I IntegralEquations
and OperatorTheory
/_ntegr. equ. oper. theory 4t (2001) 74-92
0378-620X/01/010074-19 $1.50+0.20/0
9 Birkh~iuser Verlag, Basel, 2001
DIRICHLET
OPERATORS
AND
THE
POSITIVE
MAXIMUM
PRINCIPLE
RENE L. SCHILLING
Let (A,D(A)) denote the infinitesimal generator of some strongly continuous
sub-Markovian contraction semigroup on LP(m), p /> I and m not necessarily
a-finite. We show under mild regularity conditions that A is a Dirichlet operator
in all spaces Lq(m), q ~> p. It turns out that, in the limit q --+ oo, A satisfies the
positive m a x i m u m principle. If the test functions C ~ C D(A), then the positive
m a x i m u m principle implies that A is a pseudo-differential operator associated
with a negative definite symbol, i.e., a L4vy-type operator. Conversely, we provide sufficient criteria for an operator (A, D(A)) on LP(m) satisfying the positive
m a x i m u m principle to be a Dirichlet operator. If, in particular, A on L2(m)
is a symmetric integro-differentiat operator associated with a negative definite
symbol, then A extends to a generator of a regular (symmetric) Dirichlet form
on L2(ra) with explicitly given Beurling-Deny formula.
Introduction
The Hille-Yosida Theorem gives necessary and sufficient conditions for an operator (A, D(A))
to be the infinitesimal generator of a strongly continuous contraction semigroup {Tt}t>>.oon a
Banach space ~. We will be mostly interested in sub-Markovian semigroups on ~ = L2(m),
p ~> 1, and Feller semigroups 'on the space X = Coo(X) of continuous functions on X
vanishing at infinity. ((X, ~3, m) is usually a general topological measure space, restrictions
will apply from case to case.) One of the conditions of the Hille-Yosida theorem is that A
must be diss@ative, i.e.,
IIAu - A~II:~/> A [luIt:~,
A > O, u E D(A)
where D(A) C 3~ is a dense subset. (The other condition is that for the closure A and some
,~ > 0 one has Range(k - A) = 3r
For operators (A, D(A)) on Ca(X), the dissipativity follows from the positive maz-
imum principle,
u e D(A), U(Xo) = sup u(x) & u(xo) >10
xEX
implies
Au(xo) <. O.
(PMP)
Schilling
75
In the L~(m) setting, the dissipativity can be derived from the property that (A, D(A)) is a
LP(m)-Dirichlet operator,
Au. ((~ - 1)+)p -1 dm <. O,
u e D(A).
(DO-p)
Clearly, both conditions (PMP) and (DO-p) imply more: if (A, D(A)) extends indeed to the
generator of a semigroup, this semigroup wilI be positive and sub-Markovian under (PMP)
or (DO-p). The converse is also true, generators of positive and sub-Markovian semigroups
on Coo(X) or L'(rn) necessarily fulfill (PMP) or (DO-p).
The positive maximum principle has an even more interesting consequence: whenever A maps the test-functions C ~ into the continuous or Borel measurable functions, then
AIC ~ is already a pseudo-differential operator with negative definite symbol p(x, [). This
beautiful result is due to Ph. Courr~ge [4] Theorem 2.5 below. As a matter of fact, the
structure of the symbol p(z, ~), is explicitly determined by a L~vy-Khinchine formula and A
turns out to be a L6vy-type operator, the object of desire for probabilists working on jump
processes. A survey of this theme can be found in [12, 15]. A comparable structure result
for generators of, say, semigroups given by a Dirichlet form is not known. The formula that
comes closest to such a characterization is probably the Beurling-Deny formula for symmetric
Dirichlet forms, see the monograph by M. Fukushima, Y. Oshima, and M. Takeda [7].
In this paper we will mainly be concerned with the question how the Dirichlet
operator property (DO-p) relates to the positive maximum principle (PMP). We will see
that the limiting case of (DO-p) as p --+ oo will be the positive maximum principle. Of
course, one has to take some precautions in order to guarantee that an operator on/_2(m)
admits actually a pointwise interpretation (which is necessary for (PMP)). One consequence
of our investigations could be stated as "nice (w.r.t. mapping properties gJ domains) Dirichlet
forms are generated by integro- or pseudo-differential operators", i.e., Example 1.2.4. in [7]
is the rule and not the exception.
Apart from such structural considerations, existence and knowledge of a symbol
p(x, ~) can be quite important for probabilists. For L~vy processes, p(x, ~) is the characteristic exponent which is even independent of x. It is by now classical that Fourier analysis
and the characteristic exponent can be used to exhibit many probabilistic properties of a
L6vy process. Similar statements are true for Feller processes and, in fact, for any process
with known symbol. For example, the symbol can be used to describe properties such as the
Hausdorff dimension, the regularity, the growth etc. of the sample paths, see [22, 21, 24].
Moreover, so-called global properties such as heat-kernel estimates and life-time questions are
determined by the symbol, see [12, Chapter 6], [8], [23]. Potential theoretic considerations
can be found in [6].
N O T A T I O N . We denote by aVb resp. aAb the maximum resp. minimum of a, b E N. For a
real-valued function u we write u • for the positive/negative parts, u • (x) := max{4-u(x), 0}.
If X is locally compact, Coo(X) stands for the set of all continuous functions that vanish at
infinity. All other notations are standard or should be self-explanatory.
ACKNOWLEDGEMENT.
I want to thank Niels Jacob, Swansea, for sending me the
manuscript of his forthcoming monograph [14]. He drew my attention to LP-Dirichlet op-
76
Schilling
erators, and I am grateful for many valuable discussions and the permission to use yet
unpublished material. Financial support by the Nuffield foundation, grant NAL/OOO56/G,
is gratefully acknowledged.
