. Introduction The purpose of this paper is to study the Cauchy{Problem of second order dierential operators with measurable coecients of type (.1) Lu

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1 (NONSYMMETRIC) DIRICHLET OPERATORS ON L 1 : EXISTENCE, UNIQUENESS AND ASSOCIATED MARKOV PROCESSES Wilhelm Stannat Universitat Bielefeld, Fakultat fur Mathematik, Postfach 1131, 3351 Bielefeld, Germany stannat@mathematik.uni-bielefeld.de Abstract. Let L be a nonsymmetric operator of type Lu = P a j u + P b i u on an arbitrary subset U R d. We analyse L as an operator on L 1 (U; ) where is an invariant measure, i.e., a possibly innite measure satisfying the equation L = (in the weak sense). We explicitely construct, under mild regularity assumptions, extensions of L generating sub{markovian C {semigroups on L 1 (U; ) as well as associated diusion processes. We give sucient conditions on the coecients so that there exists only one extension of L generating a C {semigroup and apply the results to prove uniqueness of the invariant measure. Our results imply in particular that if ' 2 H 1;2 loc (Rd ; dx), ' 6= dx{a.e., the generalized Schrodinger operator ( + 2'?1 r' r; C 1 (Rd )) has exactly one extension generating a C {semigroup if and only if the Friedrich's extension is conservative. We also give existence and uniqueness results for a corresponding class of innite dimensional operators acting on smooth cylinder functions on a separable real Banach space. Mathematics Subject Classication (1991): Primary: 31 C 25, 47 D 7 Secondary: 35 K 5, 47 B 44, 6 J 35, 6 J 6 Running Head: (Nonsymmetric) Dirichlet operators on L 1 Typeset by AMS-TEX 1

2 . Introduction The purpose of this paper is to study the Cauchy{Problem of second order dierential operators with measurable coecients of type (.1) Lu = dx i;j=1 a j u + dx i=1 b i u dened on C 1 (U), U R d open, on the space L 1 (U; ), where is an invariant measure, i.e., a possibly innite measure satisfying Lu 2 L 1 (U; ) for all u 2 C 1 (U) and (.2) Lu d = for all u 2 C 1 (U) : Here the (a ij ) 1i;jd are supposed to be locally strictly elliptic and b i 2 L 2 loc (Rd ; ), 1 i d. We do not assume that L is symmetric or that the rst order part is a small perturbation of the second order part in any classical sense. Results obtained by V.I. Bogachev, N. Krylov and M. Rockner (cf. [BKR1], [BKR2] and [BR]) show that it is natural to suppose that is absolutely continuous w.r.t. the Lebesgue measure dx and that the density admits a representation ' 2, ' 2 H 1;2 loc (U). Moreover, we will assume that a ij, 1 i; j d, is contained locally in the weighted Sobolev R space H 1;2 (U; ) (= the closure of C1 (U) in L 2 (U; ) w.r.t. the norm given by jruj 2 + u 2 d). We are mainly concerned with the following three problems: (i) Construct and analyse extensions L of L generating C {semigroups (T t ) t on L 1 (U; ) that are sub{markovian (i.e., f 1 implies T t f 1). Henceforth, any extension of L generating a C {semigroup will be called maximal. (ii) Find conditions on (a ij ) 1i;jd, (b i ) 1id and so that there is only one maximal extension. (iii) Construct diusion processes with transition probabilities given by (T t ) t. Concerning problem (i) we will construct in Theorem 1.5 below a maximal extension L generating a sub{markovian C {semigroup (T t ) t in such a way that the space D(L) b of all bounded functions in the domain D(L) is contained locally in the weighted Sobolev space H 1;2 (U; ). This implies that, although L is not symmetric, this maximal extension is still associated with a bilinear form. More precisely, the following representation holds: (.3) dx i;j=1 a i j v d + dx i=1 (b i? b i )@ i u v d =? Lu v d for all u 2 D(L) b and v 2 H 1;2 (U; ) (= the space of all elements v 2 H 1;2 (U; ) with compact support contained in U). Here (.4) b i = dx j=1 (@ j a ij + 2a j '=') ; 1 i d ; 2

3 (cf. Theorem 1.5). Since L is a Dirichlet operator, i.e., the generator of a sub{ Markovian semigroup, D(L) b is dense in D(L) w.r.t. the graph norm, so that our result implies that L is completely determined by the rst order object (.3). We would like to emphasize that the construction of (L; D(L)) is purely analytic and uses only results from semigroup theory as well as Dirichlet form techniques adapted to the nonsymmetric case. The question whether is (T t ){invariant (i.e., R T t u d = R u d for all u 2 L 1 (U; )) is of particular interest for applications. Clearly, is (T t ){invariant if and only if R Lu d = for all u 2 D(L). However, it can happen, even in the case where R the measure is nite and U = R d, that is not (T t ){invariant although Lu d = for all u 2 C 1 (R d ) (cf. Example 1.12). In the symmetric case (T t ){invariance of is equivalent to the conservativeness of (T t ) t and the latter has been well{studied by many authors (cf. [D], [FOT, Section 1.6], [S] and references therein). However, in the context of an L 1 {framework, there are no nontrivial results on (T t ){invariance of in the nonsymmetric case. Proposition 1.9 completely characterizes (T t ){invariance of in terms of the bilinear form (.3). More precisely, is (T t ){invariant if and only if there exist n 2 H 1;2 loc (Rd ; ) and > such that ( n? 1)? 2 H 1;2 (Rd ; ), lim n!1 n = {a.e. and n v d + dx i;j=1 a i j v d + dx i=1 (b i? b i )@ i n v d for all v 2 H 1;2 (Rd ; ), v. This characterization allows one in particular to apply the method of Lyapunov functions to derive sucient conditions for (T t ){ invariance of (cf. Proposition 1.1 (b) and (c)). These criteria are well{known in the symmetric case (cf. [D]) but have been proved in cases only where classical regularity theory to the operator L can be applied which is not the case here. Since we are working in an L 1 {framework it is also possible to formulate sucient conditions for (T t ){invariance of in terms of integrability assumptions on the coecients (cf. Proposition 1.1 (a)). More precisely, the measure is (T t ){invariant if a ij ; b i? b i 2 L 1 (R d ; ) ; 1 i; j d : Our next result is related to the uniqueness of maximal extensions of L (dened on C 1 (U)) both in the symmetric and in the nonsymmetric case. Our main result in the symmetric case states that if the (a ij ) 1i;jd are locally Holder{ continuous then (L; C R 1 (R d )) is L 1 {unique if and only if the Friedrich's extension (= the closure of Lu v d, u; v 2 C 1 (R d ), on L 2 (U; )) is conservative (cf. Theorem 2.1). This implies in particular that the generalized Schrodinger operator ( + 2'?1 r' r; C 1 (R d )) has exactly one maximal extension if and only if the Friedrich's extension (or equivalently, the associated diusion process) is conservative. This result completes on the one hand the well{known results on Markov{ uniqueness of such operators obtained by M. Rockner and T.S. hang (cf. [R]) and illustrates on the other hand the dierence between the two notions. 3

