Conservativeness of Semigroups Generated by Pseudo Differential ...

91

Potential Analysis 9: 91–104, 1998. c 1998 Kluwer Academic Publishers. Printed in the Netherlands.

Conservativeness of Semigroups Generated by Pseudo Differential Operators RENE´ L. SCHILLING ?

Mathematisches Institut, Universität Erlangen, Bismarckstraße 1 12 , D-91054 Erlangen, Germany (Received: 15 March 1996; accepted: 14 June 1996)

,

Abstract. Assume that the pseudo differential operator q (x; D) generates a Fellerian or subMarkovian semigroup. Under some natural additional conditions on the symbol q (x; ) we prove that the operator q (x; D) is conservative if and only if q (x; 0) 0.

,

,

Mathematics Subject Classifications (1991): Primary: 60J35; Secondary: 47D07, 47G30, 35S05. Key words: Conservativeness, Feller semigroup, Markov semigroup, pseudo differential operator.

1. Introduction A (C0 )-semigroup fTt gt>0 of contraction operators on a Banach space of real functions (X; k k) is said to be positive and to have the sub-Markov property if 0 6 Tt u 6 1 (in X ) whenever u 2 X and 0 6 u 6 1. If (X; k k) = (L2(Rn ; dm); kkL2 ) (one could also consider some open Rn ), we call fTt gt>0 a sub-Markovian semigroup, if (X; k k) = (C1 (Rn ); k k1 ), the Banach space of continuous functions vanishing at infinity, we call fTt gt>0 a Feller semigroup. In either case, the operators Tt are positive and continuous, and there exist representing kernels pt (x; ) of sub-probability measures s.t.

Tt u(x) =

Z

Rn

u(y)pt (x; dy); x 2 Rn

(1.1)

holds for, say, compactly supported continuous functions u. Clearly, (1.1) allows us to extend the operators Tt to all bounded measurable functions Bb (Rn ) and we will do so without changing our notation. Therefore, the following definition of conservativeness makes sense. DEFINITION 1.1. A sub-Markovian semigroup fTt gt>0 is said to be conservative if Tt 1 = 1 holds almost surely. A Fellerian semigroup fTt gt>0 is said to be conservative if Tt 1 = 1 everywhere. ? Financial support by DFG post-doctoral fellowship Schi 419/1–1 is gratefully acknowledged. Part of this work was done while the author was HCM-fellow at the University of Warwick, Coventry. He would like to thank, in particular, D. Elworthy for his hospitality.

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RENE´ L. SCHILLING

If A is the infinitesimal generator of the sub-Markovian or Fellerian semigroup

fTt gt>0 , we call A conservative whenever fTt gt>0 is and we will use both notions

interchangeably. An intuitive meaning of conservativeness can be given with the help of stochastic processes. It is well-known that both Fellerian semigroups and – whenever there is a quasi-regular semi-Dirichlet form in the sense of [12] associated – also subMarkovian semigroups give rise to stochastic processes. The relation between them is, in the Feller case, given by Tt 1B (x) = Px (Xt 2 B ) for all x 2 Rn and all Borel sets B Rn , and in the sub-Markovian context given by Tet u(x) = E x u(Xt ) for all x outside a set of capacity zero and all u 2 L2 (Rn ; dm). Here, e stands for the quasi-continuous modification, cf. [6, 12]. Therefore, we find 1 = Tt 1(x) = E x 1 = Px (Xt

2 Rn ) a:s:

Thus, a conservative process fXt gt>0 has a.s. infinite life-time. For a thorough discussion of the life-time formalism and its consequences we refer to the monograph [6]. Here we will present an example that sheds some light on our subsequent considerations. Let fSt gt>0 denote a sub-Markovian convolution semigroup on L2(Rn ). In this special setting, fSt gt>0 is also a Feller semigroup. The process fXt gt>0 corresponding to a convolution semigroup is known to be a Lévy process, cf. [1, Chap. 8], that is a Rn -valued and stochastically continuous process with stationary and independent increments. Note, that Xt is spatially homogeneous in the sense that law(Xt jX0 = x) = law(Xt + xjX0 = 0). The infinite divisibility of its law gives the Fourier transform a particularly simple structure E x eiXt

