FUKUSHIMA, MASATOSHI

LEJAN, YVES

Osaka City University Repository

Fukushima, M. and Lejan, Y.
Osaka J. Math.
28 (1991), 837-845
ON QUASI -SUPPORTS OF SMOOTH MEASURES
AND CLOSABILITY OF PRE-DIRICHLET FORMS
MASATOSHI FUKUSHIMA AND YVES LEJAN
(Received June 14, 1991)
1. Introduction
Consider a singular perturbation of the Laplacian Δ on the rf-dimensional
Euclidean space R d :
L= -Δ+L
Γ
.
Here L
Γ
 is a linear operator "living on" a closed subset TdRd which might be
of zero Lebesgue measure and as irregular as a fractal set. The problem is
how and when we can give L a proper sense. One way to formulate this is to
introduce a perturbed bilinear form
, S) = D(f, g)+β
τ
(f I
 Γ
, £ I r) , f, geC7(B?) ,
where D is the Dirichlet integral and 6T is a closable pre-Dirichlet form on
L2(Γ; μ) for some positive Radon measure μ on Γ such that C%(Rd)\Fc:*D[βΓ].
If 8 is proven to be closable on L2(Rd), the L2-sρace based on the Lebesgue
measure dx, then the associated self-adjoint operator on L2(Rd) may be thought
of as a realization of L.
Some sufficient conditions for the closability of the perturbed pre-Diri-
Λ
chlet form 6 are known (M. Fukushima [2; §2.1], J.F. Brasche and W. Kar-
wowski [1]). It is plausible that 6 ought to be closable on L2(Rd) under the
sole potential theoretic assumption that μ charges no set of zero (Newtonian)
capacity. A purpose of the present paper is to affirm this in a more general
context as will be stated in §2 and proven in §4.
The proof in §4 involves the notion of the quasi-support of a measure
and its characterization crucially. The quasi-notions have appeared in poten-
tial theory in diverse contexts. Another aim of the present paper is to show in
§ 3 the existence of the quasi-support along with its useful characterizations in
terms of classes of quasi-continuous functions.
This work was done during the stay of the first author in Europe in 1990 suported by EEC sci-
entific fund
838 M. FUKUSHIMA AND Y.
Using the characterization, the quasi-support of a smooth measure is seen
to be identical with the probabilistic notion of the support of the additive func-
tional associated with the measure. The latter notion has been adopted re-
cently by M. Fukushima, K. Sato and S. Taniguchi[3] to describe the closable
part of a pre-Dirichlet form and to give a necessary and sufficient criterion for
the closability. In §5, we restate some basic results of [3] in terms of the quasi-
support together with alternative proof using a characterization of §3 instead
of the usage of additive functionals. In particular, a characterization due to
M. Rϋckner and N. Wielens[5] for the closability is recovered. The arguments
in §4 can be regarded as a reduction of those in §5 to a simpler specific situa-
tion.
2. Statements on closability of perturbed pre-Dirichlet forms
In what follows, we fix a locally compact separable metric space X. C0(X)
denotes the family of continuous functions on X with compact support. Suppose
that a pair (<?, C) satisfies the following conditions:
(1) C is a dense subalgebra of CQ(X) such that, for any compact set K and
relatively compact open set GuK, there exists u^C with u=l on K,
u=0 on X~ G and 0<u<l on X. Furthermore, for any £>0, there
exists a real function φz(t) with φt(t) = l, ί^[0, 1], — ε^φ^ή^l+ε,
t(=R and 0^φt(t')—φf(t)^t'—t for t<tf such that φz(C)aC.
(2) 6 is a non-negative definite symmetric bilinear form on C such that,
for each £>0 and for some function <p
e
 satisfying the condition in (1),
Then we call 6 or the pair (<?, C) a pre-Dirichlet form over X.
Denote by <3tt the family of positive Radon measure on X and let
Jli1 == {m^ <3tt: supp m = X}.
where supp m denotes the topological support of m. A pre-Dirichlet form
(<?, C) is called closable on L\X\ m) for m^<5M,' if S(u
n
, w
Λ
)->0 whenever u
n
^Cy
{u
n
} is <?-Cauchy and w
Λ
->0 in L2(X\ m). If this is the case, the closure (<?, 3)
of (<?, C) on L2(X\ m) is a regular Dirichlet form on L\X\ m). Conversely,
given a Dirichlet form (69 3) on LZ(X\ m\ the restriction of G to any subfamily
Cd3 satisfying condition (1) is a pre-Dirichlet form closable on L\X\ m). In
this case, G satisfies a stronger condition than (2) in the sense that the statement
"for some function φ" in (2) can be strengthened to "for any function φ" .
