PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 132, Number 9, Pages 2743–2751 S 0002-9939(04)07283-1 Article electronically published on April 21, 2004
CONSERVATIVENESS OF DIFFUSION PROCESSES WITH DRIFT
KAZUHIRO KUWAE
(Communicated by Richard C. Bradley)
Abstract. We show the conservativeness of the Girsanov transformedp dif- fusion process by drift b ∈ Lp(Rd → Rd)withp ≥ 4/(2 − 2δ(|b|2)/λ)or p>4d/(d+2), or p =2if|b|2 is of the Hardy class with sufficiently small coef- ficient of energy δ(|b|2) <λ/2. Here λ>0 is the lower bound of the symmetric measurable matrix-valued function a(x):=(ai,j (x))i,j appearing in the given Dirichlet form. In particular, our result improves the conservativeness of the transformed process by b ∈ Ld(Rd →Rd)whend ≥ 3.
1. Statement of result In this note, we prove the conservativeness of diffusion processes on Rd by a Rd 3 7→ d Girsanov transformation. Let x a(x):=(ai,j (x))i,j=1 be a symmetric (Rd ⊗Rd)-valued measurable function and Rd 3 x 7→ b(x)anRd-valued measurable function with expression b(x)=(b1(x),b2(x), ··· ,bd(x)). We assume the uniform ellipticity of a on Rd: there exist constants Λ ≥ λ>0 such that
2 2 d d λ|ξ| ≤ha(x)ξ,ξiRd ≤ Λ|ξ| for all ξ ∈ R ,x∈ R . h· ·i | | h i1/2 Here , Rd stands for the Euclidean inner product with ξ := ξ,ξ Rd .Weconsider the following quadratic form (Ea,H1(Rd)): Z a 1 1 d E (u, v):= ha(x)∇u(x), ∇v(x)iRd dx for u, v ∈ H (R ), 2 Rd where H1(Rd):={u ∈ L2(Rd) | the distributional derivatives ∂u/∂xi,i=1, 2, ··· , d are in L2(Rd)}.Then(Ea,H1(Rd)) is a strongly local regular Dirichlet form on L2(Rd)(see[8]). Definition 1.1 (Hardy class function). A measurable function f on Rd is said to a 1 d be of the Hardy class (write f ∈ Hd) with respect to (E ,H (R )) if there exist
Received by the editors June 18, 2002 and, in revised form, December 20, 2002. 2000 Mathematics Subject Classification. Primary 60J45; Secondary 31C25. Key words and phrases. Semi-Dirichlet form, Dirichlet form, diffusion process, Kato class func- tion, Hardy class function, Sobolev inequality, Novikov’s condition, supermartingale, exponential martingale, conservativeness, Girsanov transformation. The author was partially supported by a Grant-in-Aid for Scientific Research (C) No. 13640220 from the Japanese Ministry of Education, Culture, Sports, Science and Technology.
c 2004 American Mathematical Society 2743
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constants δ(|f|) ∈ (0, ∞)andγ(|f|) ∈ [0, ∞) such that Z 2| | ≤ | | Ea | | k k2 ∈ 1 Rd (1.1) u f dm δ( f ) (u, u)+γ( f ) u L2 for u H ( ), Rd where m(dx):=dx is the d-dimensional Lebesgue measure on Rd. ⊂ 1 Rd ∈ p/2 Rd ≥ ≥ Clearly, Hd Lloc( ). If f L ( )withp d 3, in particular, if d/2 d f ∈ L (R )withd ≥ 3, then f ∈ Hd and the constant δ(|f|)canbetakentobe arbitrarily small (cf. [22],[24]). Definition 1.2 (Measures of finite energy integrals, [8]). A positive Radon measure µ on Rd is said to be of finite energy integral with respect to (Ea,H1(Rd)) (write µ ∈ S0) if and only if there exists a positive constant C depending on µ such that Z q | | ≤ Ea ∈ ∞ Rd u dµ C 1 (u, u)foru C0 ( ). Rd R a ∈ 1 Rd Ea a Then for each α>0thereexistsUαµ H ( ) such that α(Uαµ, v)= Rd vdµ ∈ ∞ Rd a ∈ for v C0 ( ). Uαµ is called the α-potential of µ S0. Furthermore, we set { ∈ | Rd ∞ a ∈ ∞ Rd } S00 := µ S0 µ( ) < ,U1 µ L ( ) . Let Capa be the 1-capacity associated with (Ea,H1(Rd)): Capa(G):= {Ea | ∈ 1 Rd ≥ } a inf 1(u, u) u H ( ),u 1a.e.onG for any open set G,andCap(A):= inf{Capa(G) | G is open with A ⊂ G} for any subset A of Rd. A subset N of Rd is said to be exceptional if Capa(N)=0. Definition 1.3 (Smooth measures, [8]). A Borel measure µ on Rd is said to be smooth with respect to (Ea,H1(Rd)) (write µ ∈ S)ifµ charges no exceptional set and there exists a sequence {Fn} of closed sets such that µ(Fn) < ∞ for each n ∈ N and a (1.2) lim Cap (K \ Fn)=0 n→∞ for any compact set K of Rd.