= e ,t

;

( )

x; 2 Rn ;

(1.2)

where the continuous negative definite characteristic exponent the Lévy-Khinchine formula

is described by

( ) = ` + ib + (; Q ) +

Z

1 , e,ix ,

ix 1 + kxk2 (dx); 2 Rn ; 1 + kxk2 kxk2

(1.3)

x6=0 with ` > 0; b 2 Rn ; Q 2 Rnn non-negative definite, andpthe finite jump (Lévy) measure on Rn nf0g. For further reference we note that j j is subadditive and locally bounded, hence, j ( )j 6 c (1 + k k2 ) holds for some constant c . Many other properties of negative definite functions are discussed in [1]. With some routine calculations, cf. [3], one easily checks that

St u(x) =

Z

Rn

eix e,t ( ) u ^( ) d, ;

u 2 Cc1(Rn ); x 2 Rn

(1.4)

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CONSERVATIVENESS, SEMIGROUPS AND PSEUDO DIFFERENTIAL OPERATORS

and, see also [1, Chap. 12]

Au(x) = , (D)u(x) =,

Z

Rn

( )û( ) d, ; u 2 Cc1(Rn ); x 2 Rn

eix

(1.5)

R are valid, where u ^( ) = Rn e,ix u(x) d,x denotes the Fourier transform and d,x = (2),n=2 dx is normalized Lebesgue measure. It is clear from (1.4) and (1.5)

that fSt gt>0 (or A or fXt gt>0 ) is conservative if and only if (0) = 0. Since the work of Courrège [4] it is known that the generator of a Fellerian semigroup such that Cc2 (Rn ) is contained in its domain is necessarily of the form

,q(x; D)u(x) = ,

Z

Rn

u() d, ; eix q (x; )^

u 2 Cc1(Rn ); x 2 Rn ;(1.6)

where q : Rn Rn ! C is locally bounded and 7! q (x; ) is continuous negative definite, that is, admits for every x 2 Rn a Lévy–Khinchine representation (1.3). We will call operators of the form (1.6) pseudo differential operators and q (x; ) the symbol of the operator. Note that (1.5) is a special, namely spatially homogeneous or constant coefficient, case of (1.6). In a series of papers Jacob [10] and Hoh [7, 8] – see also the references listed there – gave sufficient conditions for the symbol such that ,q (x; D )jCc1 (Rn ) extends to a generator of a Feller semigroup fTt gt>0 . In [10] the problem was analytically treated, regarding ,q (x; D ) as a perturbation of a constant coefficient (Lévy) generator ,q1(D ). Basically the following four assumptions were used

q : Rn Rn ! R

is continuous and q (x; )

is negative definite;

(1.7)

q(x; ) = q(x0 ; ) + (q(x; ) , q(x0; )) q1( ) + q2(x; );

(1.8)

2 2

0 (1 + a ( )) 6 1 + q1() 6 0 (1 + a ( )) for some constant 0 and a fixed continuous negative definite a2 ( ) s:t: for large j j; 2 Rn a2() > cjjr holds with some constants c > 0; r0 > 0;

(1.9)

1

0

j@x q2(x; )j 6 (x)(1 + a2 ()) for 2 N n0 ; jj 6 m; with 2 L1 (Rn )

(1:10:m)

for sufficiently large m (depending on the dimensionPn) and with some additional assumptions on the smallness of the perturbation jj6m k kL1 regarding 0 and r0 .

