AN EXTENDED GENERATOR AND SCHR¨ODINGER EQUATIONS

The generator of a Borel right process is extended so that it maps functions to smooth measures. This extension may be de(cid:12)ned either probabilistically using martingales or analytically in terms of certain kernels on the state space of the process. Then the associated Schr¨odinger equation with a (signed) measure serving as potential may be interpreted as an equation between measures. In this context general existence and uniqueness theorems for solutions are established. These are then specialized to obtain more concrete results in special situations.


Introduction
During the past 20 years or so there has been considerable interest in equations of the form (1.1) Here Λ is a linear operator and µ a signed measure. Classically Λ was the Laplacian on a domain in R n (possibly with boundary conditions) and µ was absolutely continuous with respect to Lebesgue measure, say with density p. Then (1.1) has the form (∆ + p)u = f and is often called the Schrödinger equation with potential p. Hence in the literature (1.1) is often called the (generalized) Schrödinger equation for Λ. The situation in which Λ = ∆ was generalized first to Λ a reasonable second order elliptic partial differential operator, then when Λ is an operator derived from a Dirichlet form, a Markov process or a harmonic space. Of course even when Λ = ∆ and µ is measure one must give a precise meaning to (1.1). For a sampling of the literature during this period see [BHH87], [FL88], [ABR89], [Ma91], [H93], [CZ95], [G99] and the references contained therein.
There seem to be (at least) three somewhat different but interrelated techniques for approaching (1.1) in the literature which might be described as: (i) perturbation of Dirichlet forms, (ii) perturbation of Markovian semigroups, (iii) perturbation of harmonic spaces.
It seems to me that in all three approaches the basic idea is to define Λ + µ as an operator in some space of functions and then interpret (1.1) in some weak sense. For example in method (ii) one assumes the µ corresponds to a continuous additive functional A and then defines Λ + µ as the generator of the Feynman-Kac semigroup acting in some reasonable function space, and then interprets (1.1) in an appropriate weak sense. Here X = (X t ) is the underlying Markov process and E x the expectation when X 0 = x. This approach is carried out in detail for very general Markov processes in my paper [G99].
In the present paper we introduce a rather different approach to (1.1). Namely we extend the domain of Λ so that it maps functions into measures and then interpret (1.1) as an equation between measures with the right side being the measure f m where m is a prescribed underlying measure-Lebesgue measure in the classical case. It then seems natural to replace the right side of (1.1) by a measure ν. So in fact we shall investigate the equation This approach has several advantages over the perturbation approachmethod(ii) above-used in [G99]. It seems more direct and natural (to me) and, more importantly, it is technically simpler and one obtains more general results under somewhat weaker hypotheses. Also the method lends itself to study (1.2) on suitable subsets of the state space of X. This allows a consideration of "boundary conditions" for (1.2) and leads to a notion of "harmonic functions" for Λ + µ. This aspect will be explored in a subsequent paper. The relationship between u and the value of Λu = λ is that a certain process, Y , involving u • X t and the continuous additive functional A associated with the measure λ be a martingale. See Theorem 3.9 and Definition 4.1 for the precise statement. Of course the idea of using martingales to extend the generator goes back at least to Dynkin [D65]. This was extended further by Kunita [K69] to measures absolutely continuous with respect to a given measure. See also [CJPS80] for a discussion of various extended versions of the generator. However our point of view seems somewhat different from earlier work. It would be natural to extend the domain of our generator even further by requiring the process Y mentioned above to be a local martingale rather than a martingale. We have decided not to do this in the present paper for several reasons. Most importantly in order to write down the solution of (1.1) or (1.2) it is necessary to impose certain integrability conditions. Moreover the definition adopted in section 4 can be stated without reference to martingales. Finally the results in this paper would be needed for any localization of the definition and we decided not to complicate the basic idea with additional technicalities. In discussing harmonic functions-that is solutions of (1.1) when f = 0-in a subsequent paper it will be both natural and necessary to localize the current definition.
The remainder of the paper is organized as follows. Section 2 sets out the precise hypotheses under which we shall work and reviews some of the basic definitions that are needed. It also contains some preliminary results. Section 3 contains the equivalence of a martingale property and the analytic property that is used to define the generator. In section 4 the generator is defined and discussed. We proceed somewhat more generally than indicated so far. Namely we consider a finely open nearly Borel subset D ⊂ E and define an operator Λ D that we regard as an extension of the restriction of the generator of X to D. It maps functions on E to measures on D. In section 5 we study the equation (1.2). Again we are somewhat more general and consider Here q ≥ 0 is a parameter. We prove existence and uniqueness theorems for (1.3) under various hypotheses. Finally in section 6 we suppose that ν = f m where m is the distinguished underlying measure and f ∈ L p (m), 1 ≤ p ≤ ∞. We specialize the results of section 5 to obtain existence and uniqueness theorems depending on p. If 1 < p < ∞ our results are sharper than those obtain in [G99].
We close this introduction with some words on notation.
is a measure on (E, E) for each x ∈ F ), then we write µK for the measure A → F µ(dx)K(x, A) and Kf for the function x → E K(x, dy)f (y). The symbol ":=" stands for "is defined to be." Finally R (resp. R + ) denotes the real numbers (resp. [0, ∞[) and B(R) (resp. B(R + )) the corresponding Borel σ-algebras, while Q denotes the rationals. A reference (m.n) in the text refers to item m.n in section m. Due to the vagaries of L A T E X this might be a numbered display or the theorem, proposition, etc. numbered m.n.