It is well known that µ ∈ S if and only if there exists a sequence {Fn} of closed ∈ ∈ ∈ N sets such that IFn µ S00 (or IFn µ S0)foreachn satisfying (1.2) for any ∈ 1 Rd | | ∈ compact K.Forf Lloc( ), we see f m S. Throughout this paper, we assume the following Assumptions 1.1 and 1.2.
2 a 1 d Assumption 1.1. We assume that |b| ∈ Hd with respect to (E ,H (R )) and δ(|b|2) <λ/2. Assumption 1.2. We assume that |b|∈Lp(Rd), p ≥ 2, and when 2
License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use CONSERVATIVENESS 2745 R Eb Ea − h ∇ i ∈ ∞ Rd Set (u, v):= (u, v) Rd b(x), u(x) Rd v(x)dx for u, v C0 ( ). Then, ∈ ∞ Rd under Assumptionp 1.1, there exists M>0 such that for u, v C0 ( ), α> 2 2 α2(b):=γ(|b| ) 2/λδ(|b| ),
b |E (u, v)|≤Mkuk 1 Rd kvk 1 Rd , pH ( ) H ( ) Eb ≥ − | |2 Ea − k k2 ≥ α(u, u) (1 2δ( b )/λ) (u)+(α α2(b)) u L2 0. Eb ∞ Rd 2 Rd Hence for α>α2(b), ( α,C0 ( )) is closable on L ( ) and we denote its closure Eb 1 Rd Eb 1 Rd 2 Rd by ( α,H ( )). ( α,H ( )) is a local regular semi-Dirichlet form on L ( ) (cf. [13],[18]). { a} 2 Rd Ea 1 Rd a Let Tt t>0 be the L ( )-semigroups associated with ( ,H ( )). Then Tt a ∞ ×Rd ×Rd admits a symmetricR jointly continuous heat kernel pt (x, y)on(0, ) such a a a ∈ 2 Rd a that Pt f(x):= Rd pt (x, y)f(x)dy is a version of Tt f for f L ( )andpt (x, y) satisfies the Aronson’s estimates (see [26]): there exists an M := M(λ, Λ,d) ∈ [1, ∞) such that for all x, y ∈ Rd,t>0,
1 2 M 2 (1.3) e−M|x−y| /t ≤ pa(x, y) ≤ e−|x−y| /M t . Mtd/2 t td/2 It should be noted that the ball doubling condition and the strong Poincar´einequal- ity hold for (Ea,H1(Rd)), and the pseudo-distance (or intrinsic metric) derived from (Ea,H1(Rd)) is a complete metric compatible with the endowed topology. Hence the parabolic Harnack inequality holds for the local solution of the parabolic equa- a − ∂ a 2 Rd a tion (L ∂t)u = 0 (cf. [19],[27]). Here L is the L ( )-generator of (Tt )t>0.In { a} a the same way as [28], Pt is a strong Feller semigroup, that is, Pt f is bounded { a} continuous for bounded measurable f.Furthermore, Pt is a Feller semigroup in view of the upper Gaussian estimate and the estimation (Corollary 7.3 in [28]) of the local H¨older continuity of the local solution of the above parabolic equation a F a (see also Example 4.5.2 in [8]). Let M := (Ω,Xt, t,Px ) be the Hunt process { a} a constructed by the Feller semigroup Pt .ThenM is a doubly Feller process (cf. [4]) and is properly associated with (Ea,H1(Rd)) (cf. [8]). Note that Ma is conservative (see Example 5.7.1 in [8]).
