to a ( u 2 v ~4) = 2 J-

XXX-d

J

X

u ( ~ > ~ v ( x ) f(1( x) $(y))J(dx, dy), which, combined with

(5.4.5), leads us to the desired identity (5.4.4).

Q.E.D.

Lemma 5.4.2. Assume that the Dirichlet space (F 8) ,possesses the local property, or equivalently, that the jumping measure J vanishes identically. Then

Proof. It suffices to prove

for any u, v E Fb.This is true when u, v E F f l Co(X)by virtue of the preceding lemma. Fix v E FflC o ( X ) and take any u E F,Let . u, E 3-fl Co(x> be 8,-convergent to u. By virtue of Theorem 1.4.2, we may assume that u, is uniformly bounded. Then by the same theorem and its proof, we can

154

FURTHER ANALYSIS BY ADDITIVE FUNCTIONALS

see that uf and ufw are 8'""-weakly convergent to u2 and ( w E x b ) . Hence, the formula (5.4.2) implies

uzw respectively

On the other hand, u, converges to u q.e. by taking a subsequence if necessary (Theorem 3.1.4). Furthermore, the total variation of the signed measure j ~ ( ~ - - ~ .is, > dominated by

in view of the formula (5.4.3). Hence,

J, I fu.1

+J

If1

If,

fun dp(,n.vo -

f,f u dp(,,,, 5

- -

lu, - uI I ~ P ( ~ . ~ , I 0, n 00, which, together with (5.4.7), implies (5.4.6) for u E r b and v E 3 n Co(X). In the same way, (5.4.6) extends to any zi E F b . Q.E.D.

l&(un-u,v)l

X

We can now prove a transformation rule of energy measures. Let @(x) = @(x,, x,, . . . ,x,) be any continuously differentiable real function on Rm(@ E C1(Rm)) vanishing at the origin. For any ul, u2, - ,urnE Fb, the composite function @(u) = @(u,,u2, , u,) then belongs to Tband

where V is an m-dimensional finite cube containing the range of u(x) = are uniformly bounded (u,(x),. . . ,u,(x)), x E X.If the derivatives on R", then @(u) = @ ( u , , u 2 , , u,) belongs to F f o r any u l , u 2 , . , u, E r a n d

-

These can be shown in the same manner as the proof of Theorem 1.4.2, (ii).

Theorem 5.4.1. Assume that the Dirichlet space the local property. Then

(Fa, ) possesses

155

STOCHASTIC CALCULUS RELATED TO THE DIRICHLET FORM

--

for any @ E C1(Rm)with @(o)= 0 and for any u,, u2, , urnE Xb. This formula remains true for any u l , u,, * , urnE F p r o v i d e d that Ox, are uniformly bounded in addition.

-

Proof. Let d be the family of all @ satisfying the first statement of the theorem. If @, Y E d,then by Lemma 5.4.2, dpcoc,,y(,)~,,-

v E Yb, that is, the product @Ybelongs to d.Since &contains the coordinate functions, it contains all polynomials vanishing at the origin. Take any @ E C'(Rrn)with @(o)= 0 and any u l , u 2, - ,urnE y b . Let V be a finite cube containing the range of u(x) = (u,(x),u2(x), , u,(x)). Then there exists a sequence {@(')(x)} of polynomials vanishing at @, )@ :: 1 5 i 5 m, uniformly on V.") the origin such that @(k) Due to the inequality (5.4,8), @(")(u)is then 8,-convergent to @(u) as k 03. Moreover, @(")(u(x)) is uniformly bounded and converges to @(u(x)), x E X. Therefore, by letting k tend to infinity, we can get (5.4.10) in the same way as in the last part of the proof of the preceding lemma. The same reasoning works in the proof of the second assertion of the theorem. Indeed, if we let ujk) = ((-k) V u J A k, 1 5 i 5 m, then is B -convergent to @(u). Q.E.D. (5.4.9) implies that @(u('))

-

-

-.

-

-

+

Turning to the next task of introducing the notion of a stochastic integral, consider the space Ac = { A - B : A , B E A:, E,(A,) < a ,EJB,) < 00, vt > 0, q.e. x E X I . For M , L E A, there exists a unique element ( M , L ) of A, such that (5.4.11) E,(M,L,)

= E,((M,L),), vt

> 0, q.e. x E X.

+

In fact, we may set ( M , L ) = (1/2) { ( M L ) - ( M ) - ( L ) } .See the paragraph right after (5.2.6) for the uniqueness. Let us consider the family = {M E J&' :,qM,( E S ) is a Radon measure} ATl is a linear space containing the family &?of MAF's of finite energy. G a unique signed Radon measure ,u
.