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RENE´ L. SCHILLING

Using a martingale problem approach, Hoh [7, 8] was able to improve on this result, requiring only P2 L1 (Rn ) instead of 2 L1 (Rn ), thus discarding the smallness condition on jj6m k kL1 . With both approaches we also get a L2-sub-Markovian semigroup – again denoted by fTt gt>0 – if, e.g., ,q (x; D )jCc1 (Rn ) is a symmetric operator on L2 (Rn ), see [10, 8]. Comparing the perturbed situation with the Lévy case discussed above, one is led to conjecture that the generator ,q (x; D ) of a Feller semigroup is conservative if and only if q (x; 0) 0. In fact, Hoh assumes q (x; 0) 0 and concludes via the well-posedness of the martingale problem that its solution fXt gt>0 (or fTt gt>0 ) is conservative, cf. [8, Rem. before Lem. 3.3]. Using Oshima’s conservativeness criterion [13], Hoh and Jacob [9] gave the corresponding conservativeness condition for a pseudo differential operator ,q (x; D ) which generates a L2 -sub-Markovian semigroup. They, however, assumed that q(x; ) is of a particular structure. That q (x; 0) 0 is also necessary for the conservativeness of a Feller generator ,q (x; D ) was recently shown by Jacob [11] under the assumption that the twice differentiable and uniformly continuous functions Cu2 (Rn ) are contained in D(,q(x; D)). In practice, this amounts to showing that ,q(x; D) generates a strong Feller semigroup. In this note we will show that q (x; 0) 0 is a necessary and sufficient condition for the conservativeness of ,q (x; D ) – both as generator of a Fellerian and subMarkovian semigroup – without assuming any more but (1.7)–(1:10:n + 1). Our results are summarized in the following Theorem. THEOREM 1.2. Assume that ,q (x; D ) given by (1.6) generates a Feller semigroup fTt gt>0 and that the symbol q (x; ) satisfies (1.7)–(1.9). (i) If (1.10.0) holds with 0 2 L1 (Rn ) \ L1 (Rn ), then q (x; 0) 0 implies that ,q(x; D) is conservative. (ii) If (1.10.n + 1) holds and if q (x; 0) is bounded, then q (x; 0) 0 follows from the conservativeness of ,q (x; D ).

Assume that ,q (x; D ) as in (1.6) generates a sub-Markovian semigroup fTt gt>0 and that the symbol q (x; ) satisfies (1.7)–(1.10.n+1) with 0 2 L1 (Rn )\L1 (Rn ). Then ,q (x; D ) is conservative if and only if q (x; 0) 0.

The assumptions (1.7)–(1.10.m) made in Theorem 1.2 are neither artificial nor additional. In fact, they are a bit weaker or coincide with the assumptions used by Jacob and/or Hoh to guarantee the existence of an extension of ,q (x; D )jCc1 (Rn ) to a Fellerian or sub-Markovian generator. We defer the proof of Theorem 1.2 to Sections 2 and 3 below. Note that the result stated under (i) is already contained in [8]. Our proof mimicks the methods developed there (in particular [8, Lem. 3.3]) but is simpler as it does not rely on the well-posedness of the martingale problem.

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NOTATION. We write d,x = (2 ),n=2 dx for the normalized Lebesgue measure on Rn and e (x) = e,ix . Cc (Rn ); C1 (Rn ); Cb (Rn ), and Bb (Rn ) denote the continuous functions with compact support, vanishing at infinity, bounded, and the bounded Borel measurable functions, respectively; superscripts refer to differentiability properties. All other notations are standard or should be clear from the context. 2. Sufficient Conditions In order to overcome the restrictions assumed in [11], we need some terminology which we borrow from [5, p. 111 and p. 166]. The bounded pointwise limit (bp-limit) of a sequence of bounded measurable functions fuk gk2N Bb (Rn ) is defined by bp , lim

u k!1 k

=u

if

lim u (x) = u(x); k!1 k

supk kuk k1

0 on the Banach space (X; k k1); X Bb(Rn ). Then the following implications hold: (i) ! (ii) ! (iii) with (i) (1; 0) 2 bp-closure f(u; v ) 2 X X : u 2 D (A) & v = Aug. R (ii) (1; 0) 2 f(u; v ) 2 Bb (Rn ) Bb (Rn ) : Tt u , u = 01 Ts v dsg. (iii) A is conservative, i.e., Tt 1 = 1.