4 The results on existence and uniqueness of maximal extensions of (L; C 1 (R d )) can be applied to obtain results on uniqueness of the invariant measure. As an example we prove, based on a regularity result for invariant measures obtained by V.I. Bogachev, N. Krylov and M. Rockner in [BKR2] (cf. also [ABR]), in the particular case a ij = ij, b i 2 L p loc (Rd ; dx), 1 i; j d, for some p > d 2 and P d b i=1 i(x)x i M(jxj 2 ln(jxj 2 + 1) + 1) for some M that there exists R at most one probability measure satisfying b i 2 L 1 loc (Rd ; ), 1 i d, and Lu d = for all u 2 C 1 (R d ) (cf. Proposition 2.8). This result complements a recent result on uniqueness of the invariant measure obtained by S. Albeverio, V.I. Bogachev and M. Rockner in [ABR]. Our last result in the nite dimensional case is related to the existence of diusions whose transition semigroups are given by (T t ) t. Using the framework of generalized Dirichlet forms (cf. [St1]) one can construct, only using the explicit description (.3) of the maximal extension L, in a similar way to the construction of associated Markov processes in the classical theory of Dirichlet forms (cf. [MR]), associated Markov processes having a resolvent that is quasi{strong Feller (cf. Theorem 3.5). A standard technique in the classical theory of Dirichlet forms can then be carried over to the more general nonsymmetric case to show that such Markov processes are in fact diusions (cf. Proposition 3.6). All our methods we developed for the analysis of L and the construction of associated diusions are independent of the dimension and do not use any nite dimensional specialities. This way they can help to prove new results on existence and uniqueness of Dirichlet operators in innite dimensions. As a particular example we consider in Section 4 (nonsymmetric) operators R of type L = L + r, where L is the generator of a gradient form of type EhAru; rvi d acting on smooth cylinder functions on a real separable Banach space E. The results improve the results obtained in [St2] and at the same time the class of operators under consideration is much more general. Acknowledgment: I would like to thank Professor Michael Rockner for his strong interest and steady encouragement. I am also grateful to Andreas Eberle for valuable comments. 1. Existence and Invariance in the finite dimensional case a) Existence under global assumptions. Let U R d be open, a nite positive measure on B(U) with supp() U. Suppose that d dx and that the density admits a representation ' 2, where ' 2 H 1;2 (U). We denote by k k p, p 2 [1; 1], the usual norm on L p (U; ). If W L p (U; ) is an arbitrary subspace, denote by W the subspace of all elements u 2 W such that supp(juj) is a compact subset contained in U, and by W b the subspace of all bounded elements in W. Finally, let W ;b = W \ W b. R Let (D ; H 1;2 (U; )) be the closure of hru; rvi d ; u; v 2 C 1 (U), on L 2 (U; ). Let H 1;2 1;2 loc (U; ) be the space of all elements u such that u 2 H (U; ) for all 4

5 2 C 1 (U). Let A = (a ij ) 1i;jd with (1.1) a ij = a ji 2 H 1;2 loc (U; ) ja ij 2 L 2 (U; ) ; 1 i; j d; be uniformly strictly elliptic, i.e., for some constant > (1.2)?1 jhj 2 ha(x)h; hi jhj 2 for all h 2 R d ; x 2 U : Let (1.3) B 2 L 2 (U; R d ; ) such that (1.4) L A u + hb; rui d = for all u 2 C 1 (U); where L A u := P d i;j=1 a ij@ j u ; u 2 C 1 (U). Our rst result in this section is on existence of closed extensions of L A + B r on L 1 (U; ) generating C {semigroups that are sub{markovian. Recall that in the symmetric case, i.e., B = B = (b 1; : : : ; b d ), where (1.5) b i = dx j=1 (@ j a ij + a ij 2@ j '=') ; 1 i d ; one can use bilinear form techniques to obtain such an extension. In fact, the bilinear form E (u; v) = haru; rvi d ; u; v 2 H 1;2 (U; ) ; is a well{dened symmetric Dirichlet form on L 2 (U; ). Let (L ; D(L )) be the associated generator and (Tt ) t the associated semigroup. It is easy to see that our assumptions imply that C 1 (U) D(L ) and L u = L A u + hb ; rui, u 2 C 1 (U). Since Tt is sub{markovian and symmetric it can be extended uniquely to a sub{ Markovian contraction Tt on L 1 (U; ). It is easy to see that (Tt ) t is a C { semigroup on L 1 (U; ) and that the corresponding generator (L ; D(L )) is the closure of (L ; D(L )) on L 1 (U; ). In the more general nonsymmetric case there is no symmetric bilinear form associated with L A + B r. Note that on the other hand we have that L A u + hb; rui = L u + h; rui, u 2 C 1 (U), where := B? B is such that 2 L 2 (U; R d ; ) and (1.6) h; rui d = for all u 2 C 1 (U); R R R since h; rui d = L A u + hb; rui d? L u d = for all u 2 C 1 (U). Hence L A + B r is associated with a rst order perturbation of the symmetric bilinear form E. Clearly, (1.6) extends to all u 2 H 1;2 (U; ), in particular, h; ruiv d =? h; rviu d for all u; v 2 H 1;2 (U; ) b : Therefore, Lu := L u + h; rui, u 2 D(L ) b, is an extension of L A u + hb; rui, u 2 C 1 (U). 5

6 Proposition 1.1. Let (1.1){(1.4) be satised. Then: (i) The operator Lu = L u+h; rui, u 2 D(L ) b, is dissipative, hence in particular closable, on L 1 (U; ). The closure (L; D(L)) generates a sub{markovian C { semigroup of contractions (T t ) t. (ii) D(L) b H 1;2 (1.7) E (u; v)? (U; ) and h; ruiv d =? Lu v d ; u 2 D(L) b ; v 2 H 1;2 (U; ) b : In particular, (1.8) E (u; u) =? Lu u d ; u 2 D(L) b : Recall that a densely dened linear operator (A; D(A)) on a Banach space X is called dissipative if for any u 2 D(A) there exists a normalized tangent functional ` (i.e., an element ` 2 X (= the topological dual of X) satisfying k`k X = kuk X and `(u) = kuk 2 X ) such that `(Au). Note that the sub{markovian C {semigroup of contractions (T t ) t on L 1 (U; ) can be restricted to a semigroup of contractions on L p (U; ) for all p 2 (; 1) by the Riesz{Thorin Interpolation Theorem (cf. [ReSi, Theorem IX.17]) and that the restricted semigroup is strongly continuous on L p (U; ). The corresponding generator (L p ; D(L p )) is the part of (L; D(L)) on L p (U; ), i.e., D(L p ) = fu 2 D(L) \ L p (U; )jlu 2 L p (U; )g and L p u = Lu, u 2 D(L p ). For the proof of 1.1 we need additional information on the closure (L ; D(L )) of (L ; D(L )) on L 1 (U; ): Lemma 1.2. (i) D(L ) b H 1;2 (U; ). (ii) lim t! T t u = u in H 1;2 (U; ) for all u 2 D(L ) b. (iii) E (u; v) =? R L u v d for all u 2 D(L ) b, v 2 H 1;2 (U; ) b. (iv) Let ' 2 C 2 (R), '() =, and u 2 D(L ) b. Then '(u) 2 D(L ) and L '(u) = ' (u)l u + ' (u)haru; rui : Proof. Let u 2 L(L ) b. Then T t u 2 D(L ) H 1;2 E (T t u? T s u; T t u? T s u) =? =? (U; ) if t > and L (T t u? T s u)(t t u? T s u) d (T t L u? T s L u)(t t u? T s u) d 2kuk 1 kt t L u? T s L uk 1 ; s; t > : 6

7 Therefore (Tt u) t> is an H 1;2 (U; ){Cauchy sequence, which implies that u 2 H 1;2 (U; ) and lim t! Tt u = u in H 1;2 (U; ). Hence (i) and (ii) are proved. Let v 2 H 1;2 (U; ) b. Then which proves (iii). E (u; v) = lim E (T t! t u; v) = lim?(l T t! t u; v) L 2 (U;) = lim t!? T t L u v d =? L u v d (iv) Since u 2 H 1;2 (U; ) b we obtain that '(u) 2 H 1;2 (U; ) b i '(u) = ' (u)@ i u. Hence for all v 2 H 1;2 (U; ) b by (iii) E ('(u); v) = = =? haru; rvi' (u) d haru; r(v' (u))i d? haru; rui' (u)v d (' (u)l u + ' (u)haru; rui)v d : Since ' (u)l u + ' (u)haru; rui 2 L 1 (R d ; ) the assertion now follows from [BH, I ]. Proof (of 1.1). (i) Step 1: Let u 2 D(L ) b then R Lu1 fu>1g d. To show this let " 2 C 2 (R), " >, be such that " (t) = if t 1, " 1, "(t) = 1 if t 1 + " and ". Then " (u) 2 D(L ) b by 1.2, " (u) and thus (1.9) L " (u) d = L u "(u) d : L u "(u) d + " (u)haru; rui d Since lim "! "(u) = 1 fu>1g and k "(u)k 1 1 it follows from Lebesgue's Theorem that L u 1 fu>1g d = lim L u "! "(u) d : Similarly, h; rui1 fu>1g d = lim "! = lim "! h; rui "(u)d h; r " (u)i d = by (1.5). Hence R Lu 1 fu>1g d and Step 1 is proved. 7