Definition 1.4 (Dynkin and Kato class functions). A measurable function f on d a 1 d R is said to be of the Dynkin class with respect to (E ,H (R )) (write f ∈ Dd) Ea 1 Rd ∈ k a | |k (resp. Kato class with respect to ( ,H ( )) (write f Kd))R if cα := Rα f ∞ < a ∞ −αt a ∞ →∞ for some α>0 (resp. limα cα =0).HereRα := 0 e Pt dt are the resolvents associated with Ma. A measurable function f on Rd is said to be of Ea 1 Rd ∈ loc the local Dynkin class with respect to ( ,H ( )) (write f Dd )(resp.local Ea 1 Rd ∈ loc ∈ Kato class with respect to ( ,H ( ))) (write f Kd ) if and only if IK f Dd d (resp. IK f ∈ Kd) for any compact set K in R .
We see Kd ⊂ Dd ⊂ Hd and if f ∈ Kd, δ(|f|) can be taken to be arbitrarily small (see [1], [25]). It is well known that for 2p>d≥ 2orp ≥ d =1,Lp(Rd) ⊂ p Rd ⊂ ⊂ p Rd ⊂ loc ⊂ loc Lunif( ) Kd Dd. Hence Lloc( ) Kd Dd (see Theorem 1.4(iii) in [1]). 1 d We can see that for f ∈ Dd ∩ L (R ), |f|m ∈ S00 in view of Problem 4.2.1 in [8].
Definition 1.5 (Smooth measures in the strict sense, [8]). A Borel measure µ on Rd is said to be smooth in the strict sense with respect to (Ea,H1(Rd)) (write d µ ∈ S1) if there exists a sequence {En} of Borel sets increasing to R such that
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∈ ∈ N IEn µ S00 for each n and a d P ( lim τE = ∞)=1,x∈ R , x n→∞ n { | ∈ } where τEn := inf t>0 Xt / En is the first exit time from En. ⊂ ⊂ 1 Rd loc ⊂ 1 Rd ∈ loc Since Dd Hd Lloc( ), Dd Lloc( ), then for f Dd we have |f|m ∈ S1.ItiswellknownthatS00 ⊂ S1 ⊂ S and there exists a one-to-one + correspondence between S (resp. S1) and the equivalence class of Ac of PCAF ad- + mitting exceptional sets (resp. Ac,1 of PCAF in the strict sense) under the Revuz correspondence (see Theorem 5.1.4 and 5.1.7 in [8]). We consider a Fukushima decomposition in the strict sense for coordinate func- tions ei(x):=xi (i =1, 2, ··· ,d), x =(x1,x2, ··· ,xd): − a ∈ Rd Xt X0 = Mt + Nt,Px -a.s. for all x . 1 2 ··· d 1 2 ··· d i Here Mt =(Mt ,Mt , ,Mt ), Nt =(Nt ,Nt , ,Nt ), Mt is a local CAF in the i strict sense and an MAF locally of finite energy, and Nt is a local CAF in the strict sense and a CAF locally of zero energy (see Theorem 2 in [7]). Under Ma,we consider the following multiplicative functional Lt: Z Z t t −1 ∗ 1 −1 ∗ Lt(= Lt(b)) := exp[ (a b) (Xs)dMs − (ba b )(Xs)ds]. 