156

FURTHER ANALYSIS BY ADDITIVE FUNCTIONALS

Lemma 5.4.3. If M , L E AI, f E L2 X ; pcM,)and g E L2(X;p(,)), thenf-g is integrable with respect to the absolute variation Ip((M,L)I of c(w.L) and

Proof. Making rhe expression dp(M,L) = kldv, dp(M)= k2dv and Y = p(* p(,> lp(M.L)I, we get from (5.4.11) and

+ + (5.4.12), dp(oM+bt)= (a’k2 + 2abk, + b2k,)du for any a, b E R. Therefore, the set B, = U {x E X : a2k2(x)+ 2abk,(x) + b2k3(x) dp,,, = k3dv with

o.b:rational

< 0)

is Y-negligible and we have for each a, P E R a2f(x)’k2(x) 2aPI f(x)g(x)k,(x)I P2g(x)’k3(x)>= 0 for every x Integrating this with respect to Y ,

+

+

EX -

B,.

Q.E.D.

Theorem 5.4.2. Given M E A1and f unique element fa M E A? such that

E L2(X;P ( ~ ) there ,

exists a

The mappingf-tf. M is linear and continuous from L2(X;p(M))into the space (A?, e).

j

Proof. By virtue of Lemma 5.4.3, +JXf+(M.L)j

5

1

Ilfj/LZ(p(M))

JJ~(L>, LE

Hence, Theorem 5.2.1 implies Theorem 5.4.2 together with the inequality dqpq d (1IJT) I I L 2 ( p ( M , P Q.E.D.

-

11s

The f M E A? of Theorem 5.4.2 is called the stochastic integral. This terminology is legitimate by the following lemma.

Lemma 5.4.4. Let M,fand f . M be as in Theorem 5.4.2. Then (5.4.15) dj4f.M L) = f . d , ( , M . ~ L ) , E J&

157

STOCHASTIC CALCULUS RELATED TO THE DIRICHLET FORM

Moreover, the following approximation holds forf f Co(X): (5.4.16) lim Ex( {(f.Mjjd) - W M ) , }’) ldl-0

= 0,

f

> 0, 9.e.

X E

x

where

A denotes the partition 0 = to < t,

(4

<*

- ti-11.

- - < t, = t and

l A I = max ILiS

Proof. Let B be a common properly exceptional set for M and ( M ) such that the relation (5.2.7) holds for every x E X - B. Applying the same argument as in [67; 111, ThiorGm 41 to the Hunt process MI x--B and its AF’s M and ( M ) in the ordinary sense, we can see forf E Co(X)that there exists a unique martingale AF of M 1 x - B in the ordinary sense’) satisfying for each x E X - B,

(5.4.17)

lim E X ( { ( f . M ) j d)

ldl-0

at}’)

= 0,

Ex(@;) = E x ( ~ f f ( X a ) ) 2 d ( M ) , ) .

It follows from this that e ( @ ) = lim 5 1 Em(&:) =

a

tl0

and, consequently, E 2. On the other hand, (5.4.1 1) and (5.4.17) imply

+J

x

f 2dp(M,<

00

for any L E &and q.e. x E X. When L E 2, p ( ~ is . ~a )bounded signed 1 for any boundmeasure; hence, lim tE,,.m‘(G,L),) = fx h(~)p<~.~)(dx) I10

ed y-excessive function have from (5.4.18)

By setting h 1)

= aR,h,

h

11

E

in view of the relation (5.1.9). Thxefore, we

C,,(X), and letting a tend to infinity, we conclude

@can be made perfect by a method of “perfection” (see footnote of pp. 131). 1 hus, M E 4.

158

FURTHER ANALYSIS BY ADDITIVE FUNCTIONALS

that (5.4.19) holds for any h E C,(X). Therefore, d,
QL E A.In particular,

1 J xfdp(.w,L)which ~(fi,L ) = 7

means

fi=

f-M. We have proved Lemma 5.4.4 when f E Co(X).On account of Lemma 5.4.3 and Theorem 5.4.2, the relation (5.4.19) for l @ = f - M readily extends to f E L2(x;puo,).Q.E.D.

Corollary. (i) For M E Al,f E L2(X; P ( ~ , )and g €5 L2(X;f 2 P < M , ) , (5.4.20)

g * ( f * M )= g f * M .