8 R R Note that Step 1 implies in particular n Lu 1 fnu>1g d for all n, hence Lu1fu>g d and consequently, Lu(1 fu>g? 1 fu<g ) d : Since kuk 1 (1 fu>g? 1 fu<g ) 2 L 1 (U; ) = (L 1 (U; )) is a normalized tangent functional to u we obtain that (L; D(L ) b ) is dissipative. It follows from [ReSi, Theorem X.48] that the closure (L; D(L)) generates a C { semigroup of contractions on L 1 (U; ) if and only if (1? L)(D(L ) b ) L 1 (U; ) dense which will be proved in the next step. Step 2: (1? L)(D(L ) b ) L 1 (U; ) dense. Let h 2 L 1 (U; ) be such that R (1? L)u h d = for all u 2 D(L ) b. Then u 7! R (1? L )u h d = R h; ruih d, u 2 D(L ) b, is continuous w.r.t. the norm on H 1;2 1;2 (U; ) which implies the existence of some element v 2 H (U; ) such that E 1 (u; v) = R (1? L )u h d for all u 2 D(L ) b. It follows that R (1? L )u(h? v) d = for all u 2 D(L ) b. Since (L ; D(L )) is a Dirichlet operator we obtain that (1? L)(D(L ) b ) L 1 (U; ) dense and consequently, h = v. In particular, h 2 H 1;2 (U; ) and E 1 (h; h) = lim E 1 (T t! t h; h) = lim (1? L )T t! t h h d = lim t! by (1.5) and therefore h =. hb; rt t hih d = hb; rhih d = It follows that the closure (L; D(L)) generates a C {semigroup of contractions (T t ) t. Step 3: (T t ) t is sub{markovian. Let (G ) > be the associated resolvent, i.e., G := (? L)?1. We will show that (G ) > is sub{markovian. Since T t u = lim!1 exp(t(g? 1))u for all u 2 L 1 (U; ) (cf. [Pa, 1.3.5]) we then obtain that (T t ) t is sub{markovian too. To this end let u 2 D(L) and u n 2 D(L ) b such that lim n!1 ku n? uk 1 + klu n? Lu n k 1 = and lim n!1 u n = u {a.e. Let ", " >, be as in Step 1. Then R Lun "(u n ) d for all n by (1.9) and thus R Lu "(u) d. Taking the limit "! we conclude that Lu 1 fu>1g d : Let f 2 L 1 (U; ) and u := G f 2 D(L). If f 1 then u1 fu>1g d = (u? Lu)1 fu>1g d f1 fu>1g d 1 fu>1g d : 8

9 Consequently, R (u? 1)1 fu>1g d which implies that u 1. If f then?nf 1, hence?nu 1 for all n, i.e., u. Hence (G ) > is sub{markovian and (i) is proved. (ii) Step 1: D(L ) b D(L) and Lu = L u + h; rui, u 2 D(L ) b. Let u 2 D(L ) b. Then Tt u 2 D(L ) b D(L) and LTt u = L Tt u + h; rtt ui = T tl u + h; rtt ui. Since lim t! Tt u = u in H 1;2 (U; ) by 1.2 it follows that lim t! LTt u = L u+h; rui in L 1 (U; ). Hence u 2 D(L) and Lu = L u+h; rui by closedness of (L; D(L)). Step 2: Let u 2 D(L) b, u n 2 D(L ) b be as in Step 3 of (i) and kuk 1 < M 1 < M 2. Then (1.1) lim n!1 fm 1 ju n jm 2 g haru n ; ru n i d = : Indeed, let ' 2 C 1 (R) be such that ' (t) := (t? M 1 ) + ^ (M 2? M 1 ), t 2 R, and '() =. Then by (1.6) fm 1 u n M 2 g haru n ; ru n i d = =? =? haru n ; r' (u n )i d L u n ' (u n ) d? Lu n ' (u n ) d!? h; r'(u n )i d Lu' (u) d = ; if n! 1, since ' (u). Similarly, lim n!1 Rf?M 2 u n?m 1 g haru n; ru n i d =. Step 3: Let (u n ) n1 be as in Step 2 and 2 Cb 2 (R) be such that (t) = t if jtj kuk and (t) = if jtj kuk Then lim n!1 (u n ) = u {a.e. and in L p (U; ) for all p 2 [1; 1), (u n ) 2 D(L ) b D(L) and lim n!1 L (u n ) = lim n!1 (u n )Lu n + (u n )haru n ; ru n i = Lu in L 1 (U; ) by 1.2, Step 1 and (1.1). Consequently, E ( (u n )? (u m ); (u n )? (u m )) =? L( (u n )? (u m ))( (u n )? (u m )) d 2k k 1 kl (u n )? L (u m )k 1! ; n; m! 1 : Hence u 2 H 1;2 (U; ) and lim n!1 (u n ) = u in H 1;2 1;2 (U; ) since H (U; ) is complete. If v 2 H 1;2 (U; ) b then E (u; v)? h; ruiv d = lim E ( (u n ); v)? n!1 =? lim n!1 L (u n )v d =? Luv d : h; r (u n )iv d Hence (1.7) is proved. Clearly, (1.8) follows from (1.7) by taking v = u and using (1.6). This completes the proof of

10 Remark 1.3. Since? satises the same assumptions as the closure (L ; D(L )) of L u?h; rui, u 2 D(L ) b, on L 1 (U; ) generates a sub{markovian C {semigroup of contractions (T t) t, D(L ) b H 1;2 (U; ) and E (u; v) + h; ruiv d =? L u v d ; u 2 D(L ) b ; v 2 H 1;2 (U; ) b : If (L ; D(L )) is the part of (L ; D(L )) on L 2 (U; ) and (L; D(L)) is the part of (L; D(L)) on L 2 (U; ) then (1.11) (Lu; v) L 2 (U;) =?E (u; v) + =?E (v; u)? = (L v; u) L 2 (U;) h; ruiv d h; rviu d for all u 2 D(L) b, v 2 D(L ) b. Since (L; D(L)) (resp. (L ; D(L ))) is the generator of a sub{markovian C {semigroup it follows that D(L) b D(L) (resp. D(L ) b D(L )) dense w.r.t. the graph norm and hence (1.11) extends to all u 2 D(L), v 2 D(L ) which implies that L and L are adjoint operators on L 2 (U; ). Remark 1.4. It is a remarkable fact that the well{known correspondences between Dirichlet operators and sub{markovian semigroups of contractions on L 2 {spaces (cf. [MR, I.4.4]) do have analogues on L 1. More precisely, let (A; D(A)) be the generator of a C {semigroup of contractions (S t ) t on L 1 (X; m). Then it is easy to see that the following statements are equivalent: (i) R Au 1 fu>1g dm for all u 2 D(A). (ii) (S t ) t is sub{markovian. Similarly, the following statements are equivalent: (i') R Au 1 fu>g dm for all u 2 D(A). (ii') (S t ) t is positivity preserving (i.e., S t f if f ). Also note that for a linear operator (A; D(A)) on L 1 (X; m) we always have that (i) implies (i') and (i') implies that (A; D(A)) is dissipative. b) Existence under local assumptions. 1.1 can be generalized in various directions. First of all, (E ; H 1;2 (U; )) can be replaced by any other sectorial Dirichlet form (E; H 1;2 (U; )) in the sense of [MR] 2 such that the E 1 1 {norm is equivalent to the norm on H 1;2 (U; ). A generalization to the innite dimensional case will be considered in Section 4. Another possible generalization in the nite dimensional case is obtained by "localizing" the assumptions on, A and B. 1