Proof. The implication (ii) ! (iii) is immediate. In order to see (i) note that the set in (ii) is bp-closed and contains Graph(A).

! (ii), 2

We can now give a sufficient condition for the generator of a Feller semigroup to be conservative. PROPOSITION 2.2. Assume that ,q (x; D ) generates a Feller semigroup fTt gt>0 where q (x; ) satisfies (1.7)– (1.10.0) with 0 2 L1 (Rn ) \ L1 (Rn ). Then q (x; 0) 0 implies that ,q (x; D ) is conservative. Proof. The requirement that k0 k1 < 1 implies

q(x; ) 6 (k0 k1 + 0)(1 + a2( )) 6 ca (k0 k1 + 0)(1 + k k2 ): 2

2 2 Choose the approximate identity gk (x) = e,kxk =2k ; k

q(x; D)gk (x) =

Z

Rn

(2.2)

2 N; x 2 Rn , and observe

eix q (x; )^ gk ( ) d,

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RENE´ L. SCHILLING

= =

Z

Rn Z

Rn

2 2 eix q (x; )k n e,k k k =2 d,

eix=k q

x; k

2 e,kk =2 d,:

(2.3)

From (2.2) we conclude

jq(x; D)gk (x)j 6 ca (k0 k1 + 0)

Z

2

6 ca (k0 k1 + 0) 2

Rn Z

kk2 1+ k

2

!

2 e,(kk =2) d,

(1 + kk2 ) e,(kk =2) d,; n 2

R

i.e., supk2N jq (x; D )gk (x)j < 1. The above estimate holds pointwise for the integrand and we may use the dominated convergence theorem in (2.3) to find lim

k!1

q(x; D)gk (x) = q(x; 0) = 0;

in bp-sense. Since also bp–limk!1 gk = 1, we conclude that (1,0) lies in the bpclosure of Graph(,q (x; D )), and, by Lemma 2.1, that ,q (x; D ) is conservative. 2 Proposition 2.2 is quite general in the sense that the only assumptions we need are

q(x; 0) 0 and q(x; ) 6 c(1 + k k2 ); (2.4) uniformly in x 2 Rn . For sub-Markovian semigroups we can even do without the second estimate. The proof is less straightforward than in the Feller case and we need some preparations.

LEMMA 2.3. Let fTt gt>0 be a contractive (C0 )-semigroup on a Banach space (X; k k) and denote by A its infinitesimal generator. Then

kAuk = sup Tt ut, u t>0 holds for all u 2 D (A). If X is reflexive, we have u 2 D(A) if and only if sup Tt u , u < 1:

Proof. Observe that

t>0

t

(2.5)

(2.6)

Tt u , u = 1 Z t T Auds t t 0 s holds for all u 2 D (A). By the triangle inequality and contractivity we get

Z Z

Tt u , u 1 t 1 t

6 k T Au k ds 6

t t 0 s t 0 kAuk ds = kAuk

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97


and (2.5) follows from limt!0 1t kTt u , uk = kAuk if u 2 D (A). Since for reflexive spaces X

T u , u < 1

t D(A) = u 2 X : limt!inf

0 t

2

is true, cf. [2, p. 88, Thm. 2.1.2(c)], (2.5) implies (2.6).

LEMMA 2.4. Assume that the function q2 (x; ) satisfies (1.10.n + 1). Then the Fourier transform w.r.t. x satisfies q^2 (; ) 2 L2 and is polynomially bounded in . Proof. It is well-known, cf. [10, Lem. 2.1] that (1.10.n + 1) implies

q^2(; ) 6 c(1 + a2 ())(1 + kk2 ),(n+1)=2 6 c0a (1 + k k2 )(1 + kk2 ),(n+1)=2 ;

(2.7)

2

2

which shows the square integrability.