1
Extensions of sub-Markovian semigroups
Extensions of sub-Markov semigroups are usually considered only for symmetric semigroups
(when restricted to L2(m)) and or-finite measure spaces (X, f13,m). With some modifications
of the usual proofs, one does not need these assumptions. Let (X, f13, m) be an arbitrary
measure space. As usual, LP(m), 1 ~< p < ec, denotes the pth power integrable functions and
L~176 is the space of (equivalence classes) of m-essentially bounded measurable functions,
that is, L~176 is the completion of the set of step functions
N
s(x)=EsjlB,(x),
sjEIR,
BjefB, j=I,2,...,N,
(1.1)
j=l
under the norm
NU][L~(m)=m-esssuplu(x)l=inf{N>O
: m({x : lu(x)] > N } ) = O } .
(1.2)
If we allow in (1.1) only Bj E f8 with m(Bj) < c~, then s(x) is called simple function. We
denote the set of simple functions by E(m).
o
DEFINITION
1.1 The set LC~(m) is the closure of E ( m ) under I].HL~(-~).
o
o
Clearly, L~(m) C L ~ ( m ) with equality L~(m) = L~~
holding if and only if m
is a finite measure; as a closed subspace of L ~ ( m ) it is itself a Banach space. It is not hard
o
to see that for every 1 4 p < c~ the set LP(m) N L ~ ( m ) is a dense subset of L~(m).
DEFINITION
1.2 A semigroup {Tt}t>o, Tt : LP(m) -+ LP(m), 1 ~< p < ec, of bounded
linear operators is called
- strongly continuous if limt-~0 [ITtu - u[iL~(m) = 0.
contractive, if each Tt is a contraction o n / 7 ( m ) .
- positive, if u 1> 0 rn-a.e, implies Ttu >i 0 m-a.e.
- sub-Markovian, if 0 <~ u ~< 1 m-a.e, implies 0 ~< Ttu ~< I m-a.e.
-
The next proposition shows how we can extend (non-symmetric) sub-Markovian
Q
semigroups on LP(m) to the space L~
1.3 Let {Tt}t)o be a strongly continuous, sub-Markovian contraction
semigroup on LP(m) for some 1 <. p < co. Then there is a sub-Markovian contraction
semigroup {Tt(~
on L~176 such that T(~176
) N L~176 = TtfL~(m) N L~~
PROPOSITION
Schilling
77
o
Proof. Let u E L~(m) and let {sk}keN C E(m) be a fundamental sequence approximating
u. Since Tt is sub-Markovian, TL(SN/IlsjltL~(,j) ~< 1 m-a.e., so
1T~sj(x)
-
r, sk(~)l ~< T~(Isj - skl)(x) ~ Ilsj
-
skilLS(m),
for m-almost all x. This shows that the limit L ~ ( m ) - l i m k _ ~ Ttsk exists and is independent
of the particular sequence {Sk}kSm The operators Tt(~) := TtlE(m) II'll~ are clearly welldefined on LCr
and satisfy TtlLP(m) e L~176 = T(t~)ILP(m) N L~~
Since
o
Zt(E(m)) C Tt(ZP(m) N Z~(m)) C ZP(m) e L~176 C Z~(m).
we find that Tt(~) maps L~(m) into itself. By the very construction as uniform limit, it is
clear that T (~176inherits the sub-Markovian and contractivity properties of Tt.
[]
Note that {T(~
is not strongly continuous on the space/~oo (m) since/~~176
(m) is
o
too big. However, (D~
L~176 is a good interpolation couple and we can use the classical
M. Riesz convexity theorem in order to construct (well-behaved) intermediate semigroups
between {Tt}t)o o n / 2 ( m ) and {T(~)}t~>0 on L~
Let us first recall a version of the Riesz
convexity theorem, cf. Butzer & Berens [3, Section 3.3.2].
T H E O R E M 1.4 (M. Riesz) Let (E, ~,#), (F,~, i.,) be two a-finite measure spaces. Let
1 <~ p <~ q <~ oo and assume that T : DO(p) + Lq(#) -+ I2(~) + Lq(v) is a bounded linear
operator. Then T is bounded as an operator T : L~(#) -+ L~(v) for any r E ~p, q]. If we set
o for some 0 E [0, 1], then
7"
0
The usual proof of Theorem
T 1-0
(1.3)
1.4 works with simple functions and their density in
o
the spaces L p, resp. L ~162 An additional argument
is needed in order to pass to L ~176 Since
o
simple functions are still dense in L p (and, by definition, in L c~) for not necessarily a-finite
measures, we can drop the assumption that #, p are a-finite in our setting.
Let us apply the Riesz convexity Theorem 1.4 in the present situation.
C O R O L L A R Y 1.5 Let (X, ~ , m) be a measure space and {Tt(P)}t>~o be a strongly continuous, sub-Markovian contraction semigroup on LP(m) for some p E [1, oo). Then T (p) ILP(m)A
o
L~176 can be extended to a strongly continuous, sub-Markovian contraction semigroup
{T q%o on any space Lq(m)
with p < q <
oo.
Conversely, if a positive semigroup {T~(P)}t>.o on DO(m) extends for all q > p to a
contraction semigroup {T(tq)}t>.o on Zq(m), then {T~)}t.>o is already sub-Markovian.
Proof. Proposition 1.3 shows that T (p) can be extended to a semigroup {T(~176 0 on L~
~
o
such that Tt (P)ILP(m) N LC~(m)
= T(~176
• LC~(m).
We can therefore use the Riesz
78
Schilling
convexity theorem to show that Tt(P)ILP(m) Fq L~176 extends to Tt(q)iLq(m)A L~176 p <
(q)
q < oo with norm I]Tt l]Lq(,~)~Lq(m) ~< 1, i.e. T (q) extends to Lq(m) and is a contraction.