Preliminaries
Throughout the paper X = (Ω, F, F t , θ t , X t , P x ) will denote the canonical realization of a Borel right Markov process with state space (E, E). We shall use the standard notation for Markov processes as found, for example, in [BG68], [G90], [DM87] and [Sh88]. Briefly, X is a strong Markov process with right continuous sample paths, the state space E (with Borel sets E) is homeomorphic to a Borel subset of a compact metric space, and the transition semigroup (P t ) t≥0 of X preserves the class bE of bounded E-measurable functions. It follows that the resolvent operators U q := ∞ 0 e −qt P t dt, q ≥ 0, also preserve Borel measurability. In the present situation q-excessive functions are nearly Borel and we let E n denote the σ-algebra of nearly Borel subsets of E. In the sequel, all named subsets of E are taken to be in E n and all named functions are taken to be E n -measurable unless explicit mention is made to the contrary.
We take Ω to be the canonical space of right continuous paths ω (with values in E ∆ := E ∪ {∆}) such that ω(t) = ∆ for all t ≥ ζ(ω) := inf{s : ω(s) = ∆}. The stopping time ζ is the lifetime of X and ∆ is a cemetery state adjoined to E as an isolated point; ∆ accounts for the possibility The σ-algebras F t and F are the usual completions of the σ-algebras F • t := σ{X s : 0 ≤ s ≤ t} and F • := σ{X s : s ≥ 0} generated by the coordinate maps X s : ω → ω(s). The probability measure P x is the law of X started at x, and for a measure µ on E, P µ denotes E P x (·)µ(dx). Finally, for t ≥ 0, θ t is the shift operator: X s • θ t = X s+t . We adhere to the convention that a function (resp. measure) on E (resp. E * ) is extended to ∆ by declaring its value at ∆ (resp. {∆}) to be zero.
We fix once and for all an excessive measure m. Thus, m is a σ-finite measure on (E, E * ) and mP t ≤ m for all t > 0. Since X is a right process, we then have lim Recall that a set B is m-polar provided P m (T B < ∞) = 0, where T B := inf{t > 0 : X t ∈ B} denotes the hitting time of B. A property or statement P (x) will be said to hold quasieverywhere (q.e.), or for quasi-every x ∈ E, provided it holds for all x outside some m-polar subset of E. It would be more proper to use the term "m-quasi-everywhere," but since the measure m will remain fixed the abbreviation to "q.e." will cause no confusion. Similarly, the qualifier "a.e. m" will be abbreviated to "a.e." On the other hand, certain terms (e.g., polar) have a longstanding meaning without reference to a background measure, and so we shall use the more precise term "m-polar" to maintain the distinction. Notice that any finely open m-null set is m-polar. Consequently, any excessive function vanishing a.e. vanishes q.e. A set B ⊂ E is m-semipolar provided it differs from a semipolar set by an m-polar set. It is known that B is m-semipolar if and only if P m (X t ∈ B for uncountably many t) = 0.
See [A73]. A set B is m-inessential provided it is m-polar and E r B is absorbing. According to 12)] an m-polar set is contained in a Borel m-inessential set. Since m is excessive it follows that sets of potential zero are m-null. In particular m-polar and m-semipolar sets are m-null. and D c r D cr is semipolar. Thus D p r D is always semipolar. In many situations it is in fact m-polar and we shall say that D is m-regular when D p r D is m-polar. However we shall state explicitly when we assume D is m-regular.
Definition 2.1 A continuous additive functional, A, of (X, τ ) is a real valued process A = A t (ω) defined on 0 ≤ t < τ(ω) if τ (ω) > 0 and for all t ≥ 0 if τ (ω) = 0, for which there exists a defining set Λ ∈ F and an m-inessential set N ⊂ D p -called an exceptional set for A-such that: Note that if ω ∈ Λ and τ (ω) > 0 it follows from (ν) that A 0 (ω) = 0. If A is increasing and we define for ω ∈ Λ and t ≥ τ (ω), A t (ω) := lim s↑τ (ω) A s (ω), then for ω ∈ Λ; s, t ≥ 0. We denote the totality of all continuous additive functionals of (X, τ ) by A(D) and by A + (D) the increasing elements of A(D). If A ∈ A(D), ω ∈ Λ and t < τ(ω) define |A| t (ω) to be the total variation of s → A s (ω) on [0, t]. Then it is routine to check that |A| ∈ A + (D) with the same defining and exceptional sets. Hence A + := 1 2 [|A| + A] and with the same defining and exceptional sets and A = A + − A − . Two elements A, B ∈ A(D) are equal provided they are m-equivalent; that is they have a common defining set Λ and a common exceptional set N such that Of course we are using m-equivalence as our definition of equality in A(D).

