Stochastic Calculus for Symmetric Markov Processes

Transcription

1 Stochastic Calculus for Symmetric Markov Processes (Dedicated to S. Nakao on the occasion of his 6th birthday Z.-Q. Chen, P. J. Fitzsimmons, K. Kuwae and T.-S. Zhang (Revised Version: April 18, 27 Abstract Using time-reversal, we introduce a stochastic integral for zero-energy additive functionals of symmetric Markov processes, extending earlier work of S. Nakao. Various properties of such stochastic integrals are discussed and an Itô formula for Dirichlet processes is obtained. AMS 2 Mathematics Subject Classification: Primary 31C25; Secondary 6J57, 6J55, 6H5. Keywords and phrases: Symmetric Markov process, time reversal, stochastic integral, generalized Itô formula, additive functional, martingale additive functional, dual additive functional, Revuz measure, dual predictable projection. The research of this author is supported in part by NSF Grant DMS-626. The research of this author is supported by a foundation based on the academic cooperation between Yokohama City University and UCSD The research of this author is supported by a foundation based on the academic cooperation between Yokohama City University and UCSD, and partially supported by a Grant-in-Aid for Scientific Research (C No from Japan Society for the Promotion of Science. The research of this author is supported in part by the British PSRC 1

2 1 Introduction and Framework It is well-known that stochastic integrals and Itô s formula for semimartingales play a central role in modern probability theory. However there are many important classes of Markov processes that are not semimartingales. For example, symmetric diffusions on R d whose infinitesimal generators are elliptic operators in divergence form L = ( d i,j=1 x i a ij (x x j with merely measurable coefficients need not be semimartingales. ven when X is a Brownian motion in R d and u W 1,2 (R d := {u L 2 (R d ;dx u L 2 (R d ;dx}, the process u(x t is not a semimartingale in general. To study such processes, Fukushima obtained the following substitute for Itô s formula (see [8]: for u W 1,2 (R d, u(x t = u(x + Mt u + Nu t for t, (1.1 P x -a.s. for quasi-every x R d, where M u is a square-integrable martingale and N u is a continuous additive functional of zero energy. The decomposition (1.1 is called Fukushima s decomposition and holds for a general symmetric Markov process X and for u F, where (, F is the Dirichlet space for X. In this paper, a stochastic process ξ = {ξ t,t } under some σ-finite measure P is called a Dirichlet process if ξ has locally finite quadratic variation under P. The composite process u(x is a Dirichlet process under P m, where m is the Lebesgue measure on R d, as it has finite quadratic variation on compact time intervals. Nakao introduced a stochastic integral f(x sdn u s in [17] by using a Riesz representation theorem in a suitably constructed Hilbert space. Nakao s stochastic integral played an important role in the study of lower order perturbation of diffusion processes by Lunt, Lyons and Zhang [15] and by Fitzsimmons and Kuwae [6]. However Nakao s definition of the stochastic integral f(x sdn u, requiring u to be in the domain of the Dirichlet form of X and f to be square-integrable with respect to the energy measure of u, is too restrictive to be useful in the study of lower order perturbation for symmetric Markov processes with discontinuous sample paths, such as stable processes. Such a study requires stochastic integrals for more general integrators as well as integrands. The purpose of this paper is to present a new way of defining the stochastic integral for Dirichlet processes associated with a symmetric Markov process. Our new approach uses only the time-reversal operator for the process X, and is thereby more direct and provides additional insight into stochastic integration for Dirichlet processes. This approach enables us to define Λ(M (see (1.5 for any locally square-integrable martingale additive functional (MAF M, subject to some mild conditions. Thus it not only recovers Nakao s results in [17] but also extends them significantly. The new stochastic integral allows us to study various transforms for symmetric Markov processes, a project that is carried out in a subsequent paper [2]. A more detailed description of the current paper appears below. Let X = {Ω, F, F t,x t,θ t,ζ,p x,x } be an m-symmetric right Markov process with a Lusin state space, where m is a σ-finite measure with full support on. Its associated Dirichlet space (, F on L 2 (;m is known to be quasi-regular (see [16]. By [1], (, F is quasihomeomorphic to a regular Dirichlet space on a locally compact separable metric space. Using this quasi-homeomorphism, there is no loss of generality in assuming that X is an m-symmetric 2