Since
o
o
T~ (
o
T(P)ILP(m) n fr~
: T(t~176
n fr~
= q)]LP(m) n L~176
(1.4)
and LP(m)ALe(m) C Lq(m) is dense, it is clear that {T(tq)}t)o is a sub-Markovian semigroup
o
on Lq(m). Moreover, we have for any u E LP(m) N L~176 and q > p
IIT~ (q)u -
ull~(m) =
]~
Ir(q)u -- u] q dm
2q-P IIu IIL~o(~)IlT~(')u
q-P
-
u HLp(m)
'
which proves the strong continuity of {T(q)}t)o on a dense subset, hence everywhere.
o
For the converse we pick u E LP(m) nL~176 such that
Markov inequality we get for every e > 0
e
~<
l+e
--
bl
< ]. By the Chebychev-
l+e
l+e
o
Since u
6
LP(m) C~L~~
we can pass to the limit q --+ ec
lim
q--}oo
(m
P)u[>l+e
~< l + e
whichleadstoacontradictionunlessm({[T[P)u[>l+e})=O.
<1
Sincee>Owasarbitrary,
we have [Tt(~)uI ~ 1 m-almost everywhere. Positivity is obvious.
[]
We conclude this section with a look at the domains of the Lq-generators of the
above semigroups. Let us recall the following general result that can be found in the book
by Butzer & Berens [3, Theorem 2.1.2.(c)].
1.6 Let {Tt}t>~0 be a strongly continuous contraction semigroup on a reflexive
Banach space ~. Denote by (A, D(A)) the generator of this semigroup. Then u E D(A) if
and only if liminf t-lllTtu -ul] ~ < oo.
LEMMA
t--+O
1.7 Let {T(P)}t>.o be a strongly continuous sub-Markovian contraction semigroup on I2(m) for some p e [1, oc) and denote by {T(q)}t>lo its extension to Zq(m) for
p < q < oo. If (d (p), D(A(P))) and (A (q), D(A(q))) are the corresponding infinitesimal generators, then
THEOREM
{u e D ( d (p)) Cl L ~ ( m ) : A(~)u e n ~ ( m ) } C A D(d(q))"
q>~P
(1.6)
Schilling
79
P r o o f . Let {T(~)}t~> 0 be the extension of {T(P)}t>.0 to the set L~
according to Proposition
1.3.' Assume that u E D(A@) • L~176 is such that A(P)u E L~
Of course, A(P)'a E
LP(m) A L~176 C L~(m). Using that T(tP)u - u = ftT(P)A(P)uds, a calculation similar to
(1.5) shows
Tt(q)t - u qLq(m) ~
1
-~
<-
//
T(~P)A(P)uds pLP(m)
A(P)U,,L~(m)
p
T(~)A(P)u
ds Iq-~
s
LeO(m)
IIA(P)uII~Ptm) .
Since Lq(m) is reflexive, we can apply Lemma 1.6 to conclude that u r D(A(q)).
2
G e n e r a t o r s of s u b - M a r k o v s e m i g r o u p s o n
[]
LP(m)
Unless otherwise indicated (X, fB, m) will be an arbitrary measure space. From the theory
of Dirichlet forms it is well-known that the infinitesimal generator (A, D(A)) of a strongly
continuous contraction semigroup {Tt}t)o on L2(m) satisfies
fzAu'(u-1)
if and only if
+din<0
for a l l u e D ( A )
{Tt}t>~0 is sub-Markovian
(2.1)
cf. Ma ~ RSekner [17, Proposition 1.4.3] and [18, Theorem 1.7] for the general case. In
the symmetric case, (2.1) is due to Bouleau ~z Hirsch [2, 1.3.1, 1.3.2]. Operators (A, D(A))
satisfying the condition (2.1) are called Dirichlet operators; this notion is originally due to
Bouleau and Hirsch [1].
DEFINITION
operator, if
2.1 An operator (A, D(A)) on LP(m), 1 < p < oo, is called LP(m)-Dirichlet
xAU(x)
((u
I)+)P-I(x) m(dx) <. O,
u e D(A),
(2.2)
is satisfied.
Definition 2.1 and Theorem 2.2 (2) are taken from N. Jacob [14], see also [13, 6].
Since Theorems 2.7, 2.9 and Corollary 3.2 depend on the technique of proof of Theorem
2.2, we give a short proof of our own. Note that our method enables us to extend Theorem
2.2 also to p : i, see Corollary 2.3 below. Condition (I) of Theorem 2.2 is well-known in
abstract semigroup theory, cf. Phillips [20], Nagel [19], Liskevich ~z Semenov [16], or [13] for
comments on the origin of condition (2.2). Again, our proof is crucial for Theorem 2.2 (2).
T H E O R E M 2.2 Let {Tt}t>.o be a strongly continuous contraction semigroup of operators
on/2(m), 1 < p < co. Denote by (A, D(A)) its infinitesimal generator.
(1) {Tt}t>~o is positive if and only if f x Au. (u+) p-1 dm <. 0 for all u e D(A).
80
Schilling
(2) {T,},~>0 is s~b-M~kovian ~f a,,a only if A is ~ L~(m)-D#'ici~,let opera,tot.
P r o o f . (1) Let Tt be positive. For
Ttu- >~O. Thus,
u C LP(m) one has (u+) p-1 E Lq(m), p-1 + q-1 = 1, and
f Ttu 9 (u+)V-~dm <~ f Ttu + . (u+)P-~dm <~
dX
JX
U+ P
and we ge~ for
IIT,~+IIL~<m)ll
(~+).-11k~(m)
fX
u E D(A)
]~ Au. (u+),_l dm = lit -m~ o fj x "T t ut- u (u+) p - l d m ~ O.
