Conservativeness of Semigroups Generated by Pseudo Differential Operators

Transcription

1 Potential Analysis 9: , c 1998 Kluwer Academic Publishers. Printed in the Netherlands. Conservativeness of Semigroups Generated by Pseudo Differential Operators RENÉ L. SCHILLING? Mathematisches Institut, Universität Erlangen, Bismarckstraße 1 1, D Erlangen, Germany 2 (Received: 15 March 1996; accepted: 14 June 1996) Abstract. Assume that the pseudo differential operator,q(x; D) generates a Fellerian or sub- Markovian 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 (C 0 )-semigroup ft t g t>0 of contraction operators on a Banach space of real functions (X; kk) is said to be positive andtohavethesub-markov property if 0 6 T t u 6 1(inX) whenever u 2 X and 0 6 u 6 1. If (X; kk) = (L 2 ( ;dm); kk L2 ) (one could also consider some open ), we call ft t g t>0 a sub-markovian semigroup,if(x; kk)=(c 1 ( ); kk 1 ), the Banach space of continuous functions vanishing at infinity, we call ft t g t>0 a Feller semigroup.in either case, the operators T t are positive and continuous, and there exist representing kernels p t (x; ) of sub-probability measures s.t. T t u(x) = u(y)p t (x; dy); x 2 (1.1) holds for, say, compactly supported continuous functions u. Clearly, (1.1) allows us to extend the operators T t to all bounded measurable functions B b ( ) and we will do so without changing our notation. Therefore, the following definition of conservativeness makes sense. DEFINITION 1.1. A sub-markovian semigroup ft t g t>0 is said to be conservative if T t 1 = 1 holds almost surely. A Fellerian semigroup ft t g t>0 is said to be conservative if T t 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.

2 92 RENÉ L. SCHILLING If A is the infinitesimal generator of the sub-markovian or Fellerian semigroup ft t g t>0, we call A conservative whenever ft t g t>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 sub- Markovian semigroups give rise to stochastic processes. The relation between them is, in the Feller case, given by T t 1 B (x) =P x (X t 2 B) for all x 2 and all Borel sets B, and in the sub-markovian context given by e T t u(x) = E x u(x t ) for all x outside a set of capacity zero and all u 2 L 2 ( ;dm). Here, e stands for the quasi-continuous modification, cf. [6, 12]. Therefore, we find 1 = T t 1(x) =E x 1 = P x (X t 2 ) a:s: Thus, a conservative process fx t g t>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 fs t g t>0 denote a sub-markovian convolution semigroup on L 2 ( ). In this special setting, fs t g t>0 is also a Feller semigroup. The process fx t g t>0 corresponding to a convolution semigroup is known to be a Lévy process, cf. [1, Chap. 8], that is a -valued and stochastically continuous process with stationary and independent increments. Note, that X t is spatially homogeneous in the sense that law(x t jx 0 = x) =law(x t + xjx 0 = 0). The infinite divisibility of its law gives the Fourier transform a particularly simple structure E x e ixt = e,t () ; x; 2 ; (1.2) where the continuous negative definite characteristic exponent is described by the Lévy-Khinchine formula () = ` + ib +(; Q) + x6=0 1, e,ix, ix 1 + kxk kxk 2 kxk 2 (dx); 2 ; (1.3) with ` > 0;b 2 ;Q 2 n non-negative definite, and the finite jump (Lévy) measure on nf0g. For further reference we note that p j j is subadditive and locally bounded, hence, j ()j 6 c (1 + kk 2 ) 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 S t u(x) = e ix e,t () ^u() d, ; u 2 C 1 c (Rn ); x 2 (1.4)