(ii) For MyL

E

Aly f E Lz(X;

and g

E L2(X;P ( ~ ) ) ,

By using Theorem 5.1.3, we can translate the formula (5.4.15) for M E Al and Bore1 f E Lz(X;p ( M Jinto the following relationship for elements of Ac: (5.4.22)

( f * M ,L)= f * ( M , L), vL

E

A.

In this sense our stochastic integralfa M can be regarded as a variant of the stochastic integral due to M. Motoo and S. Watanabe. Note that (5.4.23) E , ( ~ ' f ( X J 2 d ( M ) ,< ) 00

q.e. x E X

for M and f as above. In fact, we have by Lemma 5.1.9 that a y 0f ( ~ J 2 d ( M ) 35) (1 t ) /IulyII,.~,f(x)2,,M,(~x) < O0 for any lJ E s o w The Motoo-Watanabe integral has been formulated when the finiteness condition (5.4.23) is fulfilled for every x E X rather than for q.e. x E X . See Example 5.4.1 below for a related discussion. Let us extend the notion of the stochastic integral f M to a wider class of AF's. We say that an A F M of the process M is locallv in A 1 ( M E MI, in notation) if there exists a sequence {G,,} of relatively compact 00, and a sequence { M ( " ) } open sets such that G, c G,,, and Gn' t X , n of MAF's in A, such that M , = M j " ) ,vt < f,Y--G,, P,-ass. for q.e. x E X. The class dloc of AF's is defined similarly. J
+

-

-.

STOCHASTIC CALCULUS RELATED TO THE DIRICHLET FORM

159

by choosing an appropriate defining set and exceptional set of ( M ) . ( M ) does not depend (up to the equivalence) on the special choice of {Gn}and { M ' " ) }for M. For each x E X outside the exceptional set of M and for each n, the process Mi,,, - (M)rAo,is a martingale with respect to P,. Here on = as the smooth We can still define the energy measure of M E measure p ( M )associated with the PCAF ( M ) . Due to Lemma 5.1.5, we then have

In particular, ,LL<~)is a positive Radon measure and Lemma 5.4.3 extends to M , L E &1,lOC. Therefore, the formula (5.4.14) still provides us with the stochastic ~ , f ~E ~L2(X; ~ p0). The relations integralf-M E &for any M E ~ 2 ' and (5.4.20),(5.4.21) and (5.4.22)extend to &q,loc as well. In particular, ( f . M ) , together with (f.M ) , vanishes until the first leaving time gx-G from a relatively compact open set G, P,-as. for q.e. x , whenever f vanishes on the set G. Keeping this observation in mind, we can finally define the stochastic integralf-M E JhOcfor A4 E &l,loc and f E L,2,,(X; P ( ~ , )by the relation (5.4.26)

g * ( f * M )= ( g f ) * M

which holds for any bounded Bore1 function g with compact support. In fact, it suffices to let ( F M ) ,= ((IG;f).M),, E < [ T ~ - ~ , , {G,} being any sequence of relatively compact open sets such that G, c Gntl and G, t X , n co. Here we mention a lemma which is useful in the subsequent developement. +

Lemma 5.4.5. Let C, and 93be a uniformly dense subfamily of Co(X) and an B ,-dense subfamily of X Then the family {fa MLU1 :f E C1,u cz 93} of stochastic integrals is dense in (2, e). Proof. Suppose that an MAF M E d is e-orthogonal to the above family, namely, j x f dp(M.Mlu~) = 0 v ' ~ C1,vu E 2%. This identity ex-

1 60

FURTHER ANALYSIS BY ADDITIVE FUNCTIONALS

tends to all u E F i n view of Lemma 5.4.3 and the equality pL(MCYl)(X) = 2gres(u, u). Hence, (5.4.27)

( M , MCu1) = 0 “u

E 2T

In particular, (5.4.27) holds for u = Rag, a > 0, vg E C,(X>.According to [67; 111, ThCor6me 61, we then have P,(M = 0) = 1, Y E Soo, from which follows M = 0. Q.E.D. In the remainder of this section we assume that the process M is a diffusion: Px(Xris continuous in t < i)= 1, x E X.In view of Theorem 4.5.1 the Dirichlet space ( K a ) then possesses the local property and the jumping measure J vanishes identically. Accordingly, (5.4.28)

gres(u,v) = ZYfic)(u, v), u, v E

with the form Z?@) having the property (4.5.14) stronger than the local property. For AF’s A ( l ) , A (2) and a stopping time 0,we write A;” = AjZ’ vt < J-r when this is true P,-a.s. for q.e. x E X .