3 Hunt process on a locally compact metric space such that its associated Dirichlet space (, F is regular on L 2 (;m and that m is a positive Radon measure with full topological support on. This we do throughout the sequel. Without loss of generality, we can take Ω to be the canonical path space D([, [ of right-continuous, left-limited (rcll, for short functions from [, [ to, for which is a trap (that is, if ω(t = then ω(s = for all s > t. For any ω Ω, we set X t (ω := ω(t. Let ζ(ω := inf{t X t (ω = } be the lifetime of X. As usual, F and F t are the minimal augmented σ-algebras obtained from F := σ{x s s < } and Ft := σ{x s s t}, respectively, under P x ; see the next section for more details. We sometimes use a filtration denoted by (M t on (Ω, M in order to represent several filtrations, for example, (Ft, (F t+ on (Ω, F, (F t on (Ω, F and others introduced later. We use θ t to denote the shift operator defined by θ t (ω(s := ω(t + s, t,s. Let ω be the path starting from. Then ω (s for all s [, [. Note that θ ζ(ω (ω = ω if ζ(ω <, {ω } F F t for all t > and P x ({ω } P x (X = = for x. For a Borel subset B of, τ B := inf{t > X t / B} (the exit time of B is an (F t -stopping time. If B is closed, then τ B is an (Ft+-stopping time. Also, ζ is an (Ft -stopping time because {ζ t} = {X t = } Ft, t. The transition semigroup of X, {P t,t }, is defined by P t f(x := x [f(x t ] = x [f(x t : t < ζ], t. ach P t may be viewed as an operator on L 2 (;m; collectively these operators form a strongly L 2 -continuous semigroup of self-adjoint contractions. The Dirichlet form associated with X is the bilinear form 1 (u,v := lim t t (u P tu,v m defined on the space { F := u L 2 (;m sup t> Here we use the notation (f,g m := f(xg(xm(dx. } t 1 (u P t u,u m <. For the reader s convenience, we recall the following definitions from [16] and [8]. Definition 1.1 (i An increasing sequence {F n } n 1 of closed subsets of is an -nest (or simply nest if and only if n 1 F Fn is 1 -dense in F, where 1 = + (, L 2 (,m and F Fn := {u F : u = m-a.e. on \ F n }. (ii A subset N is -polar if and only if there is an -nest {F n } n 1 such that N n 1 (\F n. (iii A function f on is said to be quasi-continuous if there is an -nest {F n } n 1 such that f Fn is continuous on F n for each n 1; we denote this situation briefly by writing f C({F n }. 3

4 (iv A statement depending on x A is said to hold quasi-everywhere (q.e. in abbreviation on A if there is an -polar set N A such that the statement is true for every x A \ N. (v A nearly Borel subset N is called properly exceptional if m(n = and P x (X t \ N for t and X t \ N for t > = 1 for every x \ N. It is known (cf. [8] that a family {F n } of closed sets is an -nest if and only if ( P x lim τ F n n = ζ = 1 for q.e. x. It is also known that a properly exceptional set is -polar and that every -polar set is contained in a properly exceptional set. very element u in F admits a quasi-continuous m-version. We assume throughout this section that functions in F are always represented by their quasi-continuous m-versions. In the sequel, the abbreviations CAF, PCAF and MAF stands for continuous additive functional, positive continuous additive functional and martingale additive functional, respectively; the definitions of these terms can be found in [8]. Let M and N c denote, respectively, the space of MAFs of finite energy and the space of continuous additive functionals of zero energy. For u F, Fukushima s decomposition holds: u(x t u(x = M u t + N u t, for every t [, [, (1.2 P x -a.s. for q.e. x, where M u M and N u N c. A positive continuous additive functional (PCAF of X (call it A determines a measure µ = µ A on the Borel subsets of via the formula [ 1 t ] µ(f = lim t t m f(x s da s, (1.3 in which f : [, ] is Borel measurable. Here lim t indicates an increasing limit as t. The measure µ is necessarily smooth, in the sense that µ charges no -polar set of X and there is an -nest {F n } of closed subsets of such that µ(f n < for each n N. Conversely, given a smooth measure µ, there is a unique PCAF A µ such that (1.3 holds with A = A µ. In the sequel we refer to this bijection between smooth measures and PCAFs as the Revuz correspondence, and to µ as the Revuz measure of A µ. If M is a locally square-integrable martingale additive functional (MAF of X on the random time interval [,ζ [, then the process M (the dual predictable projection of [M] is a PCAF (Proposition 2.8, and the associated Revuz measure (as in (1.3 is denoted by µ M. More generally, if M u is the martingale part in the Fukushima decomposition of u F, then M u,m is a CAF locally of bounded variation, and we have the associated Revuz measure µ M u,m, which is locally the difference of smooth (positive measures. For u F, the Revuz measure µ M u of M u will usually be denoted by µ u. 4