3 CONSERVATIVENESS, SEMIGROUPS AND PSEUDO DIFFERENTIAL OPERATORS 93 and, see also [1, Chap. 12] Au(x) =, (D)u(x) =, e ix ()^u() d, ; u 2 C 1 c (Rn ); x 2 (1.5) are valid, where ^u() = R 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 fs t g t>0 (or A or fx t g t>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 C 2 c (Rn ) is contained in its domain is necessarily of the form,q(x; D)u(x) =, e ix q(x; )^u() d, ; u 2 C 1 c (Rn ); x 2 ;(1.6) where q :! C is locally bounded and 7! q(x; ) is continuous negative definite, that is, admits for every x 2 alévy 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)jC 1 c (Rn ) extends to a generator of a Feller semigroup ft t g t>0. In [10] the problem was analytically treated, regarding,q(x; D) as a perturbation of a constant coefficient (Lévy) generator,q 1 (D). Basically the following four assumptions were used q :! R is continuous and q(x; ) is negative definite; (1.7) q(x; ) =q(x 0 ;)+(q(x; ), q(x 0 ;)) q 1 () +q 2 (x; ); (1.8) 1 (1 + a 2 ()) q 1 () 6 0 (1 + a 2 ()) for some constant 0 0 and a fixed continuous negative definite a 2 () s:t: for large jj; 2 a 2 () > cjj r 0 holds with some constants c>0; r 0 > 0; (1.9) j@ x q 2(x; )j 6 (x)(1 + a 2 ()) for 2 N n 0 ; jj 6 m; with 2 L 1 ( ) (1:10:m) for sufficiently large m (depending on the dimension n) and with some additional assumptions on the smallness of the perturbation P jj6m k k L 1 regarding 0 and r 0.

4 94 RENÉ L. SCHILLING Using a martingale problem approach, Hoh [7, 8] was able to improve on this result, requiring only 2 L 1 ( ) instead of 2 L 1 ( ), thus discarding the smallness condition on P jj6m k k L 1. With both approaches we also get a L 2 -sub-markovian semigroup again denoted by ft t g t>0 if,e.g.,,q(x; D)jC 1 c (Rn ) is a symmetric operator on L 2 ( ), 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 fx t g t>0 (or ft t g t>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 L 2 -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 C 2 u (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 sub- Markovian 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 ft t g t>0 and that the symbol q(x; ) satisfies (1.7) (1.9). (i) If(1.10.0) holds with 0 2 L 1 ( ) \ L 1 ( ),thenq(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 ft t g t>0 and that the symbol q(x; ) satisfies (1.7) (1.10.n+1) with 0 2 L 1 ( )\L 1 ( ). 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)jC 1 c (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.

5 CONSERVATIVENESS, SEMIGROUPS AND PSEUDO DIFFERENTIAL OPERATORS 95 NOTATION. We write d, x =(2),n=2 dx for the normalized Lebesgue measure on and e (x) =e,ix. C c ( );C 1 ( );C b ( ),andb b ( ) 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]. Thebounded pointwise it (bp-it) of a sequence of bounded measurable functions fu k g k2n B b ( ) is defined by bp, k!1 u k = u if sup k ku k k 1 < 1 and k!1 u k(x) =u(x); x 2 : (2.1) As usual, the bp-closure of a subset A B b ( ) is the smallest closed (w.r.t. bp-its) set containing A. The following Lemma is implicitly included in [5, p. 166]. LEMMA 2.1. Let A be the infinitesimal generator of a contractive (C 0 )- semigroup ft t g t>0 on the Banach space (X; kk 1 );X B b ( ). 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. (ii) (1; 0) 2f(u; v) 2 B b ( ) B b ( ): T t u, u R 1 = 0 T sv dsg. (iii) A is conservative, i.e., T t 1 = 1. Proof. The implication (ii)! (iii) is immediate. In order to see (i)! (ii), note that the set in (ii) is bp-closed and contains Graph(A). 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 ft t g t>0 where q(x; ) satisfies (1.7) (1.10.0) with 0 2 L 1 ( )\L 1 ( ).Thenq(x; 0) 0 implies that,q(x; D) is conservative. Proof. The requirement that k 0 k 1 < 1 implies q(x; ) 6 (k 0 k )(1 + a 2 ()) 6 c a 2 (k 0 k )(1 + kk 2 ): (2.2) Choose the approximate identity g k (x) =e,kxk2 =2k 2 ;k 2 N;x 2, and observe q(x; D)g k (x) = e ix q(x; )^g k () d,