11 More precisely, suppose from now on that the density ' 2 of is such that ' 2 H 1;2 loc (U). In particular, the measure can be innite. We replace assumption (1.1) by (1.1') a ij = a ji 2 H 1;2 loc (U; ) ; 1 i; j d : Instead of (1.2) we suppose that for all V relatively compact in U there exist V > such that (1.2')?1 V jhj2 ha(x)h; hi V jhj 2 for all h 2 R d ; x 2 V : By the results of [MR, Subsection II.2b] the bilinear form E (u; v) = haru; rvi d ; u; v 2 C 1 (U) ; is closable on L 2 (U; ), and we denote its closure by (E ; D(E )), the associated generator by (L ; D(L )) and the corresponding semigroup by (T t ) t. Similar to Subsection a) C 1 (U) D(L ) and L u = L A u + hb ; rui, u 2 C 1 (U). Finally we replace (1.3) by (1.3') B 2 L 2 loc(u; R d ; ) R i.e., V jbj2 d < 1 for all V relatively compact in U, and we suppose that (1.4) holds. Let B = (b 1; : : : ; b d ) be as in (1.5). Then similar to Subsection a) LA u + hb; rui = R L u + h; rui, u 2 C 1 (U), where := B? B 2 L 2 loc (U; Rd ; ) again satises h; rui d = for all u 2 C 1 (U). Therefore, Lu := L u + h; rui, u 2 D(L ) ;b, is an extension of L A u + hb; rui, u 2 C 1 (U). R For all open subsets V relatively compact in U the bilinear form haru; rvi d, u; v 2 H 1;2 (V; ), is closed. Let R (L;V ; D(L ;V )) be the corresponding generator. Since h; rui d = for all u 2 H 1;2 (V; ) the closure (LV ; D(L V )) of L V u := L ;V u + h; rui, u 2 D(L ;V ) b, on L 1 (V; ) generates a sub{markovian C {semigroup of contractions by 1.1. Let (G V ) > be the corresponding resolvent. If we dene G V f := G V (f1 V ) ; f 2 L 1 (U; ) ; > ; then G V, >, can be extended to a sub{markovian contraction on L 1 (U; ). Theorem 1.5. Let (1.1'){(1.3') and (1.4) be satised. Then there exists a (closed) extension (L; D(L)) of Lu := L u + h; rui, u 2 D(L ) ;b, on L 1 (U; ) satisfying the following properties: (a) (L; D(L)) generates a sub{markovian C {semigroup of contractions (T t ) t. (b) If (U n ) n1 is an S increasing sequence of open subsets relatively compact in U such that U = U n1 n then lim n!1 G U n f = (? L)?1 f in L 1 (U; ) for all f 2 L 1 (U; ) and >. 11

12 (c) D(L) b D(E ) and E (u; v)? h; ruiv d =? Lu v d ; u 2 D(L) b ; v 2 H 1;2 (U; ) ;b : Moreover, E (u; u)? Lu u d ; u 2 D(L) b : The proof of 1.5 is based on the following lemma. Lemma 1.6. Let V 1, V 2 be relatively compact in U and V 1 V 2. Let u 2 L 1 (U; ), u, and >. Then G V 1 u G V 2 u. Proof. Clearly, we may assume that u is bounded. Let w := G V 1 u? G V 2 u. Then w 2 H 1;2 (V 2; ) but also w + 2 H 1;2 (V 1; ) since w + G V 1 u and G V 1 u 2 H 1;2 (V 1; ). Note that R h; rw iw + d = R h; rw + iw + d = and E (w + ; w? ) since (E ; H 1;2 (V 2; )) is a Dirichlet form. Hence by (1.7) E (w + ; w + ) E (w ; w + )? = h; rw iw + d (? L V1 )G V 1 u w + d? (? L V2 )G V 2 u w + d = : Consequently, w + =, i.e., G V 1 u G V 2 u. Proof of 1.5. Let (V n ) n1 be an increasing sequence S of open subsets relatively compact in U such that V n Vn+1, n 1, and U = V n1 n. Let f 2 L 1 (U; ), f. Then lim n!1 G V n f =: G f exists {a.e. by 1.6. We will show below that (G ) > is a sub{markovian C {resolvent of contractions on L 1 (U; ). We will then show that the corresponding generator satises properties (a){(c) as stated in the Theorem. Since R G V n f d R f1 Vn d it follows that G f 2 L 1 (U; ), lim n!1 G V n f = G f in L 1 (U; ) and (1.12) G f d f d : RFor arbitrary f 2 L 1 (U; R R ) let G f := G f +? G f?. Then jg fj d G f + +G f? d jfj d by (1.12). Hence G is a contraction on L 1 (U; ). Clearly, G is sub{markovian. The family (G ) > satises the resolvent equation since (G V n ) > satises the resolvent equation for all n, lim n!1 kg V n G f? G V n G V n 1 fk 1 lim n!1 kg f? G V n fk 1 = if ; > and thus (? )G G f = lim (? n!1 )GV n G f = lim (? n!1 )GV n G V n f = lim n!1 GV n f? G V n f = G f? G f 12

13 for all ; >. Let f 2 L 1 (U; ) b. Then by (1.8) E (G V n f; G V n f) = f1 Vn G V n f d 1 kfk 1kfk 1 : Consequently, sup n1 E(G V n f; G V n f) < +1, hence G V n f 2 D(E ) by Banach{ Alaoglu, lim n!1 G V n f = G f weakly in D(E ) and (1.13) E(G f; G f) lim inf E n!1 (G V n f; G V n f) = fg f d : If v 2 H 1;2 (U; ) ;b then v 2 H 1;2 (V n; ) b for big n and hence by (1.7) (1.14) E (G f; v)? = h; rg fiv d = lim E n!1 (G V n f; v)? fv d : h; rg V n fiv d To see the strong continuity of (G ) > note that u = G V n (?L V n )u = G V n (?L)u for u 2 D(L ) ;b and for big n. Hence (1.15) u = G (? L)u : In particular, kg u? uk 1 = kg u? G (? L)uk 1 = kg Luk 1 1 kluk 1!,! 1, for all u 2 C 1 (U) and the strong continuity then follows by a 3{"{ argument. Let (L; D(L)) be the generator of (G ) >. Then (L; D(L)) extends (L; D(L ) ;b ) by (1.15). By the Hille{Yosida Theorem (L; D(L)) generates a C {semigroup of contractions (T t ) t. Since T t u = lim!1 exp(t(g? 1))u for all u 2 L 1 (U; ) (cf. [Pa, 1.3.5]) we obtain that (T t ) t is sub{markovian. We will show next that (L; D(L)) satises property (b). To this end let (U n ) n1 be S an increasing sequence of open subsets relatively compact in U such that U = U n1 n. Let f 2 L 1 (U; ), f. If n 1 then by compactness of V n there exist m such that V n U m and therefore G Vn f G Um f by 1.6. Hence Gf lim n!1 G Un f. Similarly, lim n!1 G Un f Gf hence (b) is satised. Finally we will prove that (L; D(L)) satises property (c). Let u 2 D(L) b. Then G u 2 D(E ) and by (1.13) E (G u; G u)? =? LG u G u d G Lu G u d kluk 1 kuk 1 : 13

14 Consequently, sup > E (G u; G u) < 1, hence u 2 D(E ), lim!1 G u = u weakly in D(E ) by Banach{Alaoglu and E (u; u) lim inf E (G u; G u) lim inf? G Lu G u d!1!1 =? Lu u d : If v 2 D(E ) ;b then by (1.14) E (u; v)? h; ruiv d = lim E (G u; v)? h; rg uiv d!1 = lim? G Lu v d =? Lu v d :!1 This completes the proof of 1.5. Remark 1.7. (i) Clearly, (L; D(L)) is uniquely determined by properties (a) and (b) in 1.5. (ii) Similar to (L; D(L)) we can construct a closed extension (L ; D(L )) of L u? h; rui, u 2 D(L ) ;b, that generates a sub{markovian C {semigroup of contractions (T t) t. Since for all V relatively compact in U by (1.7) and (1.6) (1.16) G V u v d = ug V; v d for all u; v 2 L 1 (U; ) b ; where (G V; ) > is the resolvent of the closure (L V; ; D(L V; )) of L ;V u? h; rui, u 2 D(L ;V ) b, it follows that (1.17) G u v d = u G v d for all u; v 2 L 1 (U; ) b ; where G = (? L )?1. (iii) Similar to the case of symmetric Dirichlet operators that admit a carre du champ (cf. [BH, I.4]) D(L) b is an algebra. Proof: Let u 2 D(L) b. Clearly it is enough to show that u 2 2 D(L) b. To this end it suces to prove that if g := 2uLu + 2hAru; rui then (1.18) L v u 2 d = gv d for all v = G 1h ; h 2 L 1 (U; ) b ; since then R G 1 (u 2?g)h d = R (u 2?g)G 1h d = R u 2 (G 1h?L G 1h) d = R u 2 h d for all h 2 L 1 (U; ) b. Consequently, u 2 = G 1 (u 2? g) 2 D(L) b. 14