Conversely, assume t h a t fx Au. (u+)p-I am
for A > 0 we have R~u := (A - A)-lu E D(A) and
<<.0 for all u E D(A). Let u <~O. Then
0>. f z AR:~u'((RAu)+)V-l dm= l f x ((R~u)+)Pdm-
fx u((~u)+)'< am
.
k
k , we can
So, (R~u) + = 0, i.e., R~ is positivity preserving. Since Tt = L p -llmk-~o(-~Rk/t)
take an a.e. convergent subsequence and conclude t h a t T~ is positive.
(2) Assume t h a t Tt is sub-Markovian and u E D(A). Since (u - 1) + ~< u + ~< lul E
LP(m) and u = (u - 1) + + u A 1, we get from p a r t (1) applied to (u - 1) + t h a t
y(T~u - u) . ((u -1)+F-l d m
<~ /x(T~(u A1) --~ A1) . ((u -1)+F-l d m
s
].)+)p_l
d77z
=
O,
and therefore fx Au. ((u - 1)+) p-1 an <. O.
Conversely, let fx Au. ((u - 1)+) p-I dm <~O. Passing from u to n . u gives fx Au.
((u - n < ) + ) p-1 dm ~< O, and for n --+ oo we get fx Au. (u+)p-1 dm <. O. P a r t (1) shows
then that Tt is positive.
Let u E LP(m), u ~< 1. The same calculation as in (1) applied to AR~u, A > O,
yields
0/>
/x
A(ARQu 9((AR~u - 1)+) p-1 dm
:
~ f x ( a R ~ - ~). ( ( ~ R ~ - 1)+) p-1 a n
>~ A f x ( ( A R ~ u - 1)+)Pdm.
Therefore, u ~< 1 implies
subsequence { kj}je N.
AR~u ~ 1, thus Ttu = limj~o~(~Rkj/t)kJu ~< 1 a.e. for a suitable
[]
If we replace in the proof of T h e o r e m 2.2 the expressions (9+)p-1 (9 = u, u - 1 , . . . )
by 1{9>o} in each step, the argument carries over to the case where p = t.
Schilling
81
C O R O L L A R Y 2.3 Let {Tt}~>~0 be a stwngly continuous contraction semigroup on L m(m)
with generator (A, D(A) ).
(1) {Tt}~>0 is positive if and only if f{u>0} d u dm <~O, u e D(d).
(2) {Tt}t>~o is sub-Markovian if and only if fib>l} d u d m <~ O, u e D(d).
The conditions in Corollary 2.3 can be seen as limit cases (as p ~ 1) of Theorem
2.2. We will now study the limiting case as p --+ c~. To do so, we need some more definitions.
Denote by B(X) the set of ~-measurable functions over X. Note that, in contrast to L ~ (m),
this set consists of functions rather than classes of functions. Therefore, pointwise assertions
are well-defined in this setting.
D E F I N I T I O N 2.4 A linear operator (A, D(A)), D(A) C B(X), satisfies the positive maximum principle (on D ( d ) ) if A: D(A) --+ B ( X ) and if the condition
u(xo) = supu(x),
u(xo) >~ 0
implies
Au(xo) < 0
xEX
holds for every u E D(A).
A remarkable consequence of the positive maximum principle is the following structure result due to Ph. Courr~ge [4].
2.5 (Courr~ge) Let ~ C ]~ be an open subset. If a linear operator A :
C2~
--+ C(~) satisfies the positive maximum principle, then A is a pseudo-differential
operator
THEOREM
Au(x) = -p(x, D)u(x) = -(2~) -~/~ _Zo p(x, ~)~(~)&~,~l d~
(~ is the Fourier transform of u) with symbol p : ~ • ]~" --+ C such that p(x, ~) is locally
bounded in (x, ~1 and enjoys for fixed x E ~ the (Ldvy-Khinchine) representation in
Here, % g, Q are measurable functions, 7 >~ O, t C ~n, Q E ~ • symmetric, positive definite,
and N(x, dy) is a kernel satisfying f(~-~)\{o} tYl~/( 1 + lYl~) N(x, do) < ~ .
R E M A R K 2.6 (A) For fixed x, the function ~ ~-~ p(x,~) appearing in Theorem 2.5 is
continuous negative definite in the sense of Schoenberg. Therefore, the pseudo-differential
operators p(x, D) are sometimes called pseudo-differential operators with negative definite
symbol. This notion is due to W. Hoh; see [8, 12] for a deeper study of such pseudo-differential
operators and their connections to Feller processes.
(B) A straightforward calculation shows that the pseudo-differential operator p(x, D) appearing in Theorem 2.5 can be rewritten in the following integro-differential form:
-p(x,D)u(x)
= -~/(x)u(x) + (g(x),Vu(x)) 4- s
j,k=l
qjk(x)agku(x )
(2.3)
82
Schilling
where u E C~(~).
Courr?~ge actually showed ([4, Corollaire 3 to Th~or~me 1.5]) that a lineal' operator
A:
-,
satisfies the positive maximum principle if and only if it is of the above given integrodifferential form with (7, g, Q, N(., dy)) as in Theorem 2.5.
We will show--under certain regularity assumptions on the generator--that L pDirichlet operators satisfy the positive maximum principle. Let LSC(X) [LSCb(X)] denote
the set of [bounded] lower semicontinuous functions on the topological space X.
T H E O R E M 2.7 Let X be a Hausdorff space with Borel (7-algebra ~ and some measure m
such that supp m = X. Let {Tt(P)}t>.o be a stro~zgly continuous, sub-Markovian contraction
semigroup on LP(m) for some 1 <~ p < (x~ and denote by (A 0), D(AO))) the infinitesimal
generator. Assume that there exists a convex function cone D C D(A 0)) ~ LSCb(X) such
that
A O) : 7) --+ LSCb(X)
and assume that i9 contains for each x E X a function Ox E D with 0~(x) > supycu(= ) 0=(y)
for all open neighbourhoods U(x) of x. (In particular, Oz(x) > O~(y) >/0 if y r x.)