0 2 0 R R R Note that Λ−1 t |b|2(X )ds ≤ t(ba−1b∗)(X )ds ≤ λ−1 t |b|2(X )ds and in gen- R0 s 0 s 0 s t −1 ∗ eral the PCAF 0 (ba b )(Xs)ds admits an exceptional Nb (see section 5.1 in [8]). a a Then Lt is a local Px -martingale and a Px -supermartingaleR multiplicative func- ∈ Rd \ | |2 ∈ t −1 ∗ tional for all x Nb.If b m S1, then the PCAF 0 (ba b )(Xs)ds admits ∅ ≥ | |2 ∈ ⊂ loc no exceptional set, that is, Nb = .Ifp>d 2, then b Kd Dd ; hence | |2 ∈ a ∈ Rd b m S1,andLt is an exponential Px -martingale for all x . Precisely to a a say, pt f(x):=Ex[f(Xt)] satisfies the following estimate (see [17]): By Aronson’s estimate (1.3) with the help of Minkovski’s inequality, we have for d ≥ 1, t>0, 0 − d − 0 0 k ak 0 ≤ 2 (1/β 1/β ) ≤ ≤ ≤∞ (1.4) pt Lβ→Lβ c(M,β,β ,d)t , 1 β β . Hence, f 2 is a Kato class function if f ∈ Lp(Rd)withp>d≥ 2. Consequently, there exists a constant c1 := c1(M,p,d,kfkLp ) > 0 such that R t{ }2 a 0 f(Xs) ds ≤ c1t ∈ p Rd ≥ sup Ex [e ] c1e for f L ( )withp>d 2, x∈Rd a ∈ Rd which implies Novikov’s criterion and the Px -martingale property for every x ([17]). The continuous supermartingale multiplicative functional Lt generates a right process in the sense of [21] (see §62 in [21]). Lemma 1.1. Suppose |b|∈Lp(Rd), p ≥ 2. The following assertions hold: (a) Assumption 1.2(a) holds for p ∈ (4d/(d +2), 4) with d ≥ 1. (b) Assumption 1.2(b) holds for p ∈ [4d/(d +2), 4) with d ≥ 3, p ∈ (2, 4) with d =2,orp ∈ [2, 4) with d =1, respectively. | |∈ p Rd ∈ ≥ Proof. (a) Suppose b RL ( )forp (4d/(d +2), 4), d 1. By (1.4), we ∞ − − d 4−p k a | |2k − ≤ αt 2 p k k2 ∞ haveR Rα b Lp/(p 2) 0 e t dt b Lp < . Hence, for some/all α> a | |2 | |2 ∞ a | |2 ∈ 1 Rd Ea a | |2 R0, Rd (Rα b ) b dm < .Equivalently,Rα b H ( )and α(Rα b ,v)= | |2 ∈ 1 Rd | |2 ∈ Rd b vdm for all v H ( ) by Problem 4.2.1 in [8], which implies b m S0.
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(b) If |b|∈Lp(Rd)forp ∈ [4d/(d +2), 4), d ≥ 3(resp.p ∈ (2, 4), d =2), 4−p p/(4−p) d 4−p p/(4−p) d then |b| ∈ L (R ) ⊂ Hd (resp. |b| ∈ L (R ) ⊂ Kd), because p/(4 − p)=4/(4 − p) − 1 ≥ d/2ford ≥ 3(resp.>d/2=1ford = 2). Similarly, p d 4−p p/(4−p) d if |b|∈L (R )forp ∈ [2, 4) with d =1,then|b| ∈ L (R ) ⊂ Kd by p/(4 − p)=4/(4 − p) − 1 ≥ 1. The following theorems are shown in [6] in more general context (cf. [15], [16], [23], [17], [20]). b a b a ◦ Let pt f(x):=Ex [Ltf(Xt)],p ˆt f(x):=Ex[Lt γtf(Xt)] for nonnegative or bounded Borel f.Hereγt is the time reverse operator: Xs(γtω)=Xt−s(ω)for s ∈ [0,t].
Theorem 1.1 (Contractionp property, [15], [16], [23], [6])p. Suppose that Assump- ≥ − | |2 ≤ | |2 −αt b tion 1.1 holds. If p 2/(2 2δ( b )/λ) (resp. p 2/ 2δ( b )/λ), then e pt −αt b −αt b −αt ˆb (resp. e pˆt ) can be extended to a contraction semigroup e Pt (resp. e Pt ) 2 p p Rd ≥ 2γ(|b| ) | |2 ≥ on L ( ) for α αp(b):= p 2/λδ( b ) (resp. α αp/(p−1)(b)). p p Note that 2/(2 − 2δ(|b|2)/λ) ≤ 2 ≤ 2/ 2δ(|b|2)/λ.