5 Let (N(x,dy,H t be a Lévy system for X; that is, N(x,dy is a kernel on (, B( and H t is a PCAF with bounded 1-potential such that for any nonnegative Borel function φ on vanishing on the diagonal and any x, x [ ] φ(x s,x s = x φ(x s,yn(x s,dydh s. s t To simplify notation, we will write Nφ(x := φ(x,yn(x,dy and (Nφ H t := Nφ(X s dh s. Let µ H be the Revuz measure of the PCAF H. Then the jumping measure J and the killing measure κ of X are given by J(dx,dy = 1 2 N(x,dyµ H(dx, and κ(dx = N(x, { }µ H (dx. These measures feature in the Beurling-Deny decomposition of : for f,g F, (f,g = (c (f,g + (f(x f(y(g(x g(yj(dx,dy + f(xg(xκ(dx, where (c is the strongly local part of. For u F, the martingale part M u t in (1.2 can be decomposed as M u t = M u,c t + M u,j t + M u,κ t for every t [, [, P x -a.s. for q.e. x, where M u,c t M u,j t M u,κ t = = lim ε is the continuous part of the martingale M u, and (u(x s u(x s 1 { u(xs u(x s >ε}1 {s<ζ} <s t ( t (u(y u(x s N(X s,dy {y : u(y u(x s >ε} u(x s N(X s, { }dh s u(x ζ 1 {t ζ}, dh s } are the jump and killing parts of M u, respectively. All three terms in this decomposition of M u are elements of M. See Theorem A.3.9 of [8]. The limit in the expression for M u,j is in the sense of convergence in the norm of the space of MAFs of finite energy, and of convergence in probability under P x for q.e. x (see [8]., 5

6 Let Nc N c denote the class of continuous additive functionals of the form N u + g(x sds for some u F and g L 2 (;m. Nakao [17] constructed a linear map Γ from M into Nc in the following way. It is shown in [17] that, for every Z M, there is a unique w F such that 1 (w,f = 1 2 µ M f +M f,κ, Z ( for every f F. (1.4 This unique w is denoted by γ(z. The operator Γ is now defined by Γ(Z t := N γ(z t Nakao showed that Γ(Z is characterized by the following equation γ(z(x s ds for every Z M. (1.5 1 lim t t g m [Γ(Z t ] = 1 2 µ M g +M g,κ, Z ( for every g F b. (1.6 Here F b := F L (;m. So in particular we have Γ(M u = N u for u F. Nakao [17] then used the operator Γ to define a stochastic integral f(x s dn u s := Γ(f Mu t 1 2 Mf,c + M f,j, M u,c + M u,j t, (1.7 where u F, f F L 2 (;µ u and (f M u t := f(x s dm u s. If we define Ñ c := {N N c N = N u + A µ for some u F and some signed smooth measure µ}, then we see by (1.5 that f(x sdn u s Ñc if u F and f F L 2 (;µ u. However, the conditions imposed on the integrand f(x t and on the integrator N u in Nakao s stochastic integral are too restrictive for certain applications, in particular the perturbation theory of general symmetric Markov processes, which requires more general integrators as well as integrands; see [2]. The purpose of this paper is to give a new way of defining Γ(M and Nakao s stochastic integral for zero energy AFs N u. For a finite rcll AF M t, it is known (see [3, Lemma 3.2] that there is a Borel function ϕ on with ϕ(x,x = for all x so that M t M t = ϕ(x t,x t for every t ],ζ[, P m -a.e. (1.8 Such a ϕ is uniquely determined up to J-negligible sets. We will call ϕ the jump function of M. When M = M u, u F, the jump function ϕ for M u can be taken to be as ϕ(x,y = u(y u(x for (x,y, with u( :=. We have a similar result for locally square-integrable MAFs on [,ζ [ (see Definition 2.5(iii for the definition of locally square-integrable MAF on [,ζ [. Let M be a locally square-integrable MAF on [,ζ [. Then there exists a jump function ϕ on for M satisfying the property (1.8 (see Corollary 2.9. Assume ( ϕ 2 1 { bϕ 1} + ϕ 1 { bϕ >1} (Xs,yN(X s,dydh s < for every t < ζ, (1.9 6