15 For the proof of (1.18) x v = G 1h, h 2 L 1 (U; ) b, and suppose rst that u = G 1 f for some f 2 L 1 (U; ) b. Let u n := G V n 1 f and v n = G V n; 1 h, where (V n ) n1 is as in the proof of 1.5. Then by 1.1 and 1.5 L V n; vn uu n d =?E (v n ; uu n )? =?E (v n u n ; u)? = = + h; ruiv n u n d + Lu v n u n d +? h; rv n iuu n d harv n ; ru n iu d + harv n ; ru n iu d + Lu v n u n d + h; ru n iv n u d L Vn u n v n u d + L Vn u n v n u d + 2 haru n ; ruiv n d haru n ; r(v n u)i d haru n ; ruiv n d haru n ; ruiv n d : Note that lim n!1 R harun ; ruiv n d = R haru; ruiv d since lim n!1 u n = u weakly in D(E ) and lim n!1 haru; ruiv 2 n = haru; ruiv 2 (strongly) in L 1 (U; ). Hence L v u 2 d = lim n!1 = lim = n!1 gv d : L V n; vn uu n d Lu v n u n d + L Vn u n v n u d + 2 haru n ; ruiv n d Finally, if u 2 D(L) b arbitrary, let g := 2(G u)l(g u)+2harg u; rg ui, >. Note that by 1.5 (c) E (G u? u; G u? u)? L(G u? u)(g u? u) d 2kuk 1 kg Lu? Luk 1! if! 1 which implies that lim!1 G u = u in D(E ) and thus lim!1 g = g in L 1 (U; ). Since u + (1? )G u 2 L 1 (U; ) b and G 1 (u + (1? )G u) = G u by the resolvent equation it follows from what we have just proved that L v(g u) 2 d = for all > and thus, taking the limit! 1, L v u 2 d = g v d gv d and (1.18) is shown. 15

16 c) Invariance. Throughout this subsection let, A and B be as in Subsection b) but we suppose from now on that U = R d. Let (L; D(L)) be the closed extension of Lu = L u + h; rui, u 2 D(L ) ;b, satisfying properties (a){(c) in 1.5 and denote by (T t ) t the associated semigroup. We say that the measure is (T t )-invariant if (1.19) T t u d = u d for all u 2 L 1 (R d ; ) : Clearly, (1.19) is equivalent to the fact that R Lu d = holds for all u 2 D(L) (or more generally for all u 2 D where D D(L) dense w.r.t. the graph norm). Note that although it is true that R Lu d = for all u 2 D(L ) ;b the measure is not (T t ){invariant in general (cf below). Denition 1.8. Let p 2 [1; 1) and (A; D) be a densely dened operator on L p (X; m). We say that (A; D) is L p {unique, if there is only one extension of (A; D) on L p (X; m) that generates a C {semigroup. It follows from [Na, Theorem A{II, 1.33] that if (A; D) is L p {unique and (A; D) the unique extension of (A; D) generating a C {semigroup it follows that D D dense w.r.t. the graph norm. Equivalently, (A; D) is L p {unique if and only if (?A)(D) L p (X; m) dense for some >. Proposition 1.9. The following statements are equivalent: (i) There exist n 2 H 1;2 loc (Rd ; ) and > such that ( n? 1)? 2 H 1;2 (Rd ; ) ;b, lim n!1 n = {a.e. and (1.2) E ( n ; v) + h; r n iv d for all v 2 H 1;2 (Rd ; ) ;b ; v : (ii) (L; D(L ) ;b ) is L 1 {unique. (iii) is (T t ){invariant. Proof. (i) ) (ii): It is sucient to show that if h 2 L 1 (R d ; ) is such that R (? L)u h d = for all u 2 D(L ) ;b it follows that h =. To this end let 2 C 1 (R d ). If u 2 D(L ) b it is easy to see that u 2 D(L ) ;b and L (u) = L u + 2hAr; rui + ul. Hence (1.21) (? L )u(h) d = = (? L )(u)h d ul h d h; r(u)ih d + 2 haru; rih d haru; rih d + ul h d : 16

17 Since jj 2 L 2 loc (Rd ; ) we obtain that u 7! R (? L )u(h) d, u 2 D(L ) b, is continuous w.r.t. the norm on D(E ). Hence there exists some element v 2 D(E ) such that E (u; v) = R (?L )u(h) d and consequently, R (?L )u(v?h) d = for all u 2 D(L ) b. Hence v = h, in particular h 2 D(E ) and (1.21) implies that (1.22) E (u; h) = h; r(u)ih d L uh d haru; rih d for all u 2 D(L ) b and subsequently for all u 2 D(E ). From (1.22) it follows that E (u; h)? h; ruih d = for all u 2 H 1;2 (Rd ; ) : Let v n := khk 1 n? h. Then v n? 2 H 1;2 (Rd ; ) ;b and E (v n ; v? n )? h; rv? n iv n d? (v? n ) 2 d ; since R h; rv? n iv n d = R h; rv? n iv? n d =. Thus v? n =, i.e., h khk 1 n. Similarly,?h khk 1 n, hence jhj khk 1 n. Since lim n!1 n = {a.e. it follows that h =. R R (ii) ) (iii): Since Lu d = for all u 2 D(L ) ;b we obtain that Lu d = for all u 2 D(L) and thus t T t u d = u d + LT s u d ds = u d for all u 2 D(L). Since D(L) L 1 (R d ; ) dense we obtain that is (T t ){invariant. (iii) ) (i): Let V n := B n (), n 1. By 1.1 the closure of L ;V n u? h; rui, u 2 D(L ;V n ) b, on L 1 (V n ; ) generates a sub{markovian C {semigroup. Let (G V n; ) > be the corresponding resolvent and n := 1? G V n; 1 (1 Vn ), n 1. Clearly, n 2 H 1;2 loc (Rd ; ) and ( n? 1)? 2 H 1;2 (Rd ; ) ;b. Fix n 1 and let w := G +1G V n; 1 (1 Vn ), >. Since w G V n; +1G V n; 1 (1 Vn ) and G V n; +1G V n; 1 (1 Vn ) = G V n; 1 (1 Vn )? G V n; +1(1 Vn ) G V n; 1 (1 Vn )? 1=( + 1) by the resolvent equation it follows that (1.23) w G V n; 1 (1 Vn )? 1=( + 1) ; > : Note that by 1.5 E 1 (w ; w ) (G V n; 1 (1 Vn )? w ; w ) L 2 (R d ;) (G V n; 1 (1 Vn )? w ; G V n; 1 (1 Vn )) L 2 (R d ;) = E 1 (w ; G V n; 1 (1 Vn )) + h; rw ig V n; 1 (1 Vn ) d E 1 (w ; w ) 1 2 (E 1 (G V n; 1 (1 Vn ); G V n; 1 (1 Vn )) p Vn kjj1 Vn k 2 ) : 17