Definition 2.3 The Revuz measure associated with
is the q-potential operator associated with A. Since A is continuous a.s. P m , it is clear that ν A charges no m-semipolar set. It is evident that ν A is carried by D and since E · t 0 e −qs dA s and V q A f vanish on D cr one may replace m by m D -its restriction to D-in (2.4) because m doesn't charge the semipolar set D p r D. Moreover it is known that ν A is σ-finite [Re70, III.1]. It is also known that ν A determines A (up to m-equivalence). See also [FG88] and [FG96] as well as [Re70] for further details on Revuz measures. Finally we have the classical uniqueness theorem.

Definition 2.6 A (positive) measure ν on D is smooth provided it is the Revuz measure of an
t<τ. Clearly It is convenient to let Q q t := e −qt Q t and let I q denote this class of functions.
the condition in (2.7) seems natural and turns out to be appropriate. Of course if D is m-regular so that D p r D is m-polar, then the conditions are equivalent. We emphasize that the exceptional m-polar set off of which h = Q q t h on D is independent of t. There is another characterization of I q in terms of the stopped process which helps explain some of the results in the next section. Let τ * = inf{t ≥ 0 : X t / ∈ D}. Then the stopped process X a t := X t∧τ * has state space E. Note that τ * = τ a.s. P x for x ∈ D. Define the stopped semigroups for q ≥ 0 It is easily verified that the a Q q t are indeed semigroups. We introduce the exit operators and P q τ * f defined similarly. Again we write P τ and P τ * when Again the exceptional set does not depend on t and since a Q q t u = u on D c the critical condition is that u = a Q q t u q.e. on D. Let I q a denote this class of functions.
Proposition 2.10 Let u be finite q.e. Then u ∈ I q a if and only if P q τ * |u| < ∞ q.e. and u = P q τ u + h q.e. on D where h ∈ I q .
Remark Recall that h = 0 and P q τ * u = u on D c . Hence one may replace the equality in (2.10) by u = P q τ * u + h q.e. Proof. Suppose u ∈ I q a . Let N be m-inessential and such that for (2.11) In addition the last equality in (2.11) also holds when u is replaced by |u|. Therefore Therefore h exists and the preceding relations hold on D r N and because N c is absorbing, if where the equalities and inequalities in this and preceding sentence hold identically on D r N . Hence h ∈ I q and one half of (2.10) is established.
Hence sup t a Q q t |h| < ∞ q.e. and h = a Q q t h q.e. Also recalling that P q τ u = P q τ * u on D and writing u * = |u| for notational simplicity Hence sup t a Q q t P q τ * |u| < ∞ q.e. and the same computation now shows a Q q t P q e. and so u ∈ I q a .

The Basic Machinery
In this section we shall develop the necessary machinery which will enable us to define the extended generator in the next section. The notation is that of the preceding section. The next result is basic.
Then one readily checks that for q.e. x and then Combining these results and the facts that for q.e. x, the measure But (3.2) and (3.3) hold everywhere if u is replaced by |u| and A by |A|.
e., completing the proof of 3.1.

Remark 3.4
In the course of the proof of (3.1) it was shown that V p |u| = E · τ 0 e −pt |u|(X t ) dt < ∞ q.e. It follows that t → t 0 e −ps u(X s ) ds on [0, τ[ is in A(D) and has Revuz measure um D where m D is the restriction of m to D. In particular um D ∈ S(D). Also as remarked in section 2, if V q |A| 1 < ∞ a.e., then it is finite q.e. and so it would suffice to suppose that V q |A| 1 < ∞ a.e. in (3.1).