Lemma 5.4.6. If the difference of u l , u2 E X i s constant on a relatively compact open set G , then (5.4.29)

M?“] = M 9 l

vt

<

Proof. From (5.4.1) and (5.4.28) we see that for any f~ 3n C,(X), f,f(x)p,,,(dx) = 0 whenever u E Fb and u is constant on a neighbourhood of Supp[fl. We can see that this statement holds for u E 3 as well. To see this, observe the inequality

1

( , / J x ~ ~ ~ d-~J(JUx)~ f ~ d ~ t c )X)IZ~ ~I ~ P ( 5 . -~ ~ )I I ~ ~ Y I( L U - - ~ u - v), which follows from either (5.4.3) or (5.4.13). It then suffices to set v = ( ( - n ) V u ) / \ n and let n tend to infinity. Therefore, p(u,-u2)= 0 on G under the assumption of the present ] MCu23)r = 0 vr < which means (5.4.29). lemma. Then ( M c u 1.. Q.E.D.

We say that a function u is locally in 3 ( u E KOc in notation) if for any relatively compact open set G there exists a function w E X s u c h that

STOCHASTIC CALCULUS RELATED TO THE DIRICHLET FORM U

= w m-a.e. on G. The space

uE

161

is defined in an analogous way. For

Fb,Ioc

KO,, there exists a unique M[ulE dlOc such that

(5.4.30)

MY3 = M Y ,

vt


~

-

~

for G and w as above. Indeed, Lemma 5.4.6 guarantees to set M Y = M,:"nl vt < where {G,} is a sequence of relatively compact open sets such that G, c G,+,, G, t X , and {un} are functions in F s u c h that u = u, on G,. In particular, we obtain from Lemma 5.4.6 (5.4.31)

u l , u2 E

To,, u, - u, = constant jMCU1I= iW"1.

We can now present our transformation rule in the space d&=.

Theorem 5.4.3. Let M be an rn-symmetric diffusion on X whose Dirichlet space (F a, ) is regular. and any u,, u,, ,urn E Fb,loc, the composite (i) For any SP E C1(Rm) function @(u) = @(u,, u2, . . ,tl,) is again locally in r b and

.-

--

with bounded dervatives and any u,, u2, , u, (ii) For any @ E C1(Rm) E KOc, the composite function @(u) is again locally in r a n d the formula (5.4.32) is satisfied. (iii) Suppose @(u) E F i n the case (i) or (3). Then

Proof. By virtue of Theorem 5.4.1, we have @(u) E Fb and J,fg&<@w."> =

5 J, fg@x;(u)c44u;.">,vA

z=1

g E C,(X), vv E

--

6for any

,u, E r b . This identity readily extends to any @ E C1(Rm)and any u,, u2 , . , u, E Fb,loc with the help of (5.4.25) and (5.4.31). The isometry (5.4.21) then gives @ E C1(Rm) with @(o)= 0 and for any u , , u2,

.

On account of Lemma 5.4.5, we have the following identity of the stochastic integrals in 2:

I62

FURTHER ANALYSIS BY ADDITIVE FUNCTIONALS

which in turn means (5.4.32) by virtue of (5.4.26). We have proved (i). (ii) can be proved in the same way. Under the )] 8 (“)(@(u),@(u)) by (5.4.28). condition of (iii), the energy of M [ @ ( ”equals Thus the expression (5.4.33) results from the calculation of the energy of both sides of (5.4.32). Q.E.D. Theorem 5.4.3 immediately leads us to the following presentation of the Beurling-Deny second formula: Theorem 5.4.4. Let X be an Euclidean domain D c R” and M be an rn-symmetric diffusion on D whose Dirichlet space (5a ) on L2(D;m) is regular. Suppose that the coordinate functions x , are locally in F ( 1 5 i d n). Then and, for any u E Cl(D), (i) C1(D)c TOc

(ii) Ci(D) c F a n d , for any u

E

CA(D),

(5.4.35) Each measure uij charges no set of zero capacity. If we make the additional assumptions that Ci(D) is 8,-dense in 9and that ui, vanishes identically for i # j , then the space A c a n be represented by the stochastic integrals:

A=[ ~ J - M ( ~ E) :LA~ ( D ;

YJ,

i= 1

(5.4.36)

1

i

n

Indeed, the right hand side of (5.4.36) is then a closed subsapce of (Jie), while its subclass {f.MCU1 = c f * i i ; - M ( i )., f Co(D), ~ u E Ch(D)} is i= 1

dense in (d& e) according to Lemma 5.4.5. Example 5.4.1

Consider the Brownian motion M

= ( X t , P,)

on R“ (see

STOCHASTIC CALCULUS RELATED TO THE DIRICHLET FORM

163

Examples5.1.1 and 5.3.l)andset B,IiJ = X t C i J Xolil,X,(" being thei-thcoordinate of the sample path X, (1 5 i 2 n). The coordinate functions xt are in H,!,,(R"), while (5.4.37)

1

MCX[I= B ( 0 ( B " ' , BV')t

€4 = 6,.r.