18 Consequently, lim!1 w = G V n; 1 (1 Vn ) weakly in D(E ) and now (1.23) implies for u 2 H 1;2 (Rd ; ) ;b, u, E 1 ( n ; u) + = lim!1 h; r n iu d = lim u d?!1 u d? E 1 (w ; u)? (G V n; 1 (1 Vn )? w )u d : h; rw iu d Finally note that ( n ) n1 is decreasing by 1.6 and therefore 1 := lim n!1 n exists {a.e. If g 2 L 1 (R d ; ) b then by (1.16) g 1 d = lim n!1 = lim = n!1 g d? g n d = lim g d? n!1 G 1 g d : g d? G V n 1 g 1 Vn d gg V n; 1 (1 Vn ) d Since is (T t ){invariant it follows that R g 1 d = for all g 2 L 1 (R d ; ) b and thus 1 = which implies (i). Remark. The proof of (iii) ) (i) in 1.9 shows that if is (T t ){invariant then there exists for all > a sequence ( n ) n1 H 1;2 loc (Rd ; ) such that ( n? 1)? 2 H 1;2 (Rd ; ) ;b, lim n!1 n = {a.e. and E ( n ; v) + h; r n iv d for all v 2 H 1;2 (Rd ; ) ;b ; v : Indeed, it suces to take n := 1? G V n; (1 Vn ), n 1. Finally let us give some sucient conditions on, A and B that imply (T t ){ invariance of. Clearly, (T t ){invariance of is equivalent to the conservativeness of the dual semigroup (T t) t of (T t ) t acting on L 1 (R d ; ). Recall that (T t) t is called conservative, if T t1 = 1 for some (hence all) t >. Since in the symmetric case (i.e., B = B ) T tjl 1 (R d coincides with T ;)b tjl 1 (R d we obtain that both ;)b notions coincide in this particular case. But conservativeness in the symmetric case has been well{studied by many authors. We refer to [D], [FOT, Section 1.6], [S] and references therein. Proposition 1.1. Each of the following conditions (a), (b) and (c) imply that there exist n 2 H 1;2 loc (Rd ; ) and > as in 1.9 (i). (a) a ij ; b i? b i 2 L 1 (R d ; ), 1 i; j d. (b) There exist u 2 C 2 (R d ) and > such that lim jxj!1 u(x) = +1 and L A u + h2b? B; rui u. 18

19 (c)?2ha(x)x; xi=(jxj 2 +1)+trace(A(x))+h(2B?B)(x); xi M(jxj 2 ln(jxj 2 +1)+1) for some M. Proof. (a) By 1.9 it is sucient R to show that (L; D(L ) ;b ) is L 1 {unique. But if h 2 L 1 (R d ; ) is such that (1? L)u h d = for all u 2 D(L ) ;b we have seen in the proof of the implication (i) ) (ii) in 1.9 that h 2 H 1;2 loc (Rd ; ) and (1.24) E 1 (u; h)? h; ruih d = for all u 2 H 1;2 (Rd ; ) : Let n 2 C 1 (R d ) be such that 1 Bn () n 1 B2n () and kjr n jk 1 c=n. Then (1.24) implies that 2 nh 2 d + E ( n h; n h) = E 1 ( nh; 2 h) + har n ; r n ih 2 d? h; r nhih 2 d + h; r n i n h 2 d dx r c2 c dx n 2 khk2 1( ja ij j d) + n khk2 1( i;j=1 and thus R h 2 d = lim n!1 R 2 n h 2 d =. i=1 jb i? b i j d) (b) Let n := u n. Then n 2 H 1;2 loc (Rd ; ), ( n? 1)? is bounded and has compact support, lim n!1 n = and E ( n ; v) + h; r n iv d for all v 2 H 1;2 (Rd ; ) ; v : Finally (c) implies (b) since we can take u(x) = ln(jxj 2 + 1) + r for r suciently big. Remark Suppose that there exist a bounded, nonnegative and nonzero function u 2 C 2 (R d ) and > such that L A u + h2b? B; rui u. Then is not (T t ){invariant. Proof: We may suppose that u 1. If would be (T t ){invariant it would follow that there exist n 2 H 1;2 loc (Rd ; ), n R 1, such that ( n? 1)? 2 H 1;2 (Rd ; ) ;b, lim n!1 n = {a.e. and E( n ; v)+ h; r n iv d for all v 2 H 1;2 (Rd ; ) ;b, v (cf. the Remark following 1.9). Let v n := ( n? u). Then v n? 2 H 1;2 (Rd ; ) ;b and E(v n ; v n? )? h; rv n? iv n d? (v n? ) 2 d ; R R since h; rv n? iv n d = h; rv n? iv n? d =. Thus v n? =, i.e., u n. Since lim n!1 n = {a.e. and u it follows that u = which is a contradiction to our assumption u 6=. 19

20 Example Let := e?x2 dx, B(x) =?2x? 6e x2, Lu := u + B u, u 2 C 1 (R), (L; D(L)) be the maximal extension having R properties (a){(c) in 1.5 and x (T t ) t be the associated semigroup. Let h(x) :=?1 e?t2 dt, x 2 R. Then h + (2B? B)h h. It follows from 1.11 that is not (T t ){invariant. 2. Uniqueness in the finite dimensional case Let U = R d. Throughout this section we assume the local assumptions (1.1'){ (1.3') of Subsection 1b) and (1.4) on, A and B. Let (L ; D(L )) be the generator corresponding to the closure (E ; D(E )) of haru; rvi d ; u; v 2 C 1 (R d ) ; (Tt ) t be the associated semigroup, B = (b 1; : : : ; b d ) be as in (1.5) and = B?B so that Lu := L u+h; rui, u 2 D(L ) ;b, is an extension of L A u+hb; rui, u 2 C 1 (R d ). In 1.5 we constructed a closed extension (L; D(L)) of (L; C 1 (R d )) generating a sub{markovian C {semigroup (T t ) t on L 1 (R d ; ). In this section we will study the question whether (L; D(L)) is the only extension generating a C { semigroup on L 1 (R d ; ), i.e., whether (L; C 1 (R d )) is L 1 {unique. We will consider the symmetric and the nonsymmetric case separately (cf. 2.1 and 2.4 below). Consider the following additional assumption on A: Suppose that for all compact V there exist L V and V 2 (; 1) such that (2.1) ja ij (x)? a ij (y)j L V jx? yj V for all x; y 2 V : Theorem 2.1. Let (a ij ) 1i;jd satisfy (2.1). Then (L ; C 1 and only if (E ; D(E )) is conservative. (R d )) is L 1 {unique if Remark 2.2. Note that 2.1 implies in particular that the generalized Schrodinger operator S ' u := u + 2'?1 hr'; rui, u 2 C 1 (R d ), is L 1 {unique if and only if the Friedrich's extension (or equivalently, the associated diusion process) is conservative (which is in particular the case if the measure is nite). Hence (S ' ; C 1 (R d )) is Markov{unique in the sense that there is exactly one self{adjoint extension on L 2 (R d ; ) which generates a sub{markovian semigroup. On the other hand, it has been shown by M. Rockner and T.S. hang in [R] that (S ' ; C 1 (R d )) is Markov{unique (in the sense described above) for all ' 2 H 1;2 loc (Rd ), ' 6= dx{a.e. For the proof of 2.1 we rst state a lemma. R Lemma 2.3. Let (a ij ) 1i;jd satisfy (2.1). Let h 2 L 1 (R d ; ) be such that (1? L )u h d = for all u 2 C 1 (R d ). Then h 2 H 1;2 loc (Rd ; ) and E1 (u; h) = for all u 2 H 1;2 (Rd ; ). R Proof. First note that C(R 2 d ) D(L ) ;b and that (1? L )u h d = for all u 2 C(R 2 d ). Let 2 C 1 (R d ) and r > be such that supp() B r (). We 2