Definition 3.5 A function f is quasi-finely continuous on D provided there exists an m-
inessential set N f such that f is finite and finely continuous on the finely open set D r N f . We abbreviate this by saying that f is q-f -continuous on D.
Proposition 3.6 Let u satisfy the hypotheses of (3.1). Then u and h are q-f -continuous on D and is a P x uniformly integrable right continuous martingale for q.e. x ∈ D.
Proof. Let N be a Borel m-inessential set which contains the union of the sets {|u| = ∞}, {|h| = ∞}, {P q τ |u| = ∞}, {V q |A| 1 = ∞} and {u = P q τ u + V q A 1 + h}. We shall first show that (Y t ) is a uniformly integrable strong martingale for each x ∈ D r N . Fix such an x and let Y τ := e −qτ u(X τ )1 {τ <∞} + τ 0 e −qs dA s . Given a bounded stopping time T we shall show that Since τ is a terminal time the last term equals This time the last term becomes Combining these calculations gives If p > q then what we have shown so far and (3.1) yield the fact that Z t := e −p(t∧τ ) u(X t∧τ ) + t∧τ 0 e −ps dB s is a P x uniformly integrable strong martingale for x ∈ D r L where L is an minessential set and B is defined in (3.1). Fix such an x. Since u is nearly Borel, Z is optional. Then the optional section theorem [DM, IV-(8.6)] implies that Z is indistinguishable from its right continuous modification. But t → t∧τ 0 e −ps dB s is continuous a.s. P x and hence t → u(X t ) is right continuous P x a.s. on [0, τ[. It now follows that u is finely continuous on D r L.
Finally the same argument shows that Y defined in (3.7) is right continuous a.s. P x for q.e.
x ∈ D. From this and the right continuity of t → u(X t∧τ ) we conclude that h is q-f -continuous on D.
Theorem 3.9 Let q ≥ 0. Given a function u, A ∈ A(D) and h ∈ I q such that |u|, P q τ |u| and V q |A| 1 are finite q.e., then u = P q τ u + V q A 1 + h q.e. on D if and only if h = lim t→∞ Q q t u q.e. on D and Y t defined in (3.7) is a P x uniformly integrable right continuous martingale for q.e. x ∈ D.
Proof. Fix an m-inessential set N as in the first sentence of the proof of (3.6). Then for In view of (3.6) this proves one half of (3.9). Conversely suppose Y is a P x uniformly integrable right continuous martingale. Then Y ∞ := lim t→∞ Y t exists a.s. and in L 1 relative to P x . But a.s. P x , where for definiteness we define Then (3.10) holds a.s. P y for all y ∈ D r N . Let k(y) = E y (Z). Then k(y) exists finite for y ∈ D r N and one easily checks that k = Q q t k on D r N . Taking expectations in (3.10), e. on D and as in the first part of the proof Q q t P q τ u → 0 and Q q t V q A 1 → 0 q.e. on D as t → ∞. Therefore k = 0 q.e. on D completing the proof of (3.9).

The Extended Generator
In this section we shall define an operator Λ D which we regard as an extension of the generator of X restricted to D. It will map (equivalence classes modulo m of) functions defined on E into S(D). However with the usual abuse of notation we shall regard it as a map from functions to S(D). Ifũ = u a.e., then we say thatũ is a version of u.

Definition 4.1 The domain D(Λ D ) of Λ D consists of functions u on E which have a versionũ
which is finite q.e. and such that there exist q ≥ 0, A ∈ A(D) and h ∈ I q with P q τ |ũ| and V q |A| 1 finite q.e. and satisfying q.e. on D Here m D is the restriction of m to D and ν A is the Revuz "measure" of A as defined in section 2.
Remark Since only um D appears in the expression for Λ D u it is clear that Λ D u depends only on the equivalence class mod m containing u. See (3.4) for the fact that um D ∈ S(D). It is often convenient to write Λ D u = qu| D − ν A rather than qum D − ν A .
In order to simplify the notation in what follows we shall suppose that we have chosenũ as a version of u; that is u itself satisfies the conditions imposed onũ and we shall drop the notatioñ u. The next result justifies the definition of Λ D .