PROPOSITION 2.5. Assume that ,q (x; D ) generates a sub-Markovian semigroup fTt gt>0 and that q(x; ) satisfies (1.7)–(1.10.n + 1). Then q(x; 0) 0 implies that ,q(x; D) is conservative. 2 2 Proof. Denote by gk (x) = e,kxk =2k ; k 2 N ; x 2 Rn , and observe g^k ( ) = kng^1 (kx). By (1.8) we have q(x; ) = q1( ) + q2(x; ) with q1(0) = 0, i.e. the convolution (Lévy) semigroup generated by ,q1 (D ) is conservative. For any v 2 Cc1 (Rn ) we have by Lemma 2.3

j(q(x; D)gk ; v)L j t j(Tt gk , gk ; v)L j 6 kvksup L =1 6 kq1 (D)gk kL + sup j(q2(x; D)gk ; v)L j:

1

2

2

2

2

kvkL2 =1

2

(2.8)

Using Plancherel’s Theorem and dominated convergence we find lim

k!1

kq1 (D)gk kL = klim kq g^ k !1 1 k L = klim kq (=k)^g1 kL = 0 !1 1 2

2

2

and it is enough to consider the second term on the right-hand side of (2.8). A routine calculation shows for any v 2 Cc1 (Rn )

(q2(x; D)gk ; v)L = 2

Z

Z

Rn Rn

q^2( , ; )^gk ()^v () d, d,;

where q^2 (; ) stands for the Fourier transform w.r.t. the first variable. Now sup

kvkL2 =1

j(q2 (x; D)gk ; v)L j 2

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RENE´ L. SCHILLING

=

sup

kvkL2 =1

Z

Z

Rn Rn

q^2( , ; )^gk

6 (2),n=2 sup kvkL2 =1 6 (2),n=2

Z

Rn

Z

Rn

()^v ( ) d, d,

q^2( , ; )^gk

() d,

L2

kv^kL

2

kq^2( , ; )kL jg^k ()j d, 2

q^ ; k

jg^1 ()j d,: L Lemma 2.4, (2.7), shows that q^2 (; n=k ) is square integrable and – uniformly for all k – polynomially bounded in . By dominated convergence we get

= (2),n=2

2 Rn

Z

2

lim q^ ; k!1 k L = kq^2(; 0)kL :

2

2

2

Another application of dominated convergence – note that (2.7) also implies kq^2(; n=k)kL2 6 c(1 + kk2 ) uniformly in k, and that (1 + k k2 )^g1 2 L1(Rn ) – yields Z

lim

k!1 Rn

q^ ; k

jg^1()j d, = L

2

Plancherel’s Theorem shows kq^2 (; 0)kL2 1

k!1 t

jg^ ()j d, q ^ ; k L 1 Rn k!1 lim

2

=

lim

Z

Z

Rn

2

2

kq^2 (; 0)kL jg^1 ()j d,: 2

= kq2(; 0)kL = 0 and we find 2

j(Tt gk , gk ; v)L j = 0; 2

for all t > 0 and any v 2 Cc1 (Rn ). Since by dominated convergence limk!1 Tt gk Tt 1, and also 0 = lim

k!1

for all v

=

j(Tt gk , gk ; v)L j = j(Tt 1 , 1; v)L j; 2

2

2 Cc1(Rn ), we conclude that Tt1 = 1 a.s.

2

3. Necessary Conditions In this section we will show that – as in the case of convolution (Lévy) semigroups – q(x; 0) 0 whenever ,q(x; D) generates a conservative Feller or sub-Markov

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99

semigroup fTt gt>0 and satisfies (1.7)–(1.10.n + 1). This follows from the more general result

d (x; ) = ,q(x; ); x; 2 Rn ; dt t t=0 where t (x; ) is the symbol of the operator Tt , t (x; ) = e, (x)Tt e (x):

(3.1)

(3.2)