−αt b Theorem 1.2 (Girsanov formula, [6]). Under Assumptions 1.1, let e Tt be the 2 Rd Eb 1 Rd 2 Rd L ( )-semigroup associated with the semi-Dirichlet form ( α,H ( )) on L ( ), ˆb b 2 Rd b b α>α2(b) and Tt the dual semigroup of Tt on L ( ).ThenTt f(x)=Pt f(x) ˆb ˆb ∈ Rd ∈ 2 Rd (resp. Tt f(x)=Pt f(x))a.e.x for f L ( ). We will expose the following:
Theorem 1.3 (Conservativeness of process). Under Assumptions 1.1 and 1.2, Lt a ∈ Rd \ | |2 ∈ is a Px -martingale for all x Nb.Iffurthermoreb m S1,thenLt is a ∈ Rd a Px -martingale for all x . In particular, the transformed process by Lt is conservative. ∈ d Rd → Rd ≥ a Corollary 1.1. For b L ( ) with d 3, Lt is a Px -martingale for all d x ∈ R \ Nb. 2 2 Proof. Since |b| is of the Hardy class and δ(|b|p) can be chosen to be arbitrarily small for b ∈ Ld(Rd → Rd), we see d>4/(2 − 2δ(|b|2)/λ) for sufficiently small δ(|b|2). So Assumptions 1.1 and 1.2(c) hold. Also, Assumptions 1.1 and 1.2(a), or (b) hold for p = d ≥ 3. √ ∈ ∞ ∞ → ∞ − Example 1.1. Let g C0 ([0, ) [0, ]) with g(r):=1/(r log r)on(0, 1/e], g(r):=0on[1, ∞)andf(x):=g(|x|). Then f ∈ Ld(Rd), but f/∈ Lp(Rd)for ≥ p>d.LetM := (Bt,Px)bethed-dimensionalP R Brownian motionR with d 3. Then ··· d t i − d t 2 for b(x):=f(x)(1, 1, , 1), Lt := exp[ i=1 0 f(Bs)dBs 2 0 f(Bs) ds]isaPx- d 2 martingale for all x ∈ R \{0}.InthiscaseNb = {0}.Notethatf is not of the Kato class function by Theorem 1.3(viii) in [1]. Also, Novikov’s condition fails for starting point 0, becauseR the low of the iteratedR logarithm of 1-dimensional Brownian motion t 2 ≥ t 2 ∞ yields E0[exp[ 0 f(Bs) ds]] E0[ 0 f(Bs) ds]= (cf. Example 5.1.1 in [8]). Example 1.2 (cf. [9], [12], [3], Example (4.2) in [5]). When d ≥ 3, the following Hardy-type inequality holds: for any r>0, Z Z u(x)2 4 (1.5) dx ≤ |∇u(x)|2dx for u ∈ H1(B (0)). | |2 − 2 0 r Br (0) x (d 2) Br(0)
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Hence Z Z 2 u(x) ≤ 4 |∇ |2 ∈ 1 Rd (1.6) 2 dx 2 u(x) dx for u H ( ). Rd |x| (d − 2) Rd
Take ϕ ∈ C∞([0, ∞)) with 0 ≤ ϕ ≤ 1andϕ ≡ 1on[0, 1/2], ϕ(s) ≤ 1/sd for s ∈ ∞ d−2 ϕ(|x|) | |2 [1, ). Let b(x):= 4+ε |x|2 x for ε>0. Then b is of the Hardy class with respect to the Dirichlet form associated to d-dimensional Brownian motion M := (Bt,Px), and δ(|b|2)=8/(4 + ε)2 < 1/2. Note that |b| ∈/ Lp(Rd)forp ≥ d, but |b|∈Lp(Rd) for d/(d+1) 2. Proof of Theorem 1.3 a ∈ + Rd Proof of Theorem 1.3. First we show that Lt is a Pgm-martingale for g C0 ( ). a ∈ Rd a · Consequently,R Lt is a Px -martingale for a.e. x .WehavePgm( ):= a · Rd Px ( )g(x)m(dx)andm(dx):=dx is the d-dimensional Lebesgue measure. Ea { } | |2 There exists an -nest Fn such that IFn b is of the Kato class satisfying | |2 ∈ 1 ··· d p Rd IFn b m S00 ([2]). Then bn := (IFn b , ,IFn b )givesanL ( )-approximation | |2 n for b.Since bn is of the Kato class function, each Lt (:= Lt(bn)) is an exponential a ∈ Rd n a n Px -martingale for all x . We put pt f(x):=Ex [f(Xt)Lt ] for nonnegative or 2 2 bounded Borel functions f.By|bn|≤|b|, δ(|bn| )(resp.