Then A (p) t19 satisfies the positive maximum principle.
P r o o f . From Section 2 we know already that {TtO)}Oo can be used to get strongly continuous, sub-Markovian contraction semigroups {T(q)}t~>0on all intermediate spaces Lq(m) with
p < q < oo such that
1.7 we have
Tt(P)ILq(m) n L~176 : Tt(q)lLq(m)n ~(m).
~9 C ~ D(A (q))
and therefore
A : - - A(')119 =
According to T h e o r e m
A(~)I19.
q)p
Note that the latter assertion makes sense, since by assumption AO)(19) C LSCa(X) consists
of functions rather than equivalence classes of functions.
Since all Tt(q) are positive operators, we get from Theorem 2.2 that for any u E 7:)
and all q > p
x d u ( x ) . (u+(x)) q-' m(dx) <~0
(2.4)
holds. Therefore,
] x A u ' ( u + ) q - l d m = / A u < O } Au'(u+)q-Idm+/A~>O} Au'(u+)q-ldm<~O
which gives
0~< (/d~>0}
Au'(u+)q-1 drn) Vq<~ (f{Au<O} (-Au)'(u+)q-1 din)1/q
(2.5)
Schilling
83
Now set ~(dx) := l{A**>o}(x)lA'u(x)l m(dx) and u(dx) :: l{du<O}(x)lAu(z)l m(dz) a~d recall
that
lira tlfIIL*(,) = # - e s s s u p If]
q--+oo
holds for all f E U ( # ) M L~
for some r / > 1 (similarly for u). This and (2.5) shows
m- esssup u+(x) <~m- esssup u+(x).
zE{du>0}
xE{du<O}
(2.6)
Assume first t h a t u E LSC(X) attains its m a x i m u m 0 < Y0 = U(Xo) = suPze x u(x)
and t h a t Xo is the only (isolated) absolute m a x i m u m such t h a t u(xo) > supuCu(~o)u(y) for
all open neighbourhoods C(xo) of Xo. Then x0 E {Au <~0}:
Indeed, suppose xo E {Au > 0}. Since u and Au are lower semicontinuous, U~ : =
{x : u(x) > Y o - r
: du(x) > 0} is for any r > 0 an open set and xo E U~. Since
supp m = X, we have m(U~) > 0 for small e > O, and by (2.6)
sup ~ + ( z ) < vo = .~- esssup ~+(~) < ~ - esssup ~+(~)
xE{ Au<~O}
zE{ Au>O}
xEI Au~O}
This is clearly a contradiction, hence, xo E {Au < 0} and Au(xo) < O.
Assume now t h a t either Y0 = 0 or t h a t Y0 >~ 0 and t h a t it is a t t a i n e d at at least
one point, say Xo and t h a t the above neighbourhood condition is violated. This means t h a t
there is either another point where the supremum is a t t a i n e d or t h a t it is reached (but not
attained). Choose any point where the value Yo is actually attained, say, Xo, and use the
fact t h a t for all n E N
v~ : = u + r~0~o e ~
where
r ; ~ : = n(l(A0~o)(xo)l + 1) <
(Recall t h a t 7:) is a convex cone and A(:D) C LSCb(X).) So, for all open neighbourhoods
U = U(zo) of xo
~(~o) = ~(~o) + ~Oxo(~O)/> sup ~(V) + ~0~o(~O) > sup ~ ( v ) + sup ~O~o(V)/> sup,~(v)
y~U
y~u
y~]
v~
where we used in the first inequality that Yo is the supremum of u and, in the second
inequality, the properties of the functions 0~. This shows t h a t v~ has at x0 its only global
isolated m a x i m u m which is nowhere else reached. Since v~(xo) > U(Xo) >~ 0 we apply our
previous consideration and get
Avn(xo) ~ 0
for all
n E IN.
But we also have
0 >1 Avn(xo) = Au(xo) + r~(AO~:o)(Xo)
Au(xo) +
(AGo)(Xo)
n (l(AGo)(Xo)l + 1)'
Since
(AO~o)(zo)
i
84
Schilling
we have
A'a(~o) = lira Av,,(~o) <.4 0
~---+oo
[]
and t h e assertion follows fox" x0 was arbitrarily chosen.
If we combine the above Theorem with the representation result of Courr~ge we
obtain two interesting special cases.
2.8 Let X C tR~ be an open set, ~3 the Borel c~-algebra, and m be a
measure on X with full support. Let {Tt}t>o be for some p ) 1 a strongly continuous subMarkovian contraction semigroup on L~(m) with generator (A, D(A) ). If the test functions
C ~ ( X ) C D(A) and if A [ C ~ ( X ) : C ~ ( X ) --+ Cb(X), then the restriction of A to C ~
is a pseudo-differential operator A I C ? ( X ) = 7 P ( x , D ) I C ~ ( X ) whose symbol is given in
Theorem 2.5.
COROLLARY
In particular, every (non-symmetric) Dirichlet form ($, D ( g ) ) , D(g) C L2(m), with
generator (A, D(A)) such that O ~ ( X ) C D(A) and A ( C ~ ( X ) ) C Cb(X), is generated by a
pseudo-differential operator - p ( x , D).