= pta,O,atj,,= 6,edx; consequently, the stochastic integral f.B"' Hence ,u~~[,~,> makes sense for every f E L2(R").Furthermore,

[ 5 fl.B"' : f r E LZ(R") 1 5 i 5 n 1 e ( @ * B l i ) ) = 7 f:

AT==

(5.4.38)

i=l

r = l IlffllZL2(R")

in accordance with (5.4.36). e) is identified with the space of vector fields f = (fl, * Thus the space (4 f n ) , f , G L2(Rn).f represents Mlul for some u E H1(Rn)if and only if f is a gradient field of an Lz-function and in this case

--

(5.4.39)

f

= grad u.

When n 2 3, grad u is well defined for any u in the extended Dirichlet space Fe and {grad u, u E Fe} is a closed subspace of the above mentioned space of 1 vector fields (see (1.5.18)). In this way the space (Ye, D) is identified with a closed subspace of (& e) (see Corollary 2 to Theorem 5.2.2). We see from (5.4.23) that

(5.4.40) E , ( J ~ / ( X ~ W ) < co q.e. x

E

R",

for any f E L2(R"). For instance, the function

belongs to L2(Rn). When a < 1, (5.4.40) holds for every x E R" and the MotooWatanabe stochastic integralf-B i i ) is well defined as an AF of M in the strict sense. 2 dw ~ ) = 1 (see Example 5.1.1) and When a 2 1, however, P o ( ~ o f ( X s ) = neither the Motoo-Watanabe integral nor the Ito intgral is defined. The stochastic integral f * B ' " in our sense is still defined by taking the polar set {O) as?its "essential' exceptional set. Let us consider again an m-symmetric diffusion M on X whose Dirichlet space ( 6 a)is regular. Each ti E
164

FURTHER ANALYSIS BY ADDITIVE FUNCTIONALS

ous version tl. Furthermore, we may extend the decomposition (5.2.20) to u E KOc by setting (5.4.41)

NP1 = a(X,) - a ( X J - MFI(', t 2 0,

and we may call NLU1 a CAF locally of zero energy. The notion of the a-spectrum still makes sense for u E KOc due to the local property of 8. An open set G is said to be an a-regular set of u E KO= if g a ( w , v) vanishes for any v E 9-n C,,(x> with Supp[v] c G and for any w E Y s u c h that w = u on a neighbourhood of Supp[v]. The complementary set of the largest a-regular set of u is denoted by a,(u) and called the a-spectrum of u E KOc ( a 2 0). It is easy to see that Theorem 5.3.4 concerning the support of A""] remains true for any u E KO,. In particular, (5.4.42)

when a&)

ti(Xt) - O(XJ= M?',

i

20

= 4.

Example 5.4.2 Let a,&), 1 5 i, j 5 n, be locally integrable functions on an Euclidean domain D c Rnsuch that ail = a,i and condition (5.2.33) is satisfied. We consider the case that m(dx) = dx. Let M be a symmetric diffusion on D whose Dirichlet space (3-, 8)is

We have seen in Example 5.2.1 the following expression of the energy measure p("> of u E Fb:

This expression readily extends to any u E F b y the same reasoning as in the proof of Lemma 5.4.6. Therefore, the expression (5.2.32) of the quadratic variation of the MAF MI"] holds for any u E KO=:

Suppose u E AOc is harmonic in the sense that the right hand side of (5.4.43) vanishes for u and for every v E C;(D), then the equality (5.4.42) is valid in virtue of Problem 3.3.4.

165

RANDON TlhlE CHANGES

If the transition function of the diffusion M were absolutely continuous with respect to the underlying measure m, then any exceptional set is polar (W.2). On account of the additivity and Markovity, we see that any harmonic u € * m satisfies for each t 2 s > 0 (5.4.45)

zi(Xt) - zi(X,)

= Mf" - Mpl,

for every x E D rather than for q.e. x for the identity (5.4.44).