If (<?, C) is a pre-Dirichlet form closable on L\X\ m) for some m^<3tt,\ we
have the associated notion of capacity which can be evaluated for compact set
Kzs
Cap (K) = infill, u): u£ΞC, u>l on K},
QUASI-SUPPORTS OF SMOOTH MEASURES 839
where 8\(u, v) is the sum of 8(u, v) and the inner product (u, v)
m
 in L\X\ m).
The set of zero capacity is called 8
λ
-polar to indicate its relevence to 8 and m.
Theorem 2.1. Let (6, C) and (8, C) be pre-Dirίchlet forms closable on
L2(X; m) and L2(X\ m) respectively for some m, rh^Jtt'. We assume that
8(u,u)>8(u,u) ,
and that fa charges no 8rpolar set. Then (8, C} is closagle on L2(X\ m).
Theorem 2.1 will be proven in §4 by using quasi-notions studied in the
next section. We now show an immediate consequence of Theorem 2.1.
Theorem 2.2. (superposition of closable forms). Let (6, C) be a pre-Dίri-
chlet form closable on L2(X m} for some m e <_5K ' . Consider a collection {T
β
 θ e Θ}
of closed subsets of X and suppose that, for each θ^θ, there exist a positive Radon
measure μ
θ
 on X charging no δi-polar set with supp μ
θ
=T
θί
 and a pre-Dίrichlet
form 8B over Γθ closable on L2(TQί μθ) with C \ ΓβC.2)[£β]. Further let (Θ, JL, v) be
an auxiliary σ-finite measure space such that β
θ
(f \
 TQ, f \ ΓΘ) and μθ(K) are, as func-
tions of θ^θ, v-ίntegrable for every f^C and every compact KdX. Then the
form 6 defined by
is a pre-Dirίchlet form closable on L2(X; m).
Proof. In view of the remark made after the definition of the pre-Diri-
Λ
chlet form, we can see that (8, C) is a pre-Dirichlet form. If we let
the m is a positive Radon measure on X charging no <?
r
polar set. Further-
more in the same manner as in the proof of [2; Theorem 2.1.3], we can show
that 8 is also closable on L2(X\ rh). Hence 8 is closable on L2(X\ m) by The-
orem 2.1.
3. Quasi-supports of smooth measures and their characteri-
zations
Let m be in <3tt,' and (<?, £F) be a regular Dirichlet form on L2(X] m). Then
we have the associated capacity Cap, and we use the terms "quasi-continuous",
"polar" and "q.e." in relation to Cap. We add "<?
r
" in front of these terms
when it is necessary to emphasize their relevence to 8 and m. Without loss
of generality, we assume that each element of 3 is quasi-continuous. Two ele-
ments of 3 represent an equivalence class of £FcL2(^Γ; m) iff they coincide q.e.
An increasing sequence {F
n
} of closed sets with lim^oo Cap (X—F
n
)=Q is said
840 M. FUKUSHIMA AND Y. LE]AN
to be a nest.
A set EC: X is called quasi-open if there exists a nest {F
n
} such that EΓlF
n
is open in F
n
 in the relative topology for each n. The complement of a quasi-
open set is called quasi-closed.
Lemma 3.1. (i) A set F&X is quasi-closed if and only if there exists a
quasi-continuous function u with F=u~\{0}) q.e.
(it) If F is quasi-closed, then, for any relatively compact open set G, there
exists a non-negative quasi-continuous function u in 3 such that w=0 q.e. on F
and u>0 on G—F.
Proof. The "if" part of (i) is evident. We give probabilistic constructions
for the rest of the proof. Consider a Hunt process M=(Xt, Px) on X associated
with the regular Dirichlet form (<?, £?) on L2(X\ πί). For any Borel set B,
denote by eB the 1 -order hitting probability of B:
eB(x) = Ex(e-°*) , x^X,
where σB— inf {£>0: Xt^B}. When Cap (J5)<oo, eB is a quasi-coninuous ver-
sion of the (l-)equilibrium potential of B ([2; Th.4.3.5]). But we can see that
eB is quasi-continuous for any Borel B. In fact, for any relatively compact E C
X, w=eB/\eE is a 1 -excessive function dominated by eE^ΞF. Hence w^3
(by [2; Lemma 3.3.2]) and w (and consequently eB) is quasi-continuous (by [2;
Th.4.3.2]).