6 96 RENÉ L. SCHILLING = e ix q(x; )k n e,k2 kk 2 =2 d, = e ix=k q x; e,kk2 =2 d, : (2.3) k From (2.2) we conclude! Rn jq(x; D)g k (x)j 6 c a 2 (k 0 k ) 1 + kk2 e,(kk2 =2) d, 6 c a 2 (k 0 k ) k 2 (1 + kk 2 ) e,(kk2 =2) d, ; i.e., sup k2n jq(x; D)g k (x)j < 1. The above estimate holds pointwise for the integrand and we may use the dominated convergence theorem in (2.3) to find k!1 q(x; D)g k(x) =q(x; 0) =0; in bp-sense. Since also bp k!1 g k = 1, we conclude that (1,0) lies in the bpclosureofgraph(,q(x; D)), and, bylemma2.1, 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 + kk 2 ); (2.4) uniformly in x 2. 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 ft t g t>0 be a contractive (C 0 )-semigroup on a Banach space (X; kk) and denote by A its infinitesimal generator. Then kauk = sup T t u, u t>0 t (2.5) holds for all u 2 D(A). IfX is reflexive, we have u 2 D(A) if and only if sup T t u, u t>0 t < 1: (2.6) Proof. Observe that T t u, u 1 = t t t 0 T s Auds holds for all u 2 D(A). By the triangle inequality and contractivity we get T t u, u t 6 1 t kt s Auk ds 6 1 t kauk ds = kauk t t 0 0

7 CONSERVATIVENESS, SEMIGROUPS AND PSEUDO DIFFERENTIAL OPERATORS 97 1 and (2.5) follows from t!0 t kt tu,uk = kauk if u 2 D(A). Since for reflexive spaces X D(A) = u 2 X : inf T t u, u t!0 t < 1 is true, cf. [2, p. 88, Thm (c)], (2.5) implies (2.6). 2 LEMMA 2.4. Assume that the function q 2 (x; ) satisfies (1.10.n + 1). Then the Fourier transform w.r.t. x satisfies ^q 2 (;) 2 L 2 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 + a 2 ())(1 + kk 2 ),(n+1)=2 6 c 0 a (1 2 + kk 2 )(1 + kk 2 ),(n+1)=2 ; (2.7) which shows the square integrability. 2 PROPOSITION 2.5. Assume that,q(x; D) generates a sub-markovian semigroup ft t g t>0 and that q(x; ) satisfies (1.7) (1.10.n + 1). Then q(x; 0) 0 implies that,q(x; D) is conservative. Proof. Denote by g k (x) = e,kxk2 =2k 2 ;k 2 N;x 2, and observe ^g k () = k n^g 1 (kx). By (1.8) we have q(x; ) = q 1 () +q 2 (x; ) with q 1 (0) = 0, i.e. the convolution (Lévy) semigroup generated by,q 1 (D) is conservative. For any v 2 C 1 c (Rn ) we have by Lemma t j(t tg k, g k ;v) L 2j 6 sup j(q(x; D)g k ;v) L 2j kvk L 2=1 6 kq 1 (D)g k k L 2 + sup j(q 2 (x; D)g k ;v) L 2j: (2.8) kvk L 2=1 Using Plancherel s Theorem and dominated convergence we find k!1 kq 1(D)g k k L 2 = k!1 kq 1^g k k L 2 = k!1 kq 1(=k)^g 1 k L 2 = 0 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 C 1 c (Rn ) (q 2 (x; D)g k ;v) L 2 = ^q 2 (, ; )^g k ()^v() d, d, ; where ^q 2 (; ) stands for the Fourier transform w.r.t. the first variable. Now sup j(q 2 (x; D)g k ;v) L 2j kvk L 2=1

