revuz measures and time__ changes

Math. Z. 199, 233-256 (1988) Mathematische Zeitschrift �9 Springer-Verlag 1988 Revuz Measures and Time Changes* P.J. Fitzsimmons and R.K. Getoor Department of Mathematics, C-012, University of California, San Diego, La Jolla, California 92093, USA I. Introduction Let X = (Xt, W) be a right Markov process with state space E, and let A = (At) be a finite continuous additive functional of X. Upon time-changing X by means of A we obtain the right process )~=(X~(o: t>0), where z(t)=inf{s: As>t}. Of course, the state space of ~" is F, the fine support of A. Given an X-excessive measure m, there is a naturally associated )~-excessive measure, the Revuz mea- sure of A relative to m, defined by (1.1) v"~(f)=l imt- l P " foXsdA . t$O As is evident from (1.1), the mapping mw-~v] is additive, positive homogeneous, and order preserving. The mapping m~--~v'~ is not injective - v] is unchanged if m is replaced by its balayage on F. Owing to our insistence that an excessive measure be a-finite, mw-~v'~ is not even surjective. However, given a target X-excessive mea- sure 2, one can define a (minimal) measure rn z with the following two properties: m~. (ii) if m ~ is not a-finite (i) if m z is a-finite then it is X-excessive and 2=v A , then the equation 2 = v~ admits no X-excessive measure m as solution. The above solution of the problem of "inverting" m~--~v'~ is the focus of the paper, and appears in Sect. 5. This result allows us to prove that m~-~v] preserves the class of "harmonic" excessive measures. The same stability result holds for other classes (e.g. potentials, conservative excessive measures . . . . ), but is much easier to verify in these latter cases (see Sect. 4). These results extend work of Kaspi and Mitro [KM]. On a more practical level, the inversion of m~--~v] provides a way to build an X-excessive measure out of a given )~-exeessive measure. In particular, if 2 is X-conservative, then m z discussed above is always X-conservative. This result, and our explicit recipe for m z generalize a result of Kaspi [K84]. * This research was supported in part by NSF Grant DMS 8419377 234 P.J. Fitzsimmons and R.K. Getoor We have decided to take the basic process X to be a right process in the sense of Sharpe [S]. This class of processes is preserved by the usual transforma- tions of Markov process theory (time change, killing). The key technical point is that the state space E is only a U-space, which precludes the use of several valuable constructs (Kuznetsov processes, exit systems) whose existence seems to require that E be a Lusin space (or at least coSouslin). This poses certain difficulties, as is evidenced by the preparatory Sects. 2-4. Section 2 contains various generalities concerning Revuz measures. We show that for any additive functional A and X-excessive measure m, the function t t~--~P"~ foXsdAs is concave. This implies that the limit in (1.1) is monotone 0 increasing; this fact seems to have gone unnoticed by previous authors. Also of particular interest is Theorem (2.22) which, roughly, states that if A is a continuous additive functional then its Revuz measure is essentially unchanged if X is replaced by a subprocess (X, M), M being a multiplicative functional of X. In Sect. 3 we study the balayage operation on excessive measures introduced by Hunt [H]. In this section we also discuss the energy functional L introduced by Meyer I-Me73]. This functional is our principal tool - indeed Meyer's repre- sentation of v] in terms of L