Now, for any quasi-closed set F, the function
u(x) = \—eF(x) , x^X ,
has the required property in (i) because F— Fr is polar (by [2; Th.4.2.3]) and
eF(x)<l q.e. x^X~F (by [2; (4.3.5)]). The properties stated in (ii) is satisfied
by
u(x) =
where v is a (quasi-continuous) bounded function in 3 such that z>>0 on G
and v=Q q.e. on X-G (eg.v(x)=E
x
(f^-^ e"ff(Xt)dt) for bounded />0, /<Ξ
L*(X;m)).
Corollary 3.2. Any m-negligίble quasi-open set is polar.
Proof. Let E be an m-negligible quasi-open set. By Lemma 3.1(i), E=
{wφO} for some quasi-continuous u. Then u—Q m — a.e. and consequently
q.e., namely, E is polar.
A measure μ^<3tt is said to be of finite energy integral if SdL\X\ μ) and
(3.1) J \v(x)\μ(
QUASI-SUPPORTS OF SMOOTH MEASURES 841
for some positive constant C. The family of all measures of finite energy in-
tegrals is denoted by SQ. A positive Borel measure μ is said to be smooth if μ
charges no polar set and there exists an incresaing sequence {F
n
} of closed sets
such that
(3.2)
 μ (x- U FH] = 0, lim Cap (K—Fn) = 0 for any compaat set K
\ «— 1 / «-»oo
and μ(F
n
)<oo for each n. S will denote the totality of smooth measures. S
contains the class c_5K0 defined by
μ charges no polar set}.
It is known ([2; Th.3.2.3]) that μ^S iff there exists an increasing sequence {F
n
}
of closed sets satisfying (3.2) and IFn μζ=SQ for each n.
For set A, BdX, we write
AdB q.e. (resp. A=E q.e.)
if the set A—B (resp. the symmetric difϊernece AQB) is polar. For μ^S, a set
Fd X is said to be a quasi-support of μ if
(a) F is quasi-closed and μ(X— JF)=0
(b) if JP is another set with property (a), then FdF q.e.
The quasi-support F of μ^S is unique up to a polar set. Let F^supp
μ be the toplogical support of μ. Since any closed set is quasi-closed, we have
FdF q.e., and by deleting a polar set from F if necessary, we can always assume
Theorem 3.3.
(i) Any μ^S admits a quasi-support.
(it) For μξ=S and quasi-closed F(ΣX, the following conditions are equivalent :
(1) F is a quasi-support of μ.
(2) u=Q μ—a.e. on X if and only if u=Q q.e. on F for any u
(3) Condition (2) holds for any quasi-continuous function u.
Proof. We first prove (ii).
(2)=>(1): For μ^S and quasi-closed F, we set
(3.3) 3Zμ= {u<=3: J |
(3.4) 3>c={tte3 :n=0 q.e. on F}
and assume that 3?μ,=2v. For any relatively compact set G, take a function u
of Lemma 3.1 (ii). Then u^3Fc and hence u^Jlμ,, which means μ(G—F)=0.
Consequently we get μ(X— F)=Q. Consider other quasi-closed set Fl with
842 M. FUKUSHIMA AND Y. LE}AN
μ(X—F1)=0. Take again a function % of Lemma 3.1 (ii) for G and Fλ. Then
u^Jlμ. and hence u^3Fc, which means F Π GdFl Π G q.e. Accordingly J?C
FI q.e. proving that F is a quasi-supo support of μ.
(l)-^(3): Suppose (1) is satisfied. Then the "if" part of condition (2)
is celarly satisfied for any Borel function u. If u is quasi-continuous and u=0
V V
μ—a.e., then the set F= {#— 0} has the property (a) and hence FC.F q.e. and
u=0 q.e. on F.
The implication (3)=Φ(2) is trivial.
(i) can be proved as follows. For any μ&S, the space 32μ defined by (3.3)
is a closed subspace of the separable Hubert space (£?, £ι) because IFίt μ&S0
for some increasing closed sets F
n
 satisfying (3.2) and 3 is continuously embed-
ded into L\X\IFn μ) for each n by (3.1). Choose a countable dense subcol-
lection {uk} of 3Zμ and let
(3.5) F=v-\{Q}) for *(*) = Σ 2
Since #^37μ, ί
1
 is quasi-closed by Lemma 3.1(i) and further μ(X— F)=0.
Hence we arrive at the equality 3Ίμ,=3?Fc for 3Fc defined by (3.4) for this F.