8 98 RENÉ L. SCHILLING = sup ^q 2 (, ; )^g k ()^v() d, d, kvk L 2=1 6 (2),n=2 sup ^q 2 (,; )^g k () d, kvk L 2=1 L 2 k^vk L 2 6 (2),n=2 k^q 2 (,; )k L 2j^g k ()j d, = (2),n=2 ^q 2 ; L j^g 1 ()j d, : k 2 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 k!1 ^q 2 ; k L 2 = k^q 2 (; 0)k L 2: Another application of dominated convergence note that (2.7) also implies k^q 2 (; n=k)k L 2 6 c(1 + kk 2 ) uniformly in k,andthat(1 + kk 2 )^g 1 2 L 1 ( ) yields k!1 ^q 2 ; k L 2 j^g 1 ()j d, = = k!1 ^q 2 ; k L 2 j^g 1 ()j d, k^q 2 (; 0)k L 2j^g 1 ()j d, : Plancherel s Theorem shows k^q 2 (; 0)k L 2 = kq 2 (; 0)k L 2 = 0 and we find 1 k!1 t j(t tg k, g k ;v) L 2j = 0; for all t>0andanyv 2 C 1 c (Rn ). Since by dominated convergence k!1 T t g k = T t 1, and also 0 = k!1 j(t tg k, g k ;v) L 2j = j(t t 1, 1;v) L 2j; for all v 2 C 1 c (Rn ), we conclude that T t 1 = 1a.s 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

9 CONSERVATIVENESS, SEMIGROUPS AND PSEUDO DIFFERENTIAL OPERATORS 99 semigroup ft t g t>0 and satisfies (1.7) (1.10.n + 1). This follows from the more general result d dt t(x; ) =,q(x; ); x; 2 ; (3.1) t=0 where t (x; ) is the symbol of the operator T t, t (x; ) =e, (x)t t e (x): (3.2) The relation (3.1) was shown by Jacob [11] under the assumption that C 2 u (Rn ), and in particular the functions e (x) = e,ix ;x; 2, is in the domain of the (Feller) generator,q(x; D).Forpractical use this meansthat one should consider strongly Fellerian semigroups ft t g t>0. The following Theorem is valid without this restriction. THEOREM 3.1. Assume that,q(x; D) generates a conservative Feller semigroup ft t g t>0 and that the symbol,q(x; ) satisfies (1.7) (1.10.n + 1) with 0 2 L 1 ( ) \ L 1 ( ).Then d dt t(; x) =,q(x; ) (3.3) t=0 holds for all x; 2. 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 C 1 c (Rn ) such that 1 B1 (0) B2 (0) and set k (x):= 1 (x=k). Clearly, k (x)! 1ask! 1 and ^ k () = k n ^ 1 (k). From Lemmas 3.2, 3.3 below we know jt s q(x; D)(e k )(x)j 6 c; c = c ; 2 ; uniformly in x 2 ;s> 0, and k 2 N. Sincee k 2 D(q(x; D)), wefind T t e (x), e (x) = k!1 (T t(e k )(x), e (x) k (x)) t Combining Lemma 3.2 and 3.3 below we get =, T s q(x; D)(e k )(x)ds k!1 0 t =, k!1 T sq(x; D)(x; D)(e k )(x)ds: k!1 T sq(x; D)(e k )(x) =T s (q(;)e )(x): 0