was the point of departure for our study. This representation, and its relationship with the balayage operation, are developed in Sect. 4. Section 6 contains an application of Theorem (2.22). Briefly, if B is a second continuous additive functional of X with fine support contained in F (the fine support of A), then Bt=B~(t) defines a continuous additive functional of X. We prove that B and /~ have the same Revuz measure. More precisely, using the obvious notation, (1.2) if 2 = v], then m_ ~4 A very special case of (1.2) can be found in Griego [Gr], and (1.2) is presumably well-known in the context of classical duality. The reader mainly interested in the properties of the mapping mw*v] might skip from Lemma 2.18 to Sect. 3. The portion of Sect. 2 that follows the proof of (2.18) requires more technical machinery than any other part of the paper and is used only in Sect. 6. Notation. We use standard notation, of which [BG] and [G] are exemplary. In particular, if (H, ~) is a measurable space then h~b~Ug (resp. hepJg) means that h is an X/g-measurable function from H to [ -oe , ~] that is bounded (resp. F0, oo]-valued); of course b p ~ = b ~ c~ p J((. Unless mention is made to the contrary, the letters f and g (with or without subscripts) denote elements of pg*. Here N* is the universal completion of g, the Borel subsets of the state space E. We write E e for the o--algebra generated by the q-excessive func- tions. If K cH, then ~ c~ K denotes the trace of ~ on K. If p is a measure, then both p( f ) and #f denote ~fd#. Revuz Measures and Time Changes 235 2. Revuz Measures Let (E, g) be a (separable) Radon space (called a U-space in [G]) and let X = (Xt, P~) be a right Markov process with state space E in the sense of Sharpe I-S]. As in [S], the semigroup (Pt) and resolvent (U q) of X need be only subMarkov- ian in this paper. The potential kernel of X is U = U ~ The class of excessive measures for (Pt) or X is denoted Exc(X): meExc(X) provided m is a a-finite measure on E and mPt0. Then mPtTm as t+0. See XII-37 of [DM], for example. We now fix a multiplicative functional M = (Mr: t > 0) of X. To be precise, we suppose that (2.1) (a) t~--~Mt(co ) is decreasing, right continuous, and has values in [0, 1] for each o9 e s (b) Mte~, t>=0; (c) mt+,(og)=mt(co)m~(otog) for all s, t>0 and ~o~f2. We do not assume that M is exact. Let S=inf{t: Mr=0} and note that S is a terminal time. Let EM-- {x ~E: P~(Mo = 1) = 1} = {x: Px(S > 0) = 1} denote the set of permanent points of M. Since EM~g*, (EM, ~ c~ E~t) is a Radon space, and serves as the state space for the subprocess (X, M). Recall that (X, M) has semigroup and resolvent given by 09 (2.2) Qtf(x)=P:~(UoXtMt); V" f (x )=P ~ ~ e - " t foXtMtdt . 0 The measures Qt(x, ") and Vq(x, .) are carried by EM for each t, q>0 and xsE. Let Exc(X, M) denote the class of excessive measure for (Qt): meExc(X, M) provided m is a o--finite measure carried by EM such that m Qt < m for all t >0. If meExc(X), then m* =mlEMEExc(X, M). We refer the reader to [BG] for fur- ther properties of a multiplicative functional - hereafter abbreviated MF. (2.3) Definition. A positive, increasing, right continuous process A =(At: t >0) is a raw additive functional (RAF) of (X, M) provided Ate~,, At< ~ if t ~, and (2.4) At+s = At + MtA~ o Ot identically in s, t>O and co~O. Note that we do not assume that Ao=0. Of course, if x~E M then a.s. px one has S > 0 and A o = 2 A o. Therefore PX(A o = 0)= 1 if x ~ EM. Also, (2.4) implies that At = As = As ^ ~ if t > S. Thus the measure dA t is carried by [0, S/~ ~] c~ [0, c~ [, but may charge S ^ ~ if S/x ~ < ~. Our first result is elementary in the extreme, but seems to have gone unno- ticed. (2.5) Proposition. Let m~Exc(X, M) and let A be a RAF of (X, M). Let qg(t) = ~o,, (t) =pm (At). I f q~ (t) < oo for one t > O, then q~ is a finite, increasing, continuous, concave function on [-0, ~[ and ~o(0)=0. 