P,-a.s. E

D . An analogous statement is valid

§5.5. Random Time Changes In this section we assume that the process M is transient, i.e.,

Thus, we start with an m-symmetric Hunt process M whose Dirichlet space (F8) ,on L2(X;m) is not only regular but also transient. According to by means $1.5, the space F g i v e s rise to the extended Dirichlet space Fe of the completion with respect to the metric B . Moreover, each element of cy32 admits a quasi-continuous version by virtue of regularity (35.3)' ofFj1.5 and the argument in 83.1. Throughout this section every function in the space Fe is considered to be quasi-continuous already. In view of condition (Fe.2) of $ 1.5 and Lemma 3.1.4, we then see that 8(u, u ) = 0 if and only if u = 0 q.e. Thus, are identified if they coincide q.e. two functions in Fe Let us fix an arbitrarily positive Radon measure p on X charging not set of zero capacity. Let A be the PCAF associated with p by Theorem 5.1.3. We denote by Y and P the support of p and the support of A respectively; i.e., Y is the smallest closed set outside which p vanishes, while (5.5.1)

P=

{X E

X - N : P,(R

= 0) = I}.

Here N is a properly exceptional set of A and R(w)= inf {t > 0: A,(w) > 01 * In general, it is hard to identify the set P with Y , but we can get the following relation by choosing a properly exceptional set N of A appropriately. Lemma 5.5.1.

is a Bore1 set,

Proof. Note that

P c Y , and p( Y - F) = 0.

= {x E X - N

: d,(x)

= 1)

with

Q,(X)

=

165

FURTHER ANALYSIS BY ADDITIVE FUNCTIONALS

Ex(e-R).Since # A ( ~ )is l-excessive on X - N , we may assume by Lemma 4.2.6 that P i s a Bore1 set by taking a larger properly exceptional set N of A if necessary. It is known that Px((Zx-,p-A), = 0) = 1, r > 0, x E X - N([14; Chap V, (3.9)]). In view of Theorem 5.1.3, we have then ,u(X - Y)= p ( X - N - P) = 0, and consequently, (5.5.2) p ( Y - P) = 0. On the other hand, we have

(5.5.3)

Cap@ - Y ) = 0.

Indeed, Theorem 5.1.3 implies that A = Z,-A and, consequently, P,(A,, = 0) = 1 q.e. HenceP,(D, 5 R) = 1 q.e. and Px(O < R) = 1 q.e. on an open set X - Y, which means (5.5.3). We can now make the set P - Y empty by choosing a larger properly exceptional set N of A including P - Y. Q.E.D. Let us consider the following orthogonal decomposition of the Hilbert space (Ye,a):

We also define the space X yby replacing Y in (5.5.4) with Y. We note that Zr is identical with X yif and only if

(5.5.5) Cap(Y - Y)= 0. However we do not know whether (5.5.5) is true in general. By virtue of the O-order variant of Theorem 4.4.1, (5.5.6)

u =~ ( P u ) , u

E ZY,

where 7 denotes the restriction t o the set P and I?&x) = Ex(#(Xap)). The hitting measure Hp(x,=)is concentrated on the set P for q.e. x E X on account of Theorem 4.2.3. Our present aim is to identify the space ZY, by means of the operations jj anh E?, with the extended Dirichlet space on Y associated with the time changed process of M. Consider the time changed process M Y = 152, A, J r , , &,, PJXEr where

RANDON TIME CHANGES

(5.5.7) tr(w) = inf {s > 0; A,(w) > t }

167

.

It is known that My is a normal, right continuous strong Markov process on B ([14; Chap V, (2.11)]). Theorem 5.5.1. (i) The transition function of the time changed process MP is p-symmetric and determines a strongly continuous transient semigroup {Tf,t > 0} on L2(Y ;p ) . (ii) Let ( F:a ,~be)the corresponding extended Dirichlet space with reference measure p. Then the operation j j determines a unitary equivalence between the Hilbert spaces (F.$ Z 7 P ) and (@, a).More specifically

(5.5.8)

F t= jjZ'

in the sense that F C 3 YE" and, conversely, for each 4 E F g , there with jju = 4 p-a.e. When 4 and u are so related, exists u E Zp (5.5.9)

aq4, 4)

=

a&, 24).