The relation (3.1) was shown by Jacob [11] under the assumption that Cu2 (Rn ), and in particular the functions e (x) = e,ix ; x; 2 Rn , is in the domain of the (Feller) generator ,q (x; D ). For practical use this means that one should consider strongly Fellerian semigroups fTt gt>0 . The following Theorem is valid without this restriction. THEOREM 3.1. Assume that ,q (x; D ) generates a conservative Feller semigroup fTt gt>0 and that the symbol ,q (x; ) satisfies (1.7)–(1.10.n + 1) with 0 2 L1(Rn ) \ L1(Rn ). Then

d (; x) = ,q(x; ) dt t t=0 holds for all x; 2 Rn .

(3.3)

Proof. The proof is a bit involved and we postpone somewhat technical details to the Lemmas 3.2, 3.3 below. Pick a 1 2 Cc1 (Rn ) such that 1B1 (0) 6 1 6 1B2 (0) and set k (x) := 1 (x=k ). ^k ( ) = kn ^1(k ). From Lemmas 3.2, 3.3 Clearly, k (x) ! 1 as k ! 1 and below we know

jTsq(x; D)(e k )(x)j 6 c; c = c ; 2 Rn ; uniformly in x 2 Rn ; s > 0, and k 2 N . Since e k 2 D (q (x; D )), we find Tt e (x) , e (x) = klim (T (e )(x) , e (x)k (x)) !1 t k = , klim !1 =,

Z

0

t

t

Z

0

Ts q(x; D)(e k )(x)ds

lim T q (x; D )(x; D )(e k )(x)ds: k!1 s

Combining Lemma 3.2 and 3.3 below we get lim T q (x; D )(e k )(x) = Ts (q (; )e )(x): k!1 s

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Thus

t (x; ) , 1 = e (x) Tt e (x) , e (x) , t t Z t (3.4) = , e,t(x) Ts (q(; )e )(x)ds: 0 Since q (; ) e is bounded for fixed 2 Rn , thus Ts (q (; ) e )(x) ! q (x; ) e (x) as s ! 0 and therefore Z t 1 T ( q ( ; ) e e )( x ) ds , q ( x; ) ( x ) s t 0

Z

t

= t (Ts (q(; ) e )(x) , q(x; ) e (x))ds 0 6 sup j(Ts , id)(q(; ) e )(x)j; 1

s6t

which tends to 0 as t ! 0. This implies lim

k!1

for all x;

t(x; ) , 1 = e (x)q(x; ) e (x); , t

2 Rn and the Theorem follows.

2

In the above proof of Theorem 3.1 we referred to the following Lemmas. LEMMA 3.2. Let ,q (x; D ) Theorem 3.1. Then

= ,q1(D) , q2(x; D); fTt gt>0 ,

and

k

be as in

lim T q (x; D )(e k )(x) = Ts (q2 (; ) e )(x) k!1 s 2 holds true and for every 2 Rn

jTsq2(x; D)(e k )(x)j 6 c; c = c uniformly in x 2 Rn ; s > 0, and k 2 N . Proof. Note that

Fx7! (q2(x; D)(e k )(x))() Z Z = n e,ix n q2 x; k + eix=k eix ^1()d,d,x R R Z Z , ix ( , ,=k ) = n ne q2 x; k + ^1 ()d,d,x R R Z = n q^2 , k + ; k + ^1()d,; R

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101


where q^2 (; ) denotes the Fourier transform w.r.t the first variable. The interchange ^1 is rapidly decreasing and of the order of integration was justified by the fact that q x; k + 6 0(x) 1 + a2 k +

2 !

6 ca 0(x) 1 +

k +

2

2

!