We can then conclude that F is a quasi-support of μ from (ii).
Corollary 3.4. The underlying measure m has the full quasi-support X.
Finally we state an important probabilistic consequence of Theorem 3.3
although we shall not use it in this paper.
Corollary 3.5. For μ^S, the support of the associated positive continuous
additive functional (PCAF) of M is a quasi-support of μ.
Proof. Denote by A a PCAF of M associated with the smooth measure
μ<=S (cf. [2; Chap. 5]). The support FA of A is defined by
FA = {xEΞX-N: Px(At>0 for any f>0) = 1},
where TV is an ecxeptional (polar) set for A. Then
F
Λ
 = {xeX-N: eFA(x) = 1}
and consequently FA is quasi-closed since eFA is quasi-continuous as was seen
in the proof of Lemma 3.1. Furthermore we can check the property (3) of The-
orem 3.3 (ii) for μ and FA in the same way as in the last paragraph of the proof
of [2;Th.5.5.1].
4. Proof of Theorem 2.1.
We prove Theorem 2.1 by a series of lemmas. Suppose that <?, <?, m and
QUASI-SUPPORTS OF SMOOTH MEASURES 843
rh satisfy the conditions of Theorem 2.1. Let (£, 3") and (β, 3) be the clo-
sures of (£, C) and (£, C) on L2(^Γ nί) and L2(-SΓ, /&) respectively, (u, v)
m
 (resp.
(w, s;)£) denotes the inner product in L\X\ rri) (resp. LZ(X\ w)). Recall that
Gι(u, v) (resp. Gι(u, v)) stands for ^(z/, v)+(u, v)
m
 (resp. (^(z/, α)+(^ v)ά). Note
further that the conditions of Theorem 2.1 are never destroyed if we replace rh
by m-\-tfi. Hence we can assume without loss of generality that
(4.1) rh > m .
Lemma 4.1. We let
(4.2) 8m(u, ^) - £(«, v)+(u, v)
m
 , n,
Then (8m, 3") is a Dirichlet form on L2(X\ ni) possessing C as a core. Moreover
this is transient, namely, there exists a strictly positive wi-integrable function g such
that
(4.3)
Proof. The first assertion is evident because G?(u, u)=βm(u, u)-\-(uy u)ά is
equivalent to 6ι(u,ιΐ) for w^£F. Since rh^Jft^ with respect to the Dirichlet
form (<?, £F) on L2(X\ m) by assumption, there exist increasing closed sets F
n
such that (3.2) holds for rh and (3.1) holds for IF
Λ
^ and some positive constant
C
n
 for each n. Hence we can find a function g with the required properties.
Lemma 4.2. The measure m has the full quasί-support X with respect to
the Dirichtet form of Lemma 4.1.
Proof. Let S be a quasi-support of m with respect to the Dirichlet form
(βm, 3) onL2(X; rn) of Lemma 4.1. E=X— S is then £Γ-quasi-open and m(E)
=0. Due to the domination of 6T over 6l9 E is also ^-quasi-open and con-
sequently £χ-polar by Corollary 3.2. Then E is ^-negligible by the assumption
and 8\ -polar by Corollary 3.2 again.
The next lemma particularly implies Theorem 2.1. Denote by (βm,β)
the extended Dirichlet space of the transient Dirichlet form of Lemma 4.1. Q
is the completion of 2 with respect to the metric Qm. We may assume that
each element of Q is <?Γ-quasi-continuous.
Lemma 4.3. (<?, S] is a Dirichlet form on L\X\ m) possessing C as its core.
Proof. If u^Q and u=0 m—a.e. on Xy then w—0 6\—q,e. on X by
Lemma 4.2 and Theorem 3.3, and hence β(u, u)—0. Therefore (GyS) can be
regarded as a symmetric form on L2(X\ m). The rest of the proof is clear from
Lemma 4.1.
844 M. FUKUSHIMA AND Y. LEJAN
5. Closable part and closability criterion in terms of quasi support
In this section, we restate some basic results of [3] in terms of the quasi-
support and we see how an analytical characterization of § 3 simplifies the argu-
ments. For m^tSM! , let (<?, 3) be a regular Dirichlet form on L\X\m) pos-
sessing as its core a set C satisfying condition (1) in §2. (<?, 3) is assumed
to be either transient or irreducible. For simplicity of presentation, we only
describe the case that (<?, £?) is transient. The irreducible case can be treated
in the same way however by considering the transient Dirichlet form <?μ( , •)
=£(•,•)+(•> )Λχί« ([3], [4]).