γ(|bn| )) can be taken as 2 2 a common constant δ(|b| )(resp.γ(|b| )). Hence we may set αp(bn)=αp(b) for all Ebn 1 Rd p>1sothat( α ,H ( )) is coercive for α>α2(b). By Theorems 1.1 and 1.2, ≥ −αt n we can confirm that for α αr(b), e pt can be extended to ap strongly continu- −αt n r Rd ≥ − | |2 ous contraction semigroup e Pt on L ( ) for all r 2/(2 2δ( b )/λ)and −αt n 2 Rd { −αt n} e Pt coincides with the L ( )-contractive semigroups e Tt associated to Ebn 1 Rd 2 Rd the semi-Dirichlet form ( α ,H (R )) on L ( )forα>α2(b). The association ∞ − n α n Ebn n means that the resolvent Gα := 0 e Tt dt satisfies α (Gαf,v)=(f,v)L2 for ∈ 2 Rd ∈ 1 Rd f L ( ), v H ( ), α>α2(b)=α2(bn). Notep that the contraction property −αt n ≥ − | |2 of e Pt holds for all r 2, because 2 > 2/(2 2δ( b )/λ). ∈ + Rd On the other hand, for g C0 ( ), Z Z t 2 t a −1 − ∗ a − −1 − ∗ Egm[ (a (bn b)) (Xs)dMs ]=Egm[ ((bn b)a (bn b) )(Xs)ds] 0 0 Z t ≤ −1 a | − |2 λ Egm[ bn b (Xs)ds] 0 ≤kgk p t for p ≥ 2. L p−2 n → →∞ a ∈ + Rd This implies that Lt Lt as n in Pgm-measure for g C0 ( ). { n} a Next we show uniform integrability of Lt n∈N under Pgm.ByItˆo’s formula, we have Z Z t t n n n n 1 n −1 ∗ Lt log Lt = (1 + log Ls )dLs + Ls (bn)a (bn) (Xs)ds. 0 2 0 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use CONSERVATIVENESS 2749 { n} n →∞ →∞ Taking an increasing sequence of stopping times Tk such that Tk as k a Pgm-a.e. and Z ∧ n t Tk n n (1 + log Ls )dLs 0 a ≥ ≤ is a Pgm-martingale, we have the following estimate if p 4, or if 2 p<4 under Assumption 1.2(a) or (b): for α>α2(b), Z ∧ n t Tk a n n 1 a n −1 ∗ Egm[Lt∧T n log Lt∧T n ]= Egm[ Ls (bn)a (bn) (Xs)ds] k k 2 0 Z λ−1 t ≤ Ea [ Ln|b |2(X )ds] 2 gm s n s Z Z 0 −1 t ≤ λ n| |2 g(Ps b )dmds ( 2 0 Rd − R λ 1 t αs −αs n 2 e kgk p ke P |b| k p ds if p ∈ [4, ∞) 2 0 p−2 s L 2 ≤ − R L λ 1eαt ˆn | |2 ∈ 2 Rd (Gαg) b dm if p [2, 4) − λ 1 eαt−1 2 kgk p kbk p if p ∈ [4, ∞). 2 α p−2 L − L λ 1eαt p/2 n (4−p)/2 ≤ kbk p k(Gˆ g)|b| k 2 if p ∈ [2, 4). 2 L α L − R 1/2 λ 1eαt a | |2 | |2 · Ea ˆn ∈ 2 Rd (Rα b ) b dm α(Gαg) if p [2, 4). p/2 −αt n ≥ ≥ Here we use the L -contraction property of e Pt for p 4, α>α2(b)( ˆn n αp/2(b)) and (Gα)α>α2(b) is the dual of (Gα)α>α2(b).By p − Ea ≤ − | |2 1Ebn ∈ 1 Rd (u, u) (1 2δ( b )/λ)) α (u, u)foru H ( ),α>α2(b), a n n we have sup ∈N E [L ∧ n log L ∧ n ] < ∞. Similarly, under Assumption 1.2(c) n,k gm t Tk t Tk with the help of Theorem 1.1, for α ≥ αp/2(b), Z ∧ n t Tk a n n 1 a n −1 ∗ Egm[Lt∧T n log Lt∧T n ]= Egm[ Ls (bn)a (bn) (Xs)ds] k k 2 Z 0 −1 t λ αs −αs n 2 ≤ e kgk p ke P |b| k p ds L p−2 s L 2 2 0 −1 αt − λ e 1 2 ≤ kgk p kbk p < ∞. 