2

6 2ca 0(x)(1 + k k ) 1 + k

;

2

2

with 0 2 L1 (Rn ). Since Tt is a pseudo differential operator with symbol t (x; ) we get

Tsq2(; D)(e k )(x) =

Z

Z

Rn Rn

eix s (x; )^ q2

, k + ; k + ^1()d,d,:

For the integrand the following estimate holds

, k + ; k + ^1() 6 q^2 , k + ; k + j^1()j

eix s (x; )^ q2

2 !,(n+1)=2

2

1+a 6C 1+ , k + k + j^1 ()j ;

(by Lemma 2.4)

n

=

2 !( +1) 2

6 C 0 2n+2(1 + kk2 ),(n+1)=2 (1 + k k2 )(n+1)=2 1 +

k

(1 + a2())

1 + a2

(using 3 Peetre’s inequality)

j^ ()j; 1 k

6 C 0 Ca2 2n+2(1 + k k2 )(n+3)=2 2

(1 + kk2 ), n

=2 (1 + kk2 )(n+3)=2 j^1 ()j:

( +1)

The right-hand side is integrable in and , and the estimate is uniform in x 2 Rn , s > 0, and k 2 N . Therefore, we can use dominated convergence and arrive at lim T q (; D )(e k )(x) k!1 s 2

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RENE´ L. SCHILLING

= = =

Z

Z

Rn Rn Z

Z

Rn Rn

Z

q^ , k + ; k + ^1()d,d, s (x; ) klim !1 2

= (0)

q2 ( , ; )^1 ()d,d, eix s (x; )^

^1()d, n

R

eix

Z

Rn

Z

Rn

eix s (x; )^ q2 ( , ; )d,

s(x; ) eix (q2(; )e )^()d,

= Ts (q2(; )e )(x) and sup sup sup jTs q2 (; D )(e k )(x)j 6 c(n; a2 ; 1 ; ):

2

x2Rn s>0 k2N

The following Lemma has essentially the same proof as Lemma 3.2. LEMMA 3.3. Let ,q (x; D ) Theorem 3.1. Then

= ,q1(D) , q2(x; D); fTt gt>0 ,

and

k

be as in

lim T q (D )(e k )(x) = Ts (q1 ( )e )(x) k!1 s 1 holds true and for every 2 Rn

jTsq1(D)(e k )(x)j 6 c; c = c ; uniformly in x 2 Rn ; s > 0, and k 2 N . Assume now, that in the situation of Theorem 3.1 the generator ,q (x; D ) is conservative, i.e., Tt 1 = 1 for all t > 0. Then t (x; 0) = Tt 1 = 1 and we find d t (x; 0) , 1 = 0: ,q(x; 0) = dt t (x; 0) = tlim ! 0 t t=0 The above calculation employs Theorem 3.1 only for = 0. It is therefore clear

from the proof of Theorem 3.1 (below formula (3.4)) that we only have to assume that q (x; 0) is bounded rather than q (x; ) 6 c(1 + k k2 ). This proves the following result. COROLLARY 3.4. Assume that ,q (x; D ) generates a conservative Feller semigroup fTt gt>0 and that the symbol ,q (x; ) satisfies (1.7)–(1.10.n + 1). Then q(x; 0) 0 holds true.

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COROLLARY 3.5. Assume that ,q (x; D ) generates a conservative L2 (Rn )-subMarkovian semigroup fTtgt>0 and that the symbol ,q (x; ) satisfies (1.7)–(1.10.n+ 1). Then q (x; 0) 0 holds true. Proof. Choose a compact set K Rn and a v 2 Cc1 (Rn ) with supp v K . Since we used the Feller property only in the last stage of the proof of Theorem 3.1, we may use (3.4) and restate it in the form 0=,

1

t

t

Z

0

Tsq(; 0)ds; v

= ,t

1

L2

(remember that = 0; e 1, and t (x; ) deduce as in the proof of Theorem 3.1

t

Z

(Ts q(; 0); v)L ds; 2

0

= 1). Using Lemma 3.6 below, we

0 = (q (; 0); v )L2 : Since K and v were arbitrary, we get continuity of q (x; ), everywhere.