Proof. (i) According to the 0-order version of Theorem 3.2.3, there exists a sequence {F,} of increasing closed sets satisfying (3.2.11) and (3.2.12) such that ZFn.p is of finite 0-order energy integral for each n. Hence, there exists a p-integrable bounded function h on Y strictly positive p-a.e. such that Fe c L ' ( Y ; hop) and

y is a bounded Bore1 function on F}. Set B,(B) = {C = ty-h: t (5.5.10) implies that the 0-order potential U ( 4 . p ) E Yeof the signed Radon measure $ - p is well-defined for every 4 c B,(B). Furthermore, we can see from Lemma 5.1.4, Theorem 5.1.3, and the resolvent equation that

Now we denote by cess M', i.e.,

{Ra,a > 0} the resolvent of the time changed pro-

168

FURTHER ANALYSIS BY ADDITIVE FUNCTIONALS

We have from (5.5.11) and (5.5.12) that U ( # - p ) (x) = fi(Ro#)(x) q.e. # E B,(y).l) Therefore,

x E X,

(5.5.13)

I

E?(R"?,(B,(y)) c 2" s(fi(Ro#),v) = (#,v>,, 4 E B,(B),

21

E 2r9,

where ( , ), denotes the inner product on L2(Y; p ) . In particuIar, Bo is p-symmetric since R",# = fi(R,#) q.e. on y. (5.5.13) readily extends to any # EB(Q) such that (4, R",#), is finite, while the resolvent equation for {R,} implies (Re#, Ro&#)p ( Y - ~ ( R ' ~ # , R o # ) p = :-yoR",4, 4)c +A !, $I., < m, 4 E B m . Hence, H(R,#) = -aH(RoR,#) H(R,#) belongs to Zp and ZT(fi(R",#), v) a(R",#, v ) = ~ (#, v), # E Bl(P), v E Z ' . Consequently, R", is p-symmetric. On account of the right continuity of the sample path of the process M', it holds for # E C,( Y ) that lim aR,#(x) = #(x) for every x E 7 and,

g

+

+

a-m

consequently, for p-a.e. x E Y. Therefore Lemma 1.4.3 applies and {Ra, a > 0} gives a strongly continuous resolvent on L2(Y; p ) . The same statement holds for the transition function of My, which is transient because R",h = U ( h - p )is finite q.e. on y, and, consequently, p-a.e. on Y. (ii) Since (Roy,a,) < cm, a, E B1(y),by (5.5.13), Theorem 1.5.3 gives

(5.5.13) and (5.5.14) readily imply that the isometry (5.5.9) holds for 6 E B,(P). Since R",(B,(P)) is dense in ( 9 - 5 , a,) and any Cauchy sequence in ( 3 - 5 , 8.) (resp. in (Z',2F)) contains a subsequence which converges p-a.e. (resp. q.e.), it only remains to show that E?(Ro(Bl(y)) is dense in ( Z y8). , Suppose v E %' is 8-orthogonal to B(R,(B,(y)).Then v = 0 p-a.e. by virtue of(5.5.13). Setf, = Ivl An. Theorem 5.1.3 then implies that the AFf,*A vanishes identically. Letting tend to infinity in the equality pE,(J: e-p'f.(X,Jdt) = 0 q.e., and noting the quasi-continuity of fnand Theorem 4.3.2, we arrive atf, = 0 q.e. on ?. Hence, v = 0 q.e. on y and v = B(p) = o q.e. on X . Q.E.D.

4 = ROBand u = H(R,$) where -

-

a

The Dirichlet space on L2(Y ; .p ) (resp. the extended Dirichlet space with reference measure p) associated with the time changed semigroup { T f , t > 0) on L2(Y;p ) appearing in the above theorem is called the 1) Cf. [14; Chap.V, (3.5)].

RANDON TIME CHANGES

169

time changed Dirichlet space on Lz(Y ;p ) (resp. the time changed extended Dirchlet space with reference measure p ) . A positive Radon measure p on Xis said to be a speed measure if Supp[p] = X and p charges no set of zero capacity. Given a speed measure p , we denote by 2 the support of the associated PCAF. X - 2 is always p-negligible by Lemma 5.5.1. But if we assume the stronger condition that

(5.5.15) Cap(X - 2 ) = 0, then the space 2T' is identified with re and we can get from Theorem 5.5.1 the following simple statement concerning the time changed Dirichlet space and the time changed process. Theorem 5.5.2. Let p be a speed measure satisfying condition (5.5.15). 8 p ) with respect to (i) The time changed extended Dirichlet space (Xc, p is identical with the original extended Dirichlet space (re, a); i.e., Fc c T g , and conversely, each u E T c admits a a E Tesuch that u = a p-a.e. When u and 11 are so related,

(5.5.16)

~ P ( uU, ) =

a(a, a).