2 α L p−2 L n a These imply uniform integrability of {L ∧ n }n,k∈N under P .SinceI{ n } t T gm Lt∧T n >K k k →∞ a is lower semicontinuous as k for each K>0, we see the uniform Pgm- { n} n 1 a →∞ integrability of Lt n∈N. Hence Lt converges to Lt in L (Pgm)asn .There- { } a fore Lt is a Pgm-martingale.R Finally we show the martingale property of Lt for ∈ Rd \ b ∞ −αt b b all x Nb.LetRαf(x):= 0 e pt f(x)dt, α>0betheresolventof(pt )t>0. b Then Rα1isα-excessive, hence, finely continuous with respect to the transformed b ∈ Rd process by Lt. By the first argument, we see that αRα1(x) = 1 for a.e. x . Note that the absolute continuity condition holds with respect to the d-dimensional d Lebesgue measure m for the transformed process on R \Nb. Indeed, suppose f =0 ∈ Rd \ a a a.e. Then for all x Nb, Px (f(Xt) = 0) = 1; hence Ex[Ltf(Xt)] = 0 for all d x ∈ R \ Nb.Thed-dimensional Lebesgue measure m has a full support with re- spect to the fine topology of the transformed (right) process (see Lemma 4.6 in License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 2750 KAZUHIRO KUWAE b ∈ Rd \ b [14]). Therefore, αRα1(x) = 1 for all x Nb, and hence pt 1(x) = 1 for all d t>0andx ∈ R \ Nb. This completes the proof. Remark 2.1. (a) In [20], the conservativeness of the transformed process is proved (Proposition 3.1 in [20]) if b ∈ Ld(Rd → Rd)withd ≥ 3 and div b ≥ 0inthe following distributional sense: Z h ∇ i ≤ ∈ ∞ Rd ≥ b, u(x) Rd dx 0foru C0 ( )withu 0. Rd Our result shows that the condition div b ≥ 0 is not needed. (b) In view of the result from uniqueness of the positive Cauchy problem (see [10],[11]), our integrability condition for b ∈ Lp(Rd → Rd)(p ≥ 2) (in particu- lar, Lp-integrability at infinity) seems to be strong. However, in [10] and [11], ∈ p Rd →Rd ≥ b Lloc( )(p>d) is imposed when d 2. So Examples 1.1 and 1.2 are not derived from [10], [11]. (c) Theorem 1.3 can be generalized in the framework of a regular local symmetric Dirichlet form (E, F)onL2(X; m) corresponding to a conservative doubly Feller diffusion process M ([4]), where X is a locally compact separable metric space and m is a positive Radon measure on X with full topological support. For example, ◦ let M ∈Mloc be a local martingale CAF locally of finite energy such that the energy measure µhMi corresponding to the quadratic variation process hMi is of the Hardy class smooth measure with δ(M):=δ(µhMi) < 1/2. 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MR 99b:31008 Department of Mathematical Sciences, Yokohama City University, Yokohama 236- 0027, Japan E-mail address: [email protected] Current address: Department of Mathematics, Faculty of Education, Kumamoto University, Kumamoto 860-8555, Japan E-mail address, starting October 1, 2004: [email protected] License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use