10 100 RENÉ L. SCHILLING Thus t (x; ), 1 t = e T te (x), e (x), (x) t t =, e,(x) T s (q(;)e )(x)ds: (3.4) t 0 Since q(;) e is bounded for fixed 2, thus T s (q(;) e )(x)! q(x; ) e (x) as s! 0 and therefore 1 t T s (q(;) e )(x) ds, q(x; ) e (x) t 0 = 1 t (T s (q(;) e )(x), q(x; ) e (x))ds t 0 6 sup j(t s, id)(q(;) e )(x)j; s6t which tends to 0 as t! 0. This implies t (x; ), 1 = e, (x)q(x; ) e (x); k!1 t for all x; 2 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) =,q 1 (D), q 2 (x; D); ft t g t>0, and k be as in Theorem 3.1.Then k!1 T sq 2 (x; D)(e k )(x) =T s (q 2 (;) e )(x) holds true and for every 2 jt s q 2 (x; D)(e k )(x)j 6 c; c = c uniformly in x 2 ;s> 0, and k 2 N. Proof. Note that F x7! (q 2 (x; D)(e k )(x))() = e,ix q 2 x; k R + n x; k + e ix=k e ix ^ 1 ()d, d, x = e,ix(,,=k) q 2 = ^q 2, k + ; k + ^ 1 ()d, ; ^ 1 ()d, d, x

11 CONSERVATIVENESS, SEMIGROUPS AND PSEUDO DIFFERENTIAL OPERATORS 101 where ^q 2 (; ) denotes the Fourier transform w.r.t the first variable. The interchange of the order of integration was justified by the fact that ^ 1 is rapidly decreasing and q 2 x; k (x) 1 + a 2 k + 6 c a 2 0 (x) 1 + k + 2! 6 2c a 2 0 (x)(1 + kk 2 ) 1 +! 2 ; k with 0 2 L 1 ( ). Since T t is a pseudo differential operator with symbol t (x; ) we get T s q 2 (;D)(e k )(x), ; k + k + ^ 1 ()d, d, : = e ix s (x; )^q 2 For the integrand the following estimate holds eix s (x; )^q 2, ; k + k + ^ 1 () 6 ^q 2, k + ; k + j ^ 1 ()j! 6 C 1 + 2,(n+1)=2, k a 2 k + j ^ 1 ()j ; (by Lemma 2.4) 6 C 0 2 n+2 (1 + kk 2 ),(n+1)=2 (1 + kk 2 ) (n+1)=2 1 + k (1 + a 2 ()) (using 3 Peetre s inequality) 1 + a 2 j ^ 1 ()j; k 6 C 0 C 2 a 2 2 n+2 (1 + kk 2 ) (n+3)=2 (1 + kk 2 ),(n+1)=2 (1 + kk 2 ) (n+3)=2 j ^ 1 ()j: 2! (n+1)=2 The right-hand side is integrable in and, and the estimate is uniform in x 2, s>0, and k 2 N. Therefore, we can use dominated convergence and arrive at k!1 T sq 2 (;D)(e k )(x)

12 102 RENÉ L. SCHILLING = e ix s (x; ) k!1 ^q 2, k + ; k + ^ 1 ()d, d, and = = = (0) e ix s (x; )^q 2 (, ; )^ 1 ()d, d, e ix s (x; )^q 2 (, ; )d, s (x; ) e ix (q 2 (;)e )^()d, ^ 1 ()d, = T s (q 2 (;)e )(x) sup x2 sup s>0 sup jt s q 2 (;D)(e k )(x)j 6 c(n; a 2 ; 1 ;): 2 k2n The following Lemma has essentially the same proof as Lemma 3.2. LEMMA 3.3. Let,q(x; D) =,q 1 (D), q 2 (x; D); ft t g t>0, and k be as in Theorem 3.1. Then k!1 T sq 1 (D)(e k )(x) =T s (q 1 ()e )(x) holds true and for every 2 jt s q 1 (D)(e k )(x)j 6 c; c = c ; uniformly in x 2 ;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., T t 1 = 1forallt>0. Then t (x; 0) =T t 1 = 1 and we find,q(x; 0) = d dt t(x; 0) t (x; 0), 1 = = 0: t=0 t!0 t 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 +kk 2 ). This proves the following result. COROLLARY 3.4. Assume that,q(x; D) generates a conservative Feller semigroup ft t g t>0 and that the symbol,q(x; ) satisfies (1.7) (1.10.n + 1). Then q(x; 0) 0 holds true.