236 P.J. Fitzsimmons and R.K. Getoor Proof Obviously 99 is increasing. Let et(x ) = W(At), so that by (2.4) we have (2.6) 99(t + s)=99(t)+mQtcs. Since ~o is increasing and m Qt cs < m (cs)=99 (s), it follows from (2.6) that if 99 (t)< oo for one t>0, then 99(t)0. We now suppose that 99(t)< oo for some t>0. Clearly 99 is right continuous, and 99(0)=0 because pm(Ao>O)=O - m is carried by EM. F rom the identity AAt=Mt_sAAsoOt_s for 0 0 for at most countably many values of t. Thus A 99 = 0, and ~0 is continu- ous. Since m~Exc(X, M), tv-~mQtc ~ is decreasing; thus for 00, 99 (t § h) - 99 (t) =< q~ (s + h) - 99 (s). Setting t = s § h this last line becomes �89 + 2 h) + 99(s)] <= 99(�89 (s + 2 h) + �89 s). Therefore 99 is "midpoint concave", hence concave (being continuous). [] Here are some easy consequences of (2.5). Suppose 99(t)< oo for some (all) t > 0. Then the right hand derivative (2.7) 99 + (t) - ~" lim [99 (t + h) - 99 (t)]/h h~.O exists for t>0 and is finite for t>0 (99+(0) may be infinite). The function 99+ is decreasing and right continuous on [0, ~ [. Being concave, 99 is absolutely continuous so (since 99 (0)= 0) (2.8) 99(t)= i 99+(s)ds, t>0. o We continue to assume that meExc(X, M) and that A is a RAF of (X, M) as in (2.5). Let m=mi +mp be the decomposit ion of m into its invariant and purely excessive parts relative to (Qt):miQt = mi, mpQt(f)~ 0 as t--+ oo whenever mp(f)O, then ~b(q) is finite for all q > 0 and o oO (2.10) qO(q) = 99i(1)+ S (1 --e -qt) v(dt), 0 where v(dt)= - llo,oot(t ) d99 + (t), and 99i(1) = lim 99+ (t)= lim t - a 99(t). t~(X) t~o9 Revuz Measures and Time Changes 237 Proof Owing to the properties of ~p we have o9 o3 q~(q)=q ~ e-qtd~p(t)=q ~ e-qtqg+(t)dt 0 0 o9 =(1--e -qt) ~o+(t)[~-- ~ (1-e-qt) dq~+(t). 0 If O(q)0, then q~(t)= i~o+(s)ds is finite for all t>0. In this 0 case, since ~o + is decreasing, we have t q~ + (t)--, 0 as t $ O, and so (2.10) will obtain once we show that lim t -~ ~o(t)= l im ~o+ (t) = q~,(1). t~oO t---~ O9 The first equality above follows immediately from the concavity of q~. As for the second, first note that ~ > q~ (t) > q~p (1) > m v Qt c i ~ 0 as t ~ ~. Consequently (2.6) forces ~op(t + 1) - q~p(t) ~ 0 as t ~ ~. But t+ l q~p(t+l)= I t q); (s) ds = q)p(t + 1) - ~op(t), and so ~o+(t)~0 as t~oo. Since ~o+(t)=cpi(1)+q~(t), we obtain lim~o+(t) =~o~(1) as desired. [] t~o9 Remarks. (a) A special case of (2.9) may be found in (6.16) of [FM]. Formula (2.10) states that q~k(q) is a subordinator exponent with no linear term; equiva- lently, q ~ (q) has a completely monotone derivative and ~ (q) --* 0 as q ~ ~. (b) Here is another special case of (2.9). Let R be a terminal time of X with R < ~ if R < ~. Then A t = 1 tm o9[ (t) leo < g < o9~ defines an adapted AF of (X, R) -X killed at R. (That is, Mr= lto,Rt(t ) in the previous discussion.) In this situa- tion EM is the set of points in E irregular for R. Suppose that meExc(X). Then m* =ml~MeExc(X, R), and pm(R <__ t) = m(E\EM) + P ' (0 < R__< t). Since r (t) = P" (0 < R < t) = P"* (At), our previous discussion applies. In particular, if q~(t)< ~ for one t>0, then the P"-distribution of R is absolutely continuous on ]0, oo[ with a decreasing density and an atom m(E\EM) at 0. If, in addition, pm(e-aR; R>0)< oo for some q>0, then for all q>0 (2.11) o9 qpm(e-qR) = qm(E\EM)+ q~i(1) + ~ (1 --e -qt) v(dt). 