The assumption made in the above Corollary that A maps C ~ ( X ) into bounded
o
continuous functions is natural if one has in mind Feller semigroups as the L~(m) analogue of sub-Markovian semigroups. By definition, a Feller semigroup is a strongly continuous semigroup of sub-IVlarkovian operators on the space (C~(X), II,llL~(m)). Since A =
Coo-limt~o(Tt- 1)/t, we have A ( C ~ ( X ) ) C C ~ ( X ) C Cb(X), and the above assumption
is reasonable. One should, however, observe that for general continuous negative definite
symbols the function
x ~-+p(x, D)u(x)
= (27r) - ~ / 2
f p(x, {)~(()e '(<~)d~,
u e cy(x),
j ~n
will not be bounded. To include these cases we give in a rather concrete situation a strenghtened version of Theorem 2.7.
2.9 Let X C tR~ be an open set, f8 the Borel cr-algebra, and m be a Radon
measure on X with full support. Let {Tt}t>o be for some p >i 1 a strongly continuous subMarkovian contraction semigroup on LP(m) with generator (A, D(A)). If C2r
C D(A)
and if A : C ~ ( X ) -+ C(X), then A [ C ~ ( X ) is a pseudo-differential operator with negative
definite symbol (el. Theorem 2.5) or an integro-differential operator as in Remark 2.6 (B).
THEOREN[
P r o o f . Since m is a Radon measure, we have C ~ ( X ) C (~r>>.xLr(m) " Therefore (u+) r E
Lq(m) for q-1 + p - 1 = 1, any r > 0, and u C C ~ ( X ) . Since Tt extends to any L~(m), r > p,
we see with the argument of Theorem 2.2 (1) t h a t
~ Ttu 9 (u+) r-1 dm <. fx ~" (u+)r-x din,
u ~ c~(x),
r>p.
Schilling
85
By assumption, u E C ~ ( X ) C D ( A ) is in the domain of the LP(m)-generator and (u+) "-1 E
Lq(m) for r > 1. So,
fx" Au' (u+)"-: dm = Jim
fxt-~0
Ttu-ut (u+)~-: dm ~< 0
(2.7)
for the limit has an interpretation as weal: limit in LP(m).
We can now tbllow the proof of Theorem 2.7: (2.7) implies for u E C~~
Since (u+) r - i E L l ( p • N L ~ ( p • where p•
= l(•
m(dx), we may argue
just as in the proof of Theorem 2.7, and deduce that A satisfies the positive m a x i m u m
principle.
Invoking Courr~ge's Theorem in the form of Remark 2.6 (B) or of Theorem 2.5
finishes the proof.
[]
We have to restrict ourselves to the space N~ or open subsets of N ~ because we do
not know a result similar to Courr~ge's Theorem 2.5 in general spaces. Note that in more
general situations the integro-differential representation (2.3) should be used.
3
A converse
to Theorem
2.7
We have seen so far that a generator of a sub-Markovian contraction semigroup satisfies
the positive m a x i m u m principle and is, under certain additional conditions, a pseudo- or
integro-differential operator. One can, in fact, partially reverse this assertion. Here is an
abstract version of such a result.
T H E O R E M 3.1 Let X be a Hausdorff space with Borel ~-algebra ~ and let m be a Radon
measure with s u p p m = X . Denote by 79 C B ( X ) a set of functions that is sufficiently rich
in the following sense
for each compact set
K c X
there ezists some
)l E 79
with
1K <. X <<.I.
Assume that the operator A defined on 79 satisfies the positive maximum principle and extends
for some p >~ 1 to an operator A : D ( A ) C LP(m) --+ 22(m). If f x ( A u ) . (u+F -1 dm <~ 0
holds for all u E D ( A ) [modification for p = 1 as in Corollary 2.@ then A satisfies
fxAu.((u-1)+)P<dm<~O,
[modification if p =
1 as in
nED(A)
s.t.
s u p p u + i s compact
(3.1)
Corollary 2.3].
P r o o f . Let u E D ( A ) such that K = s u p p u + is a compact set. For ) / E 79 with 1K ~< X ~< 1
we find u - X E D ( A ) and, by assumption,
zA(U-
X) " ( ( u - x ) + ) p - l d m <. O.
86
Schilling
Therefore,
Since we have
fy
Au. ((~ -- )i)§
dlTt, ~ J(X A)i. ((~ -- )i)§
u(x)-i
u(x) - X(z) :
dTF~.
(3.2)
for x E K
- u - ( z ) - X(x) for x r
we find (u - X) + : (u - 1) + and also supp(u - X) + C K. Using (3.2) we arrive at
where the last inequality follows from the positive m a x i m u m principle: supp(u - X) + C K ,
)t] K = 1, so A)ilK <~ O.
[]
Typically, the set 79 will consist of test-functions C ~ ( X ) . In such a situation we
can restate Theorem 3.1 in another, more striking form. This also generalizes Theorem 2.1
of [11] which was already stated in a semigroup setting.
C O R O L L A R Y 3.2 Let X be an open subset o f R ~ and m be a Radon measure on X with
full support. I r A i~ defined on C ~ ( X ) , satisfies the positive maximum principle, and maps
C ~ ( X ) into some space LV(m), p >~ 1, then the following assertions are equivalent:
(1)
fx A~. (~+)p-1 dm <. o, u e c ~ ( x ) ;
(2)
fx Au. ((u - 1)§ -I
d ~ <. O,
u e C~(X);
[modification if p = 1 as in Corollary 2.3].
If one of these conditions is satisfied, A, as an operator in LP(m), is closabIe and
its closure (-A, D(-A)) satisfies (1), (2) for all u C D(A).
P r o o f . Let A be defined on C ~ ( X ) . T h a t (2) implies (1) is clear, see the proof of Theorem
2.2 part (2). Conversely, if (1) holds, Theorem 3.1 shows that (2) is fulfilled.
Assume now that (1) is satisfied. Applying (1) to u and - u shows
fxAu.sgn(u)[u];-l dm = L Au. (u+)V-I d m - f x A U " (u-)p-l dm <~ O.