13 CONSERVATIVENESS, SEMIGROUPS AND PSEUDO DIFFERENTIAL OPERATORS 103 COROLLARY 3.5. Assume that,q(x; D) generates a conservative L 2 ( )-sub- Markovian semigroup ft t g t>0 and that the symbol,q(x; ) satisfies (1.7) (1.10.n+ 1).Thenq(x; 0) 0 holds true. Proof. Choose a compact set K and a v 2 C 1 c (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 1 t 0 =, T s q(; 0)ds; v =, 1 t (T s q(; 0);v) L 2ds; t 0 L 2 t 0 (remember that = 0; e 1, and t (x; ) = 1). Using Lemma 3.6 below, we deduce as in the proof of Theorem =(q(; 0);v) L 2 : Since K and v were arbitrary, we get q(x; 0) = 0 a.s. and then, because of the continuity of q(x; ), everywhere. 2 It remains to show the following Lemma. LEMMA 3.6. Let ft t g t>0 be a L 2 -sub-markovian semigroup. Then t!0 (T tu; v) L 2 =(u; v) L 2 ; for all u 2 C b ( ) and every v 2 C c ( ). Proof. Fixa v 2 C c ( ) and assume that supp v is contained in a compact set K. Choose some approximate identity f k g k2n ; k 2 C 1 c (Rn ); 1 Bk (0) 6 k 6 1 Bk+1(0), andsetu k = u k.then j(t t u, u; v) L 2j 6 j(t t u, T t u k ;v) L 2j + j(t t u k, u k ;v) L 2j +j((u k, u)1 K ;v) L 2j: (3.5) From the conservativeness, T t 1 = 1, we get (T tu, T t u k ;v) L 2 t!0 = j(t t(u, u k );v) L 2j t!0 6 t!0 (kuk 1 T t (1, k ); jvj) L 2 = kuk 1 t!0 (1, T t k ; jvj) L 2 = kuk 1 (1; jvj) L 2, t!0 (T t k ; jvj) L 2 = 0:

14 104 RENÉ L. SCHILLING For u k 2 C c ( ) we clearly have t!0 j(t t u k, u k ;v) L 2j = 0 and since K is compact, k!1 j((u k, u)1 K ;v) L 2j = 0. Thus, the assertion follows letting in (3.5) first t! 0andthenk!1. 2 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, Butzer P.L. and Berens, H.: Semi-Groups of Operators and Approximation, Springer, Grundlehren Math. Wiss. Bd. 145, Berlin, Courrège, Ph.: Générateur infinitésimal d un semi-groupe de convolution sur, et formule de Lévy Khinchine, Bull. Sci. Math. 2 e sér. 88 (1964), Courrège, Ph.: Sur la forme intégro-différentielle des opérateurs de C 1 K dans C satisfaisant au 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, Fukushima, M., Oshima, Y. and Takeda, M.: Dirichlet Forms and Symmetric Markov Processes, de Gruyter, Studies in Math. 19, Berlin, Hoh, W.: The martingale problem for a class of pseudo differential operators, Math. Ann. 300 (1994), Hoh, W.: Pseudo differential operators with negative definite symbols and the martingale problem, Stoch. and Stoch. Rep. 55 (1995), 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), Jacob, N.: A class of Feller semigroups generated by pseudo differential operators, Math (1994), Jacob, N.: Characteristic functions and symbols in the theory of Feller processes, Warwick Preprints 46 (1995); Potential Analysis 8 (1998), Ma,.M., Overbeck, L., and Röckner, M.: Markov processes associated with semi-dirichlet forms, Osaka J. Math. 32 (1995), Oshima, Y.: On conservativeness and recurrence criteria for Markov processes, Potential Analysis 1 (1992),