7 P x -a.s. for q.e. x, where ϕ(x,y := ϕ(x,y + ϕ(y,x for x,y. By Lemma 3.2 below, there is a unique purely discontinuous local MAF K on [,ζ [ with Define P m -a.e. on [,ζ[, K t K t = ϕ(x t,x t for t < ζ, P x -a.s. for q.e. x, Λ(M t := 1 2 (M t + M t r t + ϕ(x t,x t + K t for t [, ζ[, where r t is the time-reversal operator at time t >. Note that since X is symmetric, the measure P m, when restricted to {t < ζ}, is invariant under r t. This time reversibility plays an important role in this paper. So Λ(M is clearly well-defined on [,ζ [ under the σ-finite measure P m. It will be shown in Theorem 2.18 and Remark 3.4(ii below that Λ(M is a continuous even AF of X on [,ζ [ admitting m-null set. Note that when M = M u for some u F, ϕ(x,y = u(y u(x is antisymmetric and so ϕ =. Thus P m -a.e. on {t < ζ}, Λ(M u t := 1 2 (Mu t + M u t r t + u(x t u(x t = N u t The last identity follows by applying the time-reversal operator to both sides of (1.2 and using the fact that N u t r t = N u t P m -a.e. on [,ζ [ (cf. [5, Theorem 2.1]. It follows then for every u F, Λ(M u = Γ(M u on [,ζ [ P m -a.e. We will show in Theorem 3.6 below that this holds when M u is replaced by any M M. Therefore, under the σ-finite measure P m, Λ is a genuine extension of Nakao s map Γ. A function f is said to be locally in F (denoted as f F loc if there is an increasing sequence of finely open Borel sets {D k,k 1} with k=1 D k = q.e., and for every k 1 there is f k F such that f = f k m-a.e. on D k. For two subsets A,B of, we denote A = B q.e. if A B := (A \ B (B \ A is -polar. By definition, every f F loc admits a quasi-continuous m-version, so we may assume all f F loc are quasi-continuous. Then we have f = f k q.e. on D k. For f F loc, M f,c is well defined as a continuous MAF on [,ζ [ of locally finite energy. Moreover, for f F loc and a locally square-integrable MAF M on [,ζ [, t (f M t := f(x s dm s is a locally square-integrable MAF on [, ζ [. Here for a locally square-integrable MAF M on [, ζ [ denote by M c its continuous part, which is also a locally square-integrable MAF on [,ζ [ (see Theorem 8.23 in [1]. Definition 1.2 (Stochastic integral Suppose that M is a locally square-integrable MAF on [,ζ [ and f F loc. Let ϕ : R be a jump function for M, and assume that ϕ satisfies condition (1.9. Define on [, ζ [ f(x s dλ(m s := Λ(f M t 1 2 Mf,c, M c t (f(y f(x s ϕ(y,x s N(X s,dydh s,

8 whenever Λ(f M is well defined and the third term in the right hand side of (3.1 is absolutely convergent. The above stochastic integral is well-defined on [,ζ [ under the σ-finite measure P m and extends that of Nakao (1.7. (See Remark 3.9(i and Theorem 3.1 below. We will show in Theorem 4.7 below that it enjoys a generalized Itô formula. 2 Additive functionals In this section, we will prove some facts about additive functionals, to be used later. We begin with some details on the completion of filtrations. Let P( be the family of all probability measures on. For each ν P(, let F ν (resp. Fν t be the P ν-completion of F (resp. P ν-completion of Ft in F ν and set F := ν P( Fν and F t := ν P( Fν t. Let F m (resp. Ft m be the P m -completion of F (resp. P m-completion of Ft in F m. Although m may not be a finite measure on, we do have F F, m F t Ft m, because for g L 1 (;m with < g 1 