Up to a constant sgn(u)[u] p-1 E (LP(m)) * is the (unique if 1 < p < oo) normalized tangent
functional of u C / 2 ( m ) and it is a simple calculation to see that (A, C~~
is dissipative.
Densely defined dissipative operators are closable, cf. Ethier, Kurtz [5, L e m m a 2.11], and we
even find that range(A - A) = range(A - A).
Since, by construction, C ~ ( X ) is an operator core for A, it is clear that (1) and
(2) remain valid for A on D(A).
[]
Theorem 3~! makes three crucial assumptions: (A~I) A has to satisfy the positive
m a x i m u m principle, (A-2) A maps the test-functions into some space LV(m), and ( A - 3 ) A
Schilling
87
satisfies condition (1) of Corollary 3.2. We will now study these properties in a concrete
situation.
If A maps C ~ ( X ) into C(X) (B(X) will do according to Remark 2.6, (2.3)) a
necessary and sufficient condition for A to satisfy the positive maximum principle is given
by Courr~ge's Theorem: A has to be an integro-differential operator of the form described
in Remark 2.6 and Theorem 2.5.
Since the local (differential operator) part can be treated by standard methods, we
will from now on assume that
Au(x) :
-x)\{0}
1 + lyl 2 ]
with some kernel N(x, dy) on (x - X) \ {0} such that fy#o lYl2/( 1 + lYl2)N( x, dy) < co.
Notice that the kernel N(x, dy) will in general be singular at the point y = 0. We denote
the corresponding symbol (of the pseudo-differential representation) by
p(x,~) = f(x-x)\{0} ( 1 - e -i(''y)
ii(~,y>
u
) N(x, dy).
(3.4)
One can show, cf. [24, Lemma 2.1], that
fy#o 1 +lYl2[yI2N(.,dy) L~ < oo.
][p(., r [[LOO~< C(1 + ]r ~) if and only if
Let us now turn to assumption (A-2).
(3.5)
In what follows we introduce the kernel
u(x, dy) = N(x, x - dy) on the set Nn \ {x} resulting from a change of variable y ~-~ x - y
in (3.3) and (3.4). We will use N(x, dy) and ,(x, dy) interchangeably.
LEMMA
lip(., r
3.3 Assume that X c ]R~ and m is a Radon measure on X with full support. If
~< ~ (1 + Ir ~) and if supp N(., R n \ {0}) is compact, then p(x, D) ,naps C ~ ( X )
into LP(m) for any p >>.1.
P r o o f . By our assumptions, we find a a compact set K C X containing the support
supp N(., ]i{n \ {0}) C /-( and a cut-off function X E C~(X) such that 1i( ~< X ~< 1 and
N(z, dy) = X(x)N(x, dy). Let u e C~~
Then p(x, ~) = X(x)p(z, r and
lip(., D)~(.)II~,(~) =
(2~) -~/~ f~o x(.)p(., r162 i<''e>de
< c f~o x(.)(1 + I(Ibl~(r
= cIIxlb(~)
[ (1 + lSl:)l~(~)l de.
,JR n
The last expression is, however, finite as ~ is a rapidly decreasing function.
LEMMA
ff
3.4 Let m(dx) = dx be Lebesgue measure on N '~. If NP(', ~)l[i~ <- C (1 + I([ 2) and
N(x, dy) = n(x, y).(dy)
88
Schilling
with a bounded density 0 <. n(z,y) <<. K and a measure 7J(dy) on IR'~ \ {0} such ~hat
fy#o ~
u(dy) < 0% then the integro-d~fferential operator A given by (3.3) maps C~(R '~)
into LP(dz) for all p ) 1. In fact, one ha3
(3.6)
IIA~IIL.~<o II~IIw(R-)
where Wp2(R *') is the usual L p Sobolev space.
Proof. It is enough to establish (3.6). Let us consider first the operator A1 associated with
the kernel l{lyl>i}(y ) N(x, dy). With Minkowski's inequality for double integrals we find for
e c y ( ~ ~)
IIA,,,H~. =
f~,~,>,~ (~,(~-~,1-,,Ix/+/~_,,/~/>'~,
+ I~,1~ .' "/x.~l"(~
~.
lyl>:t] 1 + IW ~(ay) IlWtl~.
lyI>'}
~< 2Kii~IIL,f
J{Iyl>l}
,(dy) + KII~II-' f
J{M>I}
~,(d~,)
~< 2K~,({Iy I > 1})(llullL~ + I I w l I . ) ,
the latter being finite since f,jr bI~ u(dy) < oo.
We will now treat A - Ai which is given by the kernel l{M<l}(y ) N ( x , dy). Using
Taylor's Theorem we find
IId~
AlUlJLp 4
f
(u(. - y) - u(.) + (y, W(.)) ) n(., y) u(dy)
J{lyl~<l}
LP
+ f{M<.l} lY121(y'+Vu(.))lyl
2
n(., y) u(dy) L,
<. t (
IOjOku(9 - "cYD[lYj[tYkl dT dO
LP
=
+ s( f ,~,.<~}1 +lYI~
lyl* ,(dy) IIWllL,
IY[~<1}
i,k=~
which is finite since fyr ~I+M2 u(dy) < oe. The assertion now follows from the Usual density
argument.
[]
We will finally consider assumption (A-3).
Schilling
89
L E M M A 3.5 Let X C ]R'~ be an open set and m be a Radon measure on X with full
support. Assume that A is given by (3.3) where p(x, ~) is real-valued. If A is symmetric and
maps C y ( X ) into L2(m), then f x Au. u +dm < 0 for any u e Cc~(X). Moreover,
1~ •
( - A n , V)L2(m) = -~
(u(y) - u(x))(v(y) - v(x)) tJ(x, dy) m(dx).