Take any non-trivial μ^JMQ. Denote by F and F the (topological) support
and the quasi-support of μ respectively. As was noticed in §3, we may asume
that FaF. For a moment, we do not assume that F—X. Consider the extend-
ed Dirichlet space (£?
e
, S) of (£?, 6). 3
e
 is a Hubert space with inner product
6 and each element of £?„ can be assumed to be ^-quasi-continuous. Let
ff.. JΓ-F = {we£?
β
: u = 0 q.e. on F} .
This is a closed subspace of (£F
β
, <S). Denote by Pj? the orthogonal projection
on the orthogonal complement of 2
βt X-p.
Note that, if vly v2^3e and vλ= v2 μ—a.e. on F, then Vi=v2 q.e. on F by
virtue of Theorem 3.3, and consequently PpVl^=PFV2 Therefore the following
definition makes sense:
£?μ = {u^L\F\ μ): u = v μ—a.e. on F for some u
6*(u, u) = G(PFV, Ppv) for v as in the above braces.
Lemma 5.1. (£?μ, βμ) is a Dirichlet form on L\F\ μ) possessing C\F as a
core. Here C \
 F denotes the restrictions to F of elements of C.
Proof. It can be readily seen that, for
β^u, u) = inf {G(v, v): v^3
e
 v = u μ—a.e. on F}.
Denote by Tu the unit contraation 0 Vu/\ 1 of u<=3μ>. Then Tu<^3lL and
β
μ(Tu9 Tu) = inf {B(v, v}: v<=Ξ3e, v=Tu μ-a.e. on F}
<. inf {8(Tv3 TV): v^3e, v=u μ-a.e. on F}
< inf {6(v, v): v^3ϊ
e
, v=u μ—a.e. on F}
= 6(u,u),
proving that the unit contraction operates. The closedness of (£P, (?μ) on
L\F\ μ) is easily verified. We refer to [3] for the last statement about the core.
A pre-Dirichlet form (JL9 C) is called the closable part of a pre-Dirichlet
form (<?, C) with respect to μ e <3M' if (JL, C) is closable on L\X\ μ),
Jl(u, u)<6(u, u), u^C, and (Jl, C) is the maximum among those. Lemma 5.1
QUASI-SUPPORTS OF SMOOTH MEASURES 845
leads us to the next assertion (cf. [3; Lemma 4.3]).
Theorem 5.2. For μ<Ξ JM' Π <3H0, (£μ, C) is the closable part of (£, C) with
respect to μ.
Theorem 5.3. For μ^JH'Γi <J/0, (<?, C) is closable on L\X\ μ) if and only
if μ has the full quasi-support X.
Proof. By Theorem 5.2, we have the following series of equivalent con-
ditions:
(£, C) is closable on L\X\ μ)
<=> Ppf = f for any/<Ξ3
e
<*f=Q q.e. on F iff/=0 q.e. on X for any/eS,
Since F is a quasi-support of μ, we see by Theorem 3.3 that the last condition
is equivalent to
"/=0 μ-a.e. on X iff/=0 q.e. on X for any/<Ξ3V',
which is in turn equivalent to "X is a quasi-support of μ" by the same theorem.
References
[1] J.F. Brasche and W. Karwowski: On boundary theory for Schrb'dίnger operators and
stochastic processes, Operator Theory: Advances and Applications 46 (1990), 199-
208.
[2] M. Fukushima: "Dirichlet forms and Markov processes," Kodansha and North-
Holland, Tokyo and Amsterdam, 1980.
[3] M. Fukushima, K. Sato and S. Taniguchi: On the closable parts of pre-Dirichlet
forms and the fine supports of underlying measures, Osaka J. Math. 28 (1991) (to ap-
pear).
[4] K. Kuwae and S. Nakao: Time changes in Dirichlet space theory, Osaka J. Math.
38 (1991) (to appear).
[5] M. Rockner and N. Wielens: Dirichlet forms-closability and change of speed meas-
ures, in "Infinite dimensional analysis and stochastic processes," ed. by S. Albe-
verio, Pitman, Boston-London-Melbourne, 1985, pp. 119-144.
M. Fukushima
Department of Mathematical Science
Faculty of Engeneering Science
Osaka University, Toyonaka, Osaka
Y. Lejan
Laboratoire de Probabilite
Universite Paris VI
4 place Jussieu, Tour 56
75252 Paris



On quasi-supports of smooth measures and closability of pre-Dirichlet forms

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