(ii) The time changed process M' can be made to be a standard process on 2,by choosing the properly exceptional set of the associated PCAF appropriately. In fact a choice of a lager properly exceptional set N including X - 2 yields the quasi-left continuity of the time changed process up to the life time. Under the condition of Theorem 5.5.2, the following description of the corresponding time changed Dirichlet space ( F p , Z f p ) on L2(X;p ) holds:

Let p be a positive Radon measure on X which is mutually absolutely continuous with respect to the original underlying measure rn, i.e., (5.5.18) dp =f s d m with a locally m-integrable, strictly positive (m-a.e.) Bore1 function f on .'A Then p is a speed measure satisfying condition (5.5.15). Indeed the

1 70

FURTHER ANALYSIS BY ADDITIVE FUNCTIONALS

function $ A in the proof of Lemma 5.5.1 equals 1 p-a.e., and, consequently, m-a.e. By Lemma 4.2.5, dA = 1 q.e., which means (5.5.15).

Example 5.5.1. Let M be the Brownian motion on R" for n 2 3. M is transient and the corresponding function space Fe (we only consider quasi-continuous functions) is the space of BLD functions (see Example 3.1.1). Let f be a locally integrable, strictly positive (a.e.) Bore1 function on Rw. Then (5.5.19)

3-f = re n L (R";fdx) v),

U, v E

.Ff

is the Dirichlet space on L 2 ( R n ; f i x )associated with the time changed process of the Brownian motion M by means of the PCAF (5.1.19).

Example 5.5.2 (Symmetric Cauchy process on the boundary). Let M be the reflecting barrier Brownian motion on the 3-dimensional demispaced = ( x = (xl,x 2 , xJ; x g 2 0} (see Example 4.4.2). The associated regular Dirichlet space relative to L2(d) is the Sobolev space (1/2 D, H1(D)). M is transient by(5.5.19). The associated extended Dirichlet space Fe is the completion of H 1 ( D )with respect to the Dirichlet integral. The set of all quasiis denoted by Fe again. Fe then admits the following continuous functions in Fe orthogonal decomposition: (5.5.20)

Fe =g d O ' @2

where F ; O ' is the space of all BLD functions on D of potential type(see Example 3.1.1) and X i s simply the space of all harmonic functions on D with finite Dirichlet integrals. The element of Feis simply called the BLD function. As the 0-order version of (4.4.15), we have (5.5.21)

xLo = ' (u E Ke:u = 0

q.e.

on iW]

Let us consider the ?-dimensional Lebesgue measure on the boundary aD, i.e., ,ucc(dx)= dx,dx,~~,,,(dx,). [ I charges no set of zero capacity (with respect to the Dirichlet form (1/2D, H ' ( D ) )because I,.dp is of finite 1-order energy integral for each compact set K c O'D in view of the proof of Lemma 2.2.1. Recall that the reflecting barrier Brownian motion M is obtained from the 3-dimensional Brownian motion ( ( X , l ' , A':*', Px\ by means of the reflection = (Xjl', XjZ1,I Xj3' I). Denote by I ( t ) the local time at 0 of the one dimensional Brownian motion X:31(see Example 5.1.1). Let A be the PCAF of the process M corresponding to the measure ,u by Theorem 5.1.3. Then (5.5.22)

A,(w) = 4Z(t, m).

RANDON TIME CHANGES

171

In fact, by making use of the equality

we can see that the formula (iii) of Lemma 5.1.4 holds for 41(t), ,u, and the transition function of A. In particular, we have (5.5.23) Supp[p]

= Supp[A] = O'D.

Furthermore, the time changed process MaD= {A',,, Px}xsaD of M with respect to the AF A , is a symmetric Cauchy process on aD. More specifically, MaD is a spatially homogeneous Hunt process on aD (see Example 4.1.1) such that

Hence, the Dirichlet space (Fan, B a Dof ) MaDon L2(O'D)= Lz(i3D; dx,dxz)is given by

in virtue of (1.4.25). In fact, the inverse function T, of 4 4 t ) obeys the one-sided stable law of exponent 1/2, i.e., E,(exp( - at,)) = exp(- ,,/&! t ) (see Ito-McKean[49]). Consequently, E,(exp(iEX,,)) = Eo(e[E.Xs) Po(t, E ds) = e-IEI2 s/* P0(xt E ds) = e-1elt/z.

Jm

s,

(5.5.20), (55.211, and (5.5.23) imply that Theorem 5.5.1 applies with P = Y = 3D and Zpy =Z . We conclude that the space Z o f harmonic functions BJD) on D with finite Dirichlet integrals and the extended Dirichlet space (FfD, of the symmetric Cauchy process M a Don aD are unitary equivalent by the correspondence

Here y denotes the restriction to 3D and HQ(.u) = E,($(A'G~~)). An analogous relation has been derived in (1.2.19) of Example 1.2.3, when D is the unit disk on R2.