0 Formula (2.11) generalizes (8.9) of [GSt]. Let us return to the general situation and recall the following basic 238 P.J. F i tzs immons and R.K. Getoor (2.12) Definition. The Revuz measure of A relative to meExc(X, M) is the measure on E defined by (2.13) v'J(f)=~limt-lp " ~ foX~dA~. t J. 0 10, t] To see that the limit in (2.13) is monotone, note that for febpg, t~--~ ~ foX~dA~ is a RAF of (X, M); thus (2.5) applied to this RAF implies ]O,t] that t ~ P" ~ fo X~ dA s is concave. Monotonicity for general f follows by trun- cation. ]o,t] We now introduce the potential kernel oO (2.14) U~f(x)=P x ~ e-StfoXtdAt, o and write UA for UA ~ By a standard Abelian theorem, we can rewrite (2.13) as oo v~/(f)=]" lim qP" ~ e-StfoXt dA, q~ ov 0 ='~ lim qmU~f S--* o0 More generally, consider the MF M~=e-StMt. Evidently A~- S e-qSdA~ is lO,t] a RAF of (X, M s) whenever A is a RAF of (X, M). In this case we define, given m ~ Exc(X, Mq), (2.15) qv"d(f)=~limt-lP " ~ foX~e-S~dA~ t,LO ]o,t] =T lim p.mU~+Sf p~oo Of course, (2.15) is just the definition (2.12) applied to A s and M s. It is evident from (2.15) that for OO and m~Exc(X, Mq). Suppose that there is a sequence (#,) of measures on E M such that #nVq~m. Then #,UJT%"~. If, in addition, f is such that there is a sequence ( f , )~pg* with vqf,~ U J f then m(f,)Tqvm4(f). These hypotheses hold whenever V q is a proper kernel (in particular, if q > 0). Proof The proof of (8.6) in [GS84] shows that if # is a measure on EM that is a countable sum of finite measures, then (2.19) oo Tl imt-apuW( ~ f~ ~' ~ e-qt f~ t ~ 0 ]o , t ] 0 If each #, is carried by EM and #, V q ~ m, then #, V q (and so also #,) is a-finite. Thus (2.15) and (2.19) combine to yield #, UJT~v]. Now suppose that Vqfk'[ U J f Then qv](f) = Tlim #. UJ f= "r lim T lira #, Vqfk = Tlim m (fk). n n k k Since UJ f is excessive for the semigroup (e-qtQt), which has 0-potential kernel V q, the last assertion of the lemma follows from XII-17 and XII-38 of [DM]. [] To simplify things, for the rest of this section we assume that A is a RAF of X (i.e., A satisfies (2.3) with Mr= 1). We also fix a MF M. Clearly A(M)t - ~ M~ dA~ is a RAF of(X, M). For m~Exc(X, M) we define ]O,t] (2.20) Mv~ (f) =- T lim t - ' pm ~ fo Xs M~ dA~ t ~ 0 ]o , t ] =T lim q.mVJf , q--~ o9 where co (2.21) Vq f (x )=P x ~ e-qtfoXtMtdAt. 0 Clearly M m ,,,, q FA = VA(M) and V~ = UJ(M ) in our previous notation. In fact (2.20) makes sense for meExc(X). Of course, nothing is changed if one replaces m by m* =mlE~Exc(X , M). If Mr= lto,Rt(t) where R is a terminal time, then we write RV~ for Mv~. Here is the promised generalization of (2.16). See Lemma 5.1.5 in [F] for a special case. (2.22) Theorem. Let A be a continuous RAF of X, and f ix m~Exc(X). Then ~v"~ = 1~ M - v~, where E~=(x : P:'(M0 = 1)= 1) as before. Proof We shall proceed in two steps, showing first that M ,, s m VA = VA, and then that Sv"j = 1E~, �9 V"A. (Recall that S=inf{t: Mt = 0}.) 