21 have to show that h 2 H 1;2 (Rd ; ). Let L and 2 (; 1) be such that ja ij (x)? a ij (y)j Ljx? yj for all x; y 2 B r () and dene a ij (x) := a ij (( r jxj ^ 1)x) ; x 2 Rd : Then a ij (x) := a ij (x) P for all x 2 B r () and ja ij (x)? a ij (y)j 2Ljx? yj for all x; y 2 R d. Let L A d = a i;j=1 ij@ j. By [K, and 4.3.2] there exists for all f 2 C 1 (R d ) and > a unique function R f 2 Cb 2(Rd ) satisfying R f?l A R f = f and kr fk 1 kfk 1. Moreover, R f if f by [K, 2.9.2]. Since C 1 (R d ) C 1 (R d ) dense (where C 1 (R d ) is the space of all bounded continuous functions vanishing at innity) we obtain that f 7! R f, f 2 C 1 (R d ), can be uniquely extended to a positive linear map R : C 1 (R d )! C b (R d ), which is a contraction on C b (R d ), hence a sub{markovian kernel on (R d ; B(R d )), which will be denoted again by R. Let f n 2 C 1 (R d ), n 1, such that kf n k 1 khk 1 and h ~ := lim n!1 f n is a {version of h. Then lim n!1 R f n (x) = R h(x) ~ for all x 2 R d by Lebesgue's Theorem and kr hk1 ~ khk 1. Then E (R f n ; R f n ) =? (2.2) =? =?? L (R f n )R f n d L A (R f n ) 2 d? 2 + 2? L A (R f n ) 2 R f n d? L A (R f n ) 2 d? 2 har; ri(r f n ) 2 d? hb ; r(r f n )i R f n d : har; rr f n ir f n d hb ; r(r f n )ir f n d har; r(r f n )ir f n d (R f n? f n ) 2 R f n d Hence E (R f n ; R f n ) c E (R f n ; R f n ) 1=2 + M for some positive constants c and M independent of n. Consequently, sup n1 E (R f n ; R f n ) < +1, hence R ~ h 2 D(E ) and lim n!1 R f n = R ~ h weakly in D(E ). 21

22 Note that (2.3)? (R h ~? h)r ~ h ~ 2 d? = lim? n!1 = lim? n!1 = lim? n!1 (R ~ h? ~ h) ~ h 2 d (R f n? f n ) ~ h 2 d L A (R f n ) ~ h 2 d L A ( 2 R f n ) ~ h d + 4 har; rr f n i ~ h d + L A ( 2 )R f n h ~ d = lim? 2 R f n h ~ d + hb ; r( 2 R f n )i h ~ d n!1 + 4 har; rr f n i h ~ d + L A ( 2 )R f n h ~ d =? 2 (R h) ~ ~ h d + hb ; r(r h)i ~ h ~ d + hb ; ri(r h) ~ h ~ d + 4 har; r(r h)i ~ h ~ d? 4 har; ri(r h) ~ ~ h d + L A ( 2 )(R h) ~ ~ h d c E (R ~ h; R ~ h) 1=2 + M for some positive constants c and M independent of. Combining (2.2) and (2.3) we obtain that E (R h; ~ R h) ~ lim inf E (R fn ~ ; R fn ~ ) n!1? + 2? L A (R ~ h) 2 d? 2 har; ri(r ~ h) 2 d? hb ; r(r ~ h)ir ~ h d ~c E (R ~ h; R ~ h) 1=2 + ~ M har; r(r ~ h)ir ~ h d (R ~ h? ~ h) 2 R ~ h d for some positive constants ~c and ~ M independent of. Hence (R ~ h)> is bounded in D(E ). If u 2 D(E ) is the limit of some weakly convergent subsequence ( k R k ~ h)k1 22

23 with lim k!1 k = +1 it follows for all v 2 C 1 (R d ) that (u? ~ h)v d = lim = lim lim k!1 n!1 = lim lim k!1 n!1 = lim lim k!1 n!1 = lim k!1 + k!1 ( k R k ~ h? ~ h)v d ( k R k f n? f n )v d L A (R k f n )v d R k f n L (v) d? R k ~ h L (v) d? hb ; rir k ~ h v d hb ; rr k f n iv d hb ; r(r k ~ h)iv d 1 lim (khk 1 kl (v)k 1 + p kjb jvk 2 E ( k R k h; ~ k R k h) ~ 1=2 k!1 k + p khk 1 kjb jvk 2 E (; ) 1=2 ) = : Consequently, ~ h is a {version of u. In particular, h 2 H 1;2 (Rd ; ). Let u 2 H 1;2 (Rd ; ) with compact support, 2 C 1 (R d ) such that 1 on supp(juj) and u n 2 C 1 (R d ), n 1, such that lim n!1 u n = u in H 1;2 (Rd ; ). Then haru; rhi d + uh d = lim har(u n ); rhi d + u n h d = lim n!1 n!1 (1? L )u n h d = : Proof of 2.1. Clearly, if (L ; C 1 (R d )) is L 1 {unique it follows that (L ; D(L ) ;b ) is L 1 {unique too. Hence is (T t ){invariant by 1.9. Since (T t ) t is symmetric it follows that T t 1 = 1, i.e., (E ; D(E )) is conservative. R Conversely, let h 2 L 1 (R d ; ) be such that (1?L )u h d = for all u 2 C 1 (R d ). Then h 2 H 1;2 loc (Rd ; ) and E 1 (u; h) = for all u 2 H 1;2 (Rd ; ) by 2.3. In particular, (2.4) (1? L )u h d = E 1 (u; h) = for all u 2 D(L ) : Since (T t ) t is symmetric and T t 1 = 1 it follows that is (T t ){invariant. Hence (L ; D(L ) ;b ) is L 1 {unique by 1.9 and (2.4) now implies that h =. Hence (L ; C 1 (R d )) is L 1 {unique too. Our next result is on uniqueness in the nonsymmetric case. 23

24 Proposition 2.4. Let (L ; C 1 (R d )) be L 1 {unique. Suppose that there exist n 2 H 1;2 loc (Rd ; ), n 1, and > as in 1.9 (i). Then (L; C 1 (R d )) is L 1 {unique too. Proof. (L; D(L ) ;b ) is L 1 {unique by 1.9. Hence it is enough to prove that for u 2 D(L ) ;b there exist u n 2 C 1 (R d ) such that lim n!1 ku n? uk 1 + klu n? Luk 1 =. To this end let u 2 D(L ) ;b, u 6=. Since (L ; C 1 (R d )) is L 1 {unique there exist u n 2 C 1 (R d ), n 1, such that lim n!1 ku n? uk 1 + kl u n? L uk 1 = and lim n!1 u n = u {a.e. Similar to the proof of (1.1) (cf. Step 2 in the proof of 1.1) it can be shown that (2.5) lim n!1 for all kuk 1 < M 1 < M 2. fm 1 ju n jm 2 g haru n ; ru n i d = Let 2 C 1 b (R) be such that (t) = t if jtj kuk 1+1 and (t) = if jtj kuk Then (u n ) 2 C 1 (R d ), n 1, and lim n!1 L (u n ) = lim n!1 (u n )L u n + (u n )haru n ; ru n i = L u in L 1 (R d ; ) by (2.5). Consequently, E ( (u n )? u; (u n )? u) =? i.e., lim n!1 (u n ) = u in D(E ). L ( (u n )? u)( (u n )? u) d 2k k 1 kl (u n )? L uk 1! ; n! 1 ; Finally let 2 C 1 (R d ) such that 1 on supp(juj). Then (u n ) 2 C 1 (R d ) for all n and it is easy to see that lim n!1 k (u n )? uk 1 + kl( (u n ))? Luk 1 = which implies the assertion. The uniqueness results can be applied to derive results on the uniqueness of related martingale problems. According to [AR] we make the following denition: Denition 2.5. A right process M = (; F; (X t ) t ; (P x ) x2r d ) with state space R d and natural ltration (F t ) t is said to solve the martingale problem for (L; C 1 (R d )) if for all u 2 C 1 (R d ) (i) R t Lu(X s) ds, t, is (P {a.s.) independent of the {version for Lu. (ii) u(x t )? u(x )? R t Lu(X s) ds, t, is an (F t ){martingale under P = R P x d(x). Proposition 2.6. Assume that (L ; C 1 (R d )) is L 1 {unique and that there exist n 2 H 1;2 loc (Rd ; ) and > as in 1.9 (i). Let M = (; F; (X t ) t ; (P x ) x2r d ) be a right process that solves the martingale problem for (L; C 1 (R d )) such that is an subinvariant measure for M. Then E x [f(x t )] is a {version of T t f for all f 2 B b (R d ) \ L 1 (R d ; ) and is an invariant measure for M. 24