q(x; 0) = 0 a.s. and then, because of the 2

It remains to show the following Lemma. LEMMA 3.6. Let fTt gt>0 be a L2 -sub-Markovian semigroup. Then lim (T u; v )L2 = (u; v )L2 ; t!0 t for all u 2 Cb (Rn ) and every v 2 Cc (Rn ). Proof. Fix a v 2 Cc (Rn ) and assume that supp v is contained in a compact set K Rn . Choose some approximate identity fk gk2N; k 2 Cc1(Rn ); 1Bk (0) 6 k 6 1Bk+1(0) , and set uk = uk . Then

j(Tt u , u; v)L j 6 j(Tt u , Tt uk ; v)L j + j(Tt uk , uk ; v)L j +j((uk , u)1K ; v)L j: 2

2

2

(3.5)

2

From the conservativeness, Tt 1 = 1, we get

lim (Tt u , Tt uk ; v )L2

t!0

j(T (u , uk ); v)L j = tlim !0 t 6 tlim (kuk1 Tt (1 , k ); jvj)L !0 2

2

= kuk1 tlim (1 , Tt k ; jvj)L !0

2

= kuk1 (1; jvj)L , tlim (T ; jvj)L !0 t k = 0: 2

2

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RENE´ L. SCHILLING

For uk 2 Cc (Rn ) we clearly have limt!0 j(Tt uk , uk ; v )L2 j = 0 and since K is compact, limk!1 j((uk , u)1K ; v )L2 j = 0. Thus, the assertion follows letting in 2 (3.5) first t ! 0 and then k ! 1. Note that the proof of Lemma 3.6 is in line with Oshima’s criterion for conservativeness of Dirichlet forms, cf. [13]. References 1. Berg, C. and Forst, G.: Potential Theory on Locally Compact Abelian Groups, Springer, Ergebnisse der Mathematik und ihrer Grenzgebiete, II. Ser. 87, Berlin, 1975. 2. Butzer P.L. and Berens, H.: Semi-Groups of Operators and Approximation, Springer, Grundlehren Math. Wiss. Bd. 145, Berlin, 1967. 3. Courrège, Ph.: ‘Générateur infinitésimal d’un semi-groupe de convolution sur Rn , et formule de Lévy–Khinchine’, Bull. Sci. Math. 2e sér. 88 (1964), 3–30. 1 dans C satisfaisant au 4. Courrège, Ph.: ‘Sur la forme intégro-différentielle des opérateurs de CK principe du maximum’, Sém. Théorie du Potentiel (1965/66), 38pp. 5. Ethier, St.E. and Kurtz, Th.G.: Markov Processes: Characterization and Convergence, Wiley, Series in Prob. and Math. Stat., New York, 1986. 6. Fukushima, M., Oshima, Y. and Takeda, M.: Dirichlet Forms and Symmetric Markov Processes, de Gruyter, Studies in Math. 19, Berlin, 1994. 7. Hoh, W.: ‘The martingale problem for a class of pseudo differential operators’, Math. Ann. 300 (1994), 121–147. 8. Hoh, W.: ‘Pseudo differential operators with negative definite symbols and the martingale problem’, Stoch. and Stoch. Rep. 55 (1995), 225–252. 9. Hoh, W. and Jacob, N.: ‘Upper bounds and conservativeness for semigroups associated with a class of Dirichlet forms generated by pseudo differential operators’, Forum Math. 8 (1996), 107–120. 10. Jacob, N.: ‘A class of Feller semigroups generated by pseudo differential operators’, Math. Z. 215 (1994), 151– 166. 11. Jacob, N.: ‘Characteristic functions and symbols in the theory of Feller processes’, Warwick Preprints 46 (1995); Potential Analysis 8 (1998), 1–19. 12. Ma, Z.M., Overbeck, L., and Röckner, M.: ‘Markov processes associated with semi-Dirichlet forms’, Osaka J. Math. 32 (1995), 97–119. 13. Oshima, Y.: ‘On conservativeness and recurrence criteria for Markov processes’, Potential Analysis 1 (1992), 115–131.

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