Theorem 4.3 Λ D is a well-defined linear map from D(Λ D ) to S(D).
Proof. We shall first show that D(Λ D ) is a vector space. We often omit the qualifying phrase "q.e. on D" where it is clearly required. Obviously if u ∈ D(Λ D ) and α ∈ R, then αu ∈ D(Λ D ).
If p > q, then according to Lemma 3.1, u = P p τ u+V p B 1 where B is defined in Lemma 3.1. Hence u+v = P p τ (u+v)+V p C+B 1+k and the appropriate finiteness conditions are satisfied. Consequently u + v ∈ D(Λ D ).
Next we show that Λ D is well-defined. Once again we omit the "q.e. on D" in places where it is obviously required.
This implies that the PCAF's A + +C − and A − +C + have the same finite q-potential relative to (X, τ ). Hence A = C and then Remark 4.4 Because of Proposition 3.6, u-that is the versionũ in (4.1)-is q-f -continuous on D. Thus we may suppose that u is q-f -continuous on D when u ∈ D(Λ D ). We stress that elements in D(Λ D ) are defined on E although they may vanish off D.
Here are some examples. Let u = V q f where V q |f | < ∞ a.e. and hence q.e. Then u = 0 on D cr and so 1 D u = u a.e. and P q τ 1 D u = 0. Therefore Note that P q τ u itself need not vanish a.e. on D, let alone q.e. on D, unless D is m-regular. If u = U q f with U q |f | < ∞ a.e., then u = P q τ u + V q f so that u ∈ D(Λ D ) and Λ D u = (qu − f )m D . If, for example, q > 0 and f bounded, then u = U q f is in the domain of the "generator" Λ of X and Λu = qu − f . In this case Λ D u is the restriction of Λu to D. As a final example if A ∈ A(D) and for some

The Schrödinger Equation
The assumptions and notation are as in the previous sections. If q ≥ 0 and µ ∈ S(D) are fixed, we consider the equation where ν ∈ S(D). A solution u of (5.1) is an element u ∈ D(Λ D ) such that One could absorb the parameter q into µ by replacing µ by µ − qm D . But the basic data are µ and ν and one is often interested in the dependence of the solution on q and so it is preferable to keep q explicitly in (5.1) We need to introduce some notation and prepare several lemmas before discussing existence and uniqueness results for (5.1).
Let A, B ∈ A(D) and q ≥ 0. Define the following operations on functions whenever the integrals make sense: For example if A ∈ A(D), B ∈ A + (D) and f ≥ 0 then these integrals exist although they might be identically infinite. Note that in the notation of the previous sections V q,0

Lemma 5.5 Let
The last term in the display equals proving the first equality in (5.6). A similar argument using the identity e At = 1+e At t 0 e −As dA s completes the proof of (5.6). Suppose A, B ∈ A(D). If V q,|A| |B| |f |(x) < ∞ and (5.6) holds for this x with |A|, |B| and |f |, then the previous manipulations are valid and the assertion in the second sentence of (5.5) holds.

Lemma 5.7 Let A ∈ A(D), B ∈ A + (D) and A = A + − A − be the decomposition defined below (2.2). If
f are finite q.e. and q-f -continuous on D.
Proof. Let N = N A ∪ N B where N A and N B are the exceptional sets for A and B. On E r N one has The last term equals A similar computation shows that A simple integration by parts shows that the expressions in square brackets in the last two displays are equal proving the equality in (5.8). But 1 + Combining this with the last displayed expression yields the inequality in (5.8).
f are finite q.e. But the first three functions in the last sentence are q-excessive for (X, τ ) and, hence, finely continuous on D p . It follows that V q