240 P.J. Fitzsimmons and R.K. Getoor To begin suppose that S > ~ a.s., and define kernels (2.23) U~f(x )=-P x ~ e-q~foXtMt_ldMt, lO,S[ q>=0. As noted by Meyer [Me66], (2.24) U q = V q + U~ V q, and by a similar calculation, using the continuity of A, q- - q q (2.25) U~ - VJ + U~ V~. The point to be taken from (2.24) and (2.25) is: (2.26) vq f.'r V~ f => Uq f~'F UJ f. Now given q>0 and f there is a sequence (f,) such that Vqf,'FV~f Using (2.18) we see that m(f,)TMqv](f)=qV~(M)(f); for the same reason (because of (2.26)), m(f,)T%](f) . In view of (2.16) we must have MV~=v]=Sv~, the last equality being true since S > ~. In general W(S< ~)>0 is possible, in which case (2.24) and (2.25) fail. How- ever, (2.24) and (2.25) become true if U q and U q are replaced by the kernels S P" ~ e-qtf~ dt, P" I e-~*f~ o 10,s[ respectively. This change being made the previous argument allows us to con- clude that MV~ = SV~, for an arbitrary MF M. S m We shall prove that v A - l~(s) v~, where E(S) - {x: Px(S> 0) = 1} = EM. In proving this we may assume that S is exact. If not, let D - $ lim(t + S o Ot) denote t io the exact regularization of S. Then E(D)~E(S) since D>S. Moreover S=D if S >0, and the set E(D)\E(S) is of zero potential, hence m-null. It now follows easily that "v]=sv] . Thus, if we have proved that "v]=l~(o)v~, then since sv] is carried by E(S), Sm 1 Dm m SV~t =- 1E(S) VA = "E(S) 1)A = 1E(S) 1E(D) V~ = 1E(S) VA, as desired. Thus, for the rest of the proof we assume that S is exact and continue to write E M = {x: W(S > 0)= 1}. In proving that sv] = 1E, ~ v] we need replace- ments for (2.24)-(2.26); such will be provided by the last exit decomposition associated with S. (This decomposition was established by Meyer [Me74] under the hypothesis that E is Lusin, but Meyer's argument works even if E is only a separable Radon space.) For the rest of the proof we let Qt and V q denote the semigroup and resolvent of (X, S), which has state space EM. Note that EM~ e since S is exact. Define St=t-}-SoOt, t>0, and let J denote the closure in ]0, ff[ of {St: t>0} c~]0, ~[-. Evidently J is an optional, homogeneous random Revuz Measures and Time Changes 241 set. For xeE let (Q*(x, ");t>0) be the (Qt)-entrance law appearing in the last exit decomposition for J, and define 0o Wqf(x ) = S e -q tQ* f (x ) dr. o Since S is exact, E~ coincides with the set F defined in (2.2) of [Me74]. Thus each measure Q*(x, ") is carried by E~, and x~--~Q*(x, .) is g*-measurable; the same is therefore true for Wq(x, .). Also, W q l(x)< o0 for q>0, xsE . The last exit decomposition takes the form (2.27) uq f = vq f -]- U q (1E~ f ) + Uq wq f, where Ufl is the q-potential of a certain finite (adapted) AF, F, of X. (Take Laplace transforms in Cor. 1, p. 205 of [Me74].) Upon "Laplace transforming" the entrance law property of (Q*) we obtain (2.28) and (2.29) W q = W ~ + (r - q) W q W = W ~ + (r - q) W ~ V q, Wq f = 1" lim r W q +~ vq f r - - coo 0~q O, (2.30) ( r - -q)VqV,~f=P" I (e -q" -e - 'U ) foXudAu, ]o,s[ whence (2.31) and vq v~ : v" v q , q, r=>O, (2.32) V]=V,~+(r - -q)V 'V] , O 0 and g vanishing off EM. As before there is a sequence (f,), each f , bounded and vanishing off E M, such that vqf , T Vqg. In view of (2.29) and (2.33) we have 242 P.J. Fitzsimmons and R.K. Getoor Wqf, T WJg as well, hence Uqf, T UJg by (2.27) and (2.34). (N.B. Uq(1E~ f , )=0 since f, vanishes off EM.) It now follows as in the first part of the pro~of that Sv](g) = v~(g). Since Sv~ is carried by E2u, we conclude that Sv'~ = 1~ v'~. To complete the proof of the theorem we must establish (2.34). Fix g vanish- ing off EM and apply (2.27) with f = (r-q)V~ g, r > q: (2.35) (r -- q) U q V~ g = (r -- q) V q V~ g + (r - q) U~ W q V~ g. We shall examine the limit as r ~oo of each term in (2.35). First, note that (2.36) UqV~g=P'; e(~-q)tdt I e-r"g~ 0 ]t,St[ For t>0 define Gt=sup{u<__t: ueJ}. It is easy to check that {t