25 Proof. Let (p t ) t be the transition semigroup of M. Since is subinvariant for M, i.e., R p t f d R f d for all f 2 B b (R d ) \ L 1 (R d ; ), f, it follows that (p t ) t induces a semigroup of (sub{markovian) contractions (S t ) t on L 1 (R d ; ). Using [MR, II.4.3] and the fact that M is a right process it is easy to see that (S t ) t is strongly continuous. Let (A; D(A)) be the corresponding generator. If u 2 C 1 (R d ) and v 2 B b (R d ) then (p t u? u)v d = E [(u(x t )? u(x ))v(x )] = E ( = ( t t Lu(X s ) ds)v(x ) p s Lu ds)v d ; R t hence S t u? u = S slu ds in L 1 (R d ; ). It follows from the strong continuity of (S t ) t that u 2 D(A) and Au = Lu. Since (L; C 1 (R d )) is L 1 {unique by 2.4 we obtain that A = L, hence (S t ) t = (T t ) t which implies the rst assertion. The second assertion follows from 1.9. Remark 2.7. In Section 3 we will construct a diusion process M associated with (L; D(L)) in the sense that its transition probabilities are given by (T t ) t. It is easy to see that M is a solution of the martingale problem for (L; C 1 (R d )) in the sense of 2.5. Application to uniqueness of invariant measures. We want to demonstrate how the results of Section 1 and 2 can be applied to obtain results on uniqueness of the invariant measure. For simplicity suppose that a ij = ij ; 1 i; j d. Proposition 2.8. Let d 2 and B 2 L p loc (Rd ; R d ; dx) for some p > d. Suppose that there exists M such that (2.6) hb(x); xi M(ln(jxj 2 + 1) + 1) for all x 2 R d : Then there exists at most one probability measure satisfying (2.7) B 2 L 1 loc(r d ; R d ; ) and u + hb; rui d = for all u 2 C 1 (R d ) : Proof. Let 1, 2 be two probability measures satisfying (2.7) and let := Clearly, satsies (2.7) again. By [ABR, Theorem 2.5] d dx and for the density we have that 2 H 1;p loc (Rd ). Moreover, admits a strictly positive continuous modication and thus ' := p 2 H 1;2 loc (Rd ). Let B = (b 1; : : : ; b d ), b i = i ', 1 i d. 25

26 By 1.5 there exist closed extensions (L; D(L)) of Lu := u+hb; rui, u 2 C 1 (R d ), and (L ; D(L )) of L u := u + h2b? B ; rui, u 2 C 1 (R d ), on L 1 (R d ; ) generating sub{markovian C {semigroups (T t ) t and (T t) t. It follows from (1.17) (cf. 1.7) that R T t u v d = R ut tu d for all u; v 2 L 1 (R d ; ). Note that (2.6) implies the existence of some function u 2 C 2 (R d ), u, and some > such that lim jxj!+1 u(x) = +1 and Lu u. Hence is (T t){invariant by 1.1. Consequently, R T 1? 1 d =, i.e., T t1 = 1, and thus R T t u d = R ut t1 d = R u d for all u 2 L 1 (R d ; ). Thus is (T t ){invariant too. R Let (L ; D(L )) be the generator of the closure (E ; D(E )) of hru; rvi d, u; v 2 C 1 (R d ), on L 2 (R d ; ). (L ; C 1 (R d )) is L 1 {unique by 2.1 since is nite. Hence (L; C 1 (R d )) is L 1 {unique too by 1.9 and 2.4. Note that d 1 d, h := d 1 d 2 L 1 (R d ; ) and (2.8) Lu h d = Lu d 1 = for all u 2 C 1 (R d )) : Then (2.8) extends to all u 2 D(L) which implies that h 2 D(L ) b and L h =. By 1.5 h 2 D(E ) and E (h; h)? R L h h d =. Since the density d dx that D(E ) H 1;2 admits a strictly positive continuous modication it follows loc (Rd ) and E (h; h) = implies i h =, 1 i d. Consequently, h c for some constant c 2 R. Since 1 = R d 1 = R h d = c R d = c it follows that 1 = and nally 2 = 2? 1 = Associated Markov processes in the finite dimensional case Let U R d be open. Throughout this section we assume the local assumptions (1.1'){(1.3') of Subsection 1b) and (1.4) on, A and B. Let (L ; D(L )) be the generator corresponding to the closure (E ; D(E )) of haru; rvi d ; u; v 2 C 1 (U) ; (Tt ) t be the associated semigroup, B = (b 1; : : : ; b d ) be as in (1.5) and = B?B so that Lu := L u+h; rui, u 2 D(L ) ;b, is an extension of L A u+hb; rui, u 2 C 1 (U). Let (L; D(L)) be the closed extension of (L; D(L ) ;b ) satisfying (a){ (c) in 1.5, (T t ) t be the associated semigroup on L 1 (U; ) and (G ) > be the associated resolvent. Since (T t ) t is a sub{markovian semigroup of contractions it determines uniquely a semigroup of contractions on L 2 (U; ) by the Riesz{Thorin Interpolation Theorem. Clearly, (T t ) t is strongly continuous again. Let (L; D(L)) be the generator and (G ) > be the associated resolvent. Note that T t (resp. G ) coincides with T t (resp. G ) on L 1 (U; ) \ L 2 (U; ). Lemma 3.1. Let f 2 D(L). Then f 2 D(E ) and E (f; f)? R Lff d. Proof. Let g n 2 L 1 (U; ) b be such that lim n!1 kg n?(1?l)fk 2 =. Then G 1 g n 2 D(L) b D(E ) and E 1 (G 1 g n?g 1 g m ; G 1 g n?g 1 g m ) R (g n?g m )(G 1 g n?g 1 g m ) d. 26

27 Since lim n!1 kg 1 g n?fk 2 = it follows that (G 1 g n ) n1 is an E {Cauchy{sequence, hence f 2 D(E ) and E (f; f)? R Lff d. It is well{known that the general theory of Dirichlet forms can be used to construct a diusion M = ( ; F ; (Xt ) t ; (Px ) x2u ) with life time that is associated with (E ; H 1;2 (U; )) in the sense that E f(x t ) is an E {quasi continuous (= E {q.c.) {version of T t f for all f 2 B b (U) \ L 2 (U; ), t > (cf. [MR] or [FOT]). E {quasi continuity of p t f means that there exists an increasing sequence (F k ) k1 of closed subsets of U such that S k1 D(E ) Fk D(E ) dense (where we set D(E ) Fk = fv 2 D(E )jv = on F c kg ) and p t f jfk is continuous for all k. Such an increasing sequence of closed subsets (F k ) k1 is called an E {nest and it is well{known in the theory of Dirichlet forms that (F k ) k1 is an E {nest if and only if P limk!1 UnFk < + =. Here UnFk = infft > jxt 2 U n F k g denotes the rst hitting time. The construction of M is possible because the domain of E contains enough continuous functions with compact support in U, since C 1 (U) D(E ) dense. This property implies quasi{regularity of (E ; D(E )) and thus the existence of M (cf. [MR, IV.3.5 and V.1.5]). Since L is neither symmetric nor sectorial we cannot use the same framework to construct a diusion process associated with L. However, a closer look to the construction of associated stochastic processes in the framework of Dirichlet forms shows that the assumption on the symmetry of (L ; D(L )) can be removed and only information on the domain D(L ) and the resolvent (? L )?1, >, is used to construct M. This observation is used in the theory of generalized Dirichlet forms to construct stochastic processes associated with non{symmetric operators (cf. [St1]). First note that (L; D(L)) is associated with a generalized Dirichlet form by [St1, I.4.9 (ii)]. The explicit construction of L (hence L too) in 1.5 provides enough information on the domain D(L) and the resolvent (G ) > to apply the fundamental existence result in the theory of generalized Dirichlet forms (cf. [St1, IV.2.2]) to obtain a {tight special standard process M = (; F; (X t ) t ; (P x ) x2u ) with life time that is associated with (L; D(L)) in the sense that E R e?t f(x t ) dt is an E {q.c. {version of G f for all f 2 B b (U) \ L 2 (U; ), > (cf. 3.5 below) and we will show in 3.6 below that M is a diusion in the sense that P x [t! X t is continuous on [; )] = 1 E {q.e. Analytic potential theory related to L. In order to apply [St1, IV.2.2] we have to prove quasi{regularity of L, which is dened in the framework of generalized Dirichlet forms in a similar way as quasi{ regularity in the framework of (sectorial) Dirichlet forms (for details we refer to [St1]). To this end we rst introduce some notions that are well{known in the classical framework. 27