Remark 5.9
In fact part of the last assertion may be improved. Namely it is readily checked that V q,A B f is q-excessive for the subprocess of (X, τ ) corresponding to the multiplicative function [BG68,]. Consequently if V q,A B f < ∞ a.e., it is finite q.e. and q-fcontinuous on D. Therefore if A, B ∈ A(D) and f arbitrary with V q,A |B| |f | < ∞ a.e., it follows that V q,A B f is q-f -continuous on D, while a direct application of (5.7) would require V q,A + |B| |f | < ∞ a.e. We next formulate a general existence and uniqueness theorem for solutions of (5.1). Subsequently we shall investigate conditions which guarantee that its hypotheses hold.
Proof. By (5.7), V q,A + |B| 1 is, in fact, finite q.e. whenever it is finite a.e. Also if w is a function with V q,A |A| |w| < ∞ a.e., then in light of (5.9) it is finite q.e. Thus in using (5.13) one may replace a.e. by q.e. Since (5.1) is linear and the hypotheses involve |B| only, in showing that u is a solution we may, and shall, suppose ν ≥ 0 so that B ∈ A + (D). We shall omit the qualifier "q.e. on D" in those places where it is clearly required.
1 and hence V q |A| u is finite. Thus from (5.5), |A| u are finite. In the proving the uniqueness assertion we can no longer assume ν ≥ 0. By hypothesis u = V q,A B 1 satisfies (5.13). The following lemma is the key step in proving uniqueness.
We shall use (5.14) to complete the proof of Theorem 5.10 before giving the proof of (5.14). Let u 1 and u 2 be solutions of (5.1) satisfying (5.11), (5.12) and (5.13). We may suppose that u 1 and u 2 are q-f -continuous on D. Then so is v := u 1 − u 2 . By (5.14), v = V q A v and V q |A| |v| < ∞ a.e. and then q.e. on D since V q |A| |v| is q-excessive for (X, τ ). By hypothesis q.e. on D, V q,A |A| |v| < ∞. Now using (5. We shall now prove (5.14) which will complete the proof of (5.10). Since v ∈ D(Λ D ) we may choose a version of v which is q-f -continuous of D and vanishes on D c . Then P q τ v = 0 and so v = V q C 1 + h where C ∈ A(D) satisfies V q |C| 1 < ∞ and h ∈ I q . Again we omit "q.e. on D". But Q q t V q |C| 1 → 0 and hence h = 0 a.e. and then being q-f -continuous on D, q.e. on D. Now Remarks 5.15 (i) One can also establish the existence of a solution under slightly different conditions. Throughout these remarks we omit the phrase "q.e. on D". For example suppose V q,A |B| 1, V q |B| 1 and V q,A |A| V q,A |B| 1 are finite, then u = V q,A B 1 is the unique solution of (5.1) satisfying (5.11), (5.12) and (5.13). To prove that u = V q,A B 1 is a solution we may suppose ν ≥ 0. Clearly u and V q,A |A| |u| are finite. Our finiteness assumptions are strong enough to justify the following The remainder of the argument is the same as before. (ii) The hypotheses in (5.10) that imply that u = V q,A B 1 is a solution of (5.1) involve only A + ; A − could be any element of A + (D). However the uniqueness hypotheses involve A − through (5.13). The conditions for a solution in (i) above involve both A + and A − , although A + in a rather different manner.
(iii) A straightforward argument using the Fubini theorem shows that if A ∈ A(D), B and C in A + (D) and f ≥ 0, then Taking C = |A| it follows that a sufficient condition that (5.13) holds is that We shall specialize (5.10) and (5.15-i)in several directions. The next lemma contains relationships that are needed to establish the results to follow. It complements (5.7).
Lemma 5.16 Let A ∈ A(D), f ≥ 0 and q ≥ 0. Then (5.17) and Proof. If t < τ an integration by parts yields Multiply this by e −qt f (X t ), integrate from 0 to τ and then take expectations with respect to P x for x / ∈ N A -the exceptional set for A-to obtain (5.17). For (5.18), integrating by parts one has for t < τ Drop the positive term e −qt e At , let t ↑ τ and then take P x expectations for x / ∈ N A to obtain (5.18).
Remarks It follows from (5.18) that q.e. V q,A Also q.e. using (5.18) for q > 0 and (5.17) for q = 0 In what follows we often omit the "q.e. on D" in proofs where it is obviously needed, but we include it in our hypotheses.