(3.7)
with ~,(z, dy) = N(z, x - dy) and A being the diagonal in X x X.
The operator A is symmetric if and only if
fxx
x\zx
f ( x ' Y ) L'(x'dy)m(dx)= f z
f(x,y)~,(y, dx) m(dy)
•
(3.8)
holds for all f E Cc(X x X \ A).
A sufficient criterion for (3.8) to hold is that ~,(x, dy) = ~(x, y)m(dy) with a symmetric density ~(x, y) = a(y, x).
P r o o f . First observe that for a real-valued symbol p(x, ~) the term i(G y)/(1 + lyl 2) under
the integral sign in (3.4) drops out. Therefore, the integro-differential form (3.3) of the
operator reduces to
A (x) = fJy (u(y-x)- u(x)) N(x,dy)= f
d,).
dye\
r
We will now prove (3.8). Take r X E C ~ ( X ) such that supp r N supp X = ~. Then
we find using the above representation that
fxx(x)Ar
=
fxX(X)fx,,{x}r
dy)m(dx)
Interchanging X and r and using the symmetry of A therefore yields
and (3.8) follows since the test functions r174 with disjoint support generate the set C~(X x
Z \ ZX).
Let u, v ff C ~ ( X ) and r (x, y) be a sequence of functions cr~ E Cg~(X x X \ A) such
that lcr~ ~< crn N< 1 for some open neighbourhoods U~ of the diagonal, cr~(x, y) = cr~(y, x),
and sup,~e~ an = 1{~#~}. Set
A ( x , y ) := ~ ( x , y ) ( u ( y ) - u(x))v(x).
YVe apply (3.8) to the functions f,~(x, y) and use Lebesgue's dominated convergence theorem
to deduce
xx\A
(~(y) u(~))v(~).(~, dy) m(d~) = f
dxxx\A
(u(~) u(y))v(y) 12(X~dy) .~(d~)
90
Schilling
Hence,
2 fx~AU, v d m =
fx•
- u(x))v(x)
dy) m(d ) +
(u(x) - u(y))v(y) .(x, aN)
which is but (3.7). In particular, (3.7) proves that condition (3.8) implies the s y m m e t r y of
A.
We can now use (3.77 in order to show that f x A u . u + dm <<.O. Let u E C ~ ( X ) .
First, observe that
lu+(Y) - u+(x)l ~ lu(Y) - u(x)l.
We can, therefore apply formula (3.7) to the functions u and v := u +. It is now easy to
check that the product
(u(Y) - u(x)). (u+(y) - u+(x)) /> 0.
Finally, if u(x I dy) = ~(x, y) m(dx) with n(x, y) = ~(y, x), (3.8) is obviously satisfied
since u(x, dy) m(dx) = n(z, y) m(dz) m(dy)
n(y, x) m(dz) m(dy) and the L e m m a is proved.
=
[]
THEOREM
3.6 Let X C R ~ be an open set and m be a Radon measure on X with full
support. If A : C ~ ( X ) --~ L2(m) is symmetric, satisfies the positive maximum principle and
has a real-valued symbol, then (A, C ~ ( X ) ) is closable, its closure A is a Diriehlet operator
and the bilinear form s
:= (--Au, V)L2(m) extends to a regular Dirichlet form with
Beurling-Deny formula given by (3.7).
P r o o f . The assertion follows immediately from Corollary 3.2, from the above Lemma, and
from the fact that A, its closure A, and its Friedrichs extension AF give rise to the same
energetic space which is the domain of the Dirichlet form,
[]
It is surprisingly difficult to find concrete conditions that guarantee the s y m m e t r y
of a wide class of pseudo-differential operators with negative definite symbols. Various
conditions were found by Jacob and Hoh & Jacob in [t0, 91. We will add a further, quite
natural one.
C O R O L L A R Y 3.7 Let X C ]Rn be an open set and m be a Radon measure on X with
full support. Assume that the operator A defined on C ~ ( X ) satisfies the positive maximum
principle and is given by
Au(x) = f
Jx
(u(y) - u(x)) u(x, dy)
(3.9)
with a kernel u(x, dy) on R ~ \ {z} and fyr Ix - y]: (1 + Ix - yI2) -1 , ( z , dy) < co. Assume
moreover that there exist positive functions X E Cr
and n E B ( X x X \ A)
=
where
=
(31o)
Schilling
91
sup ( X ( X ) ~
x~R"
l x - y l 2 x ( Y ) g ( x , y ) m ( d y ) ) < 0%
(3.11)
r
and ,that
Jv#=
Jyr
Then A is a symmetric operator, maps the test functions into L2(m), and admits a
Friedrichs extension AF which generates a regular symmetric Dirichlet form. The BeurlingDeny formula for this Dirichlet form is given by (3.7) with jump measure J(dz, dy) =
y)x(y)
Proof. The condition f~dr f ( x -- y)~'(X, dy) = O, f is any odd function, causes that in
formula (3.4) the imaginary part vanishes. Therefore, p(x,~) will be real-valued and the
proof follows immediately from the preceding Lemmas. In this case the particular form of
t;he generator (3.9) has not to be assumed as it follows already from the positive maximum
principle, compare the proof of Lemma 3.5.
If the first part of (3.12) holds, (3.9) is well-defined for test functions C ~ ( X ) , and
the argument of Lemma 3.5 goes through without major changes. Again, the assertion
follows from the preceding Lemmas.
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School of Mathematical Sciences
University of Sussex
Brighton
Falmer
BN1 9QH
United Kingdom
r. s c h i l l i n g @ s u s s e x , ac. uk
31C25; 60J35; 47D07; 47G20; 47G30; 60J75.
Submitted: Januai3~ 6, 2000
Revised:
July 28, 2000