Remarks
The proof actually shows that u is the unique solution in the class of all solutions satisfying (5.11), (5.12) and (5.13). Since a solution is only determined a.e., here bounded really means in L ∞ (m). Also if q > 0, then Q q t u → 0 as t → ∞ is automatic for u bounded. We next give a somewhat different criterion for existence and uniqueness in the spirit of section 3.3 of [CZ95]. Given µ ∈ S(D) we define, following Chung and Zhao, F = F(D, µ) to consist of those ν ∈ S(D) for which there exist constants α and β-depending on ν-such that |ν| ≤ αm + β|µ|. (5.22) Let A correspond to µ.
Theorem 5.23 Let ν ∈ F(D, µ). (i) If q > 0 and q.e. on D, V q |A| 1 < ∞ and V q,A A + 1 is bounded, then u = V q,A B 1 is the unique bounded solution of (5.1) vanishing a.e. on D c . (ii) If q = 0 and, in addition, q.e. on D either V 1 = E · (τ ) is bounded or V A 1 is bounded and V 1 < ∞, then the same conclusion holds. If β = 0 in (5.22), then the condition V q |A| 1 < ∞ is not needed in either (i) or (ii).
|A| 1 is bounded. ¿From (5.20) if q > 0, V q,A 1 is bounded, and if q = 0 and V 1 is bounded then so is V A 1. Thus in all cases both V q,A A + 1 and V q,A 1 are bounded. The inequality (5.22) implies that d|B| t ≤ αdt + βd|A| t . Therefore V q,A |B| 1 ≤ αV q,A 1+βV q,A |A| 1 is bounded and hence so is u = V q,A B 1. Moreover V q |B| 1 ≤ αV q 1+βV q |A| 1 is finite. Clearly when β = 0, the hypothesis on V q |A| 1 is not needed. Using (5.19) again V q,A |A| V q,A |B| 1 is bounded. Therefore according to (5.15-i) u = V q,A B 1 is the unique solution of (5.1) vanishing a.e. on D c and satisfying (5.12) and (5.13). But under the present hypotheses any bounded function f satisfies (5.12) and (5.13) since when q = 0, We next give some, perhaps more familiarly, conditions guaranteeing the hypotheses of (5.23).
The usual Kato condition, for example as in [CZ95], assumes lim Proof. (i) By (5.25) if q > ω, then q.e. on D For t > 0, integrating by parts gives Thus if t > 0, Remarks If A + satisfies the Kato condition, then for q > ω , V q,A A + 1 ≤ V q,A + A + 1 and V q,A 1 ≤ V q,A + 1 are bounded q.e. Thus if V q A − 1 is finite q.e. the hypotheses in (5.23-i) hold. If µ + << m and A + satisfies the Kato condition and if V A + 1 is bounded q.e., then the hypotheses in the (5.23-ii) hold provided E · (A − τ ) < ∞ q.e. In [G99] we gave a condition that implies (5.24) that involves µ more directly. Namely let (V q ) be the resolvent of the moderate Markov dual relative to m of (X, τ ). Then condition (5.24) holds for A ∈ A + (D) provided that for some q < ∞ one has

L p Theory
In this section we shall investigate the situation where ν << m and under the assumption that f = dν dm ∈ L p (m, D), 1 ≤ p ≤ ∞. Let L p = L p (m, D) be the real L p space over (D, m). We shall need the moderate Markov dual of (X, τ ) which we denote by (X,τ ). If A ∈ A + (D) thenÂ is its dual as defined in section 4 of [G99]. In particular A andÂ have the same Revuz measure. If A ∈ A(D), thenÂ :=Â + −Â − . Also (V q ) and (Q t ) denote the resolvent and semigroup of (X,τ ). For notational simplicity we writeV q,A andV q,A B in place ofV q,Â andV q,Â B and so forth. The following duality relations are crucial. Let (f, g) := f gdm whenever the integral makes sense, not necessarily finite. In the next two propositions, A ∈ A + (D) with Revuz measure µ and f, g ≥ 0.
See [G99] Proposition 4.8 for the first assertion and Proposition 4.6 for the second. We shall say that a kernel W on D is m-proper provided there exists h > 0 on D with W h < ∞ a.e. The next result is (5.6) in [G99]. See also (5.8) in [G99].

Proposition 6.2 Let
We refer the reader to [G99] for information about the dual process but warn him that the notation is slightly different there.
We are now going to investigate the equation (5.1) when ν = f m with f ∈ L p . This means that . Then |ν| = |f |m and |ν|, hence ν + and ν − , are in S(D) provided t 0 |f |(X s ) ds < ∞ on 0 ≤ t < τ a.s. P × for q.e. x ∈ D. This certainly is the case in V q |f | < ∞ q.e. for some q ≥ 0. But m is excessive and so if q > 0, V q : L p → L p , 1 ≤ p ≤ ∞. Therefore ν = f m ∈ S(D) whenever f ∈ L p . Then equation (5.1) is written The next lemma is necessary in order to show that V q,A is well-defined on L p .

Lemma 6.4 Let
Proof. Using (6.1-i), if f ≥ 0 then (f, V q,|A| 1 J ) = (V q,|A| f, I J ) = 0. Therefore V q,|A| 1 J and V q,A 1 J vanish a.e. The second assertion is an immediate consequence of the first and (5.9).
In what follows µ ∈ S(D) and A corresponds to µ. Any additional hypotheses on µ or A will be explicitly stated. The case p = ∞ is a simple corollary of (5.23). If ν = f m with f ∈ L ∞ , then |ν| = |f |m ≤ f ∞ m and so ν satisfies (5.22) with β = 0. In this situation dB t = f (X t ) dt and V q,A B 1 = V q,A f . Thus the following proposition is an immediately consequence of (5.23) and (6.4).

Proposition 6.5
Let ν = f m with f ∈ L ∞ . Suppose that V q,A A + 1 is bounded q.e. on D and when q = 0 suppose, in addition, that q.e. on D either E · (τ ) is bounded or V A 1 is bounded and E · (τ ) < ∞. Then u = V q,A f is the unique solution of (6.3) in L ∞ that vanishes on D c .
When 1 ≤ p < ∞ the following space of functions turns out to be the space in which solutions are unique. Define W p (µ) = {u ∈ L p : u has a q-f -continuous versionũ with |ũ| p d|µ| < ∞}. (6.6) We next consider the case p = 1.
Theorem 6.7 Let ν = f m with f ∈ L 1 . Suppose thatV q,|A| |A| 1 is bounded q.e. and in addition, when q = 0 thatV |A| 1 is bounded q.e. Then u = V q,A f is the unique solution of (6.3) in W 1 (µ) vanishing a.e. on D c and with lim t→∞ e −qt Q t u = 0 a.e.
Consequently V q,A : L 1 → W 1 (µ). Thus u = V q,A f is a solution in W 1 (µ). On the other hand if w ∈ W 1 (µ) one may suppose that w itself is q-f -continuous, then using (6.2) again V q,|A| |A| |w| dm = |w|V q,|A| 1 d|µ| < ∞ becauseV q,|A| 1 is bounded q.e. and hence a.e. |µ| since |µ| does not charge m-semipolars. It now follows from (5.10) that u = V q,A f is the unique solution of (6.3) in W 1 (µ) which vanishes on D c and satisfies lim t→∞ e −qt Q t u = 0 a.e. Remarks 6.8 (i) The proof shows that u is the unique solution subject to (5.11) and (5.12) in the possibly larger class of solutions w having a q-f -continuous versionw satisfying |w|V q,|A| 1 d|µ| < ∞. (ii) Using (2.9), (2.6) and (2.3) of [G99] one can show that ifV q,|A| |A| 1 andV |A| 1 are bounded a.e., then they are bounded q.e. Hence the hypotheses in (6.7) may be relaxed to this extent.
When 1 < p < ∞ the situation is similar but we require the dual assumptions as well as those of (6.7).

Theorem 6.9
Let ν = f m with f ∈ L p , 1 < p < ∞. Suppose that V q,|A| |A| 1 andV q,|A| |A| 1 are bounded q.e. and, in addition, when q = 0, that V |A| 1 andV |A| 1 are bounded q.e. Then u = V q,A f is the unique solution of (6.3) in W p (µ) vanishing a.e. on D c and with lim t→∞ e −qt Q t u = 0 a.e.
Thus by (6.4), V q,|A| and V q,A map L p into L p and for f ∈ L p , V q,|A| |f | and V q,|A| |f | p are determined q.e. Fix f ∈ L p . Then u = V q,A f is a solution of (6.3) and u ∈ L p . From (5.7), u is q-f -continuous and using (6.2) V q,|A| |f | p d|µ| ≤ c V q,|A| |f | p d|µ| = c |f | pV q,|A| |A| 1 dm < ∞.
Hence V q,|A| and V q,A map L p into W p (µ). If w ∈ W p (µ) and w is q-f -continuous, Therefore as in the proof of (6.7) V q,|A| |A| |w| p dm ≤ c |w| pV q,|A| 1 d|µ| < ∞ and V q,|A| |A| |w| < ∞ a.e. Now an appeal to (5.10) completes the proof of (6.9).
As remarked in (6.8) it would suffice to suppose thatV q,|A| |A| 1 andV |A| 1 are bounded a.e. Of course (5.7) implies that if V q,|A| |A| 1 and V |A| 1 are bounded a.e. then they are bounded q.e.

Final Remark
Let µ = (µ + , µ − ) with µ + satisfying (5.27) relative to X (that is with D = E) and µ − smooth. Let (Q t ) and (Q t ) be the dual semigroups corresponding to A = A µ andÂ. See [G99] for the precise definitions. Then the arguments in section 5 of [SV96] are readily modified to show that if (P t ) and (P t )-the semigroups of X andX-are continuous from L 1 (m) to L ∞ (m), then (Q t ) and (Q t ) are continuous from L p (m) to L q (m) for 1 ≤ p ≤ q ≤ ∞. This was supposed to appear as an added note in [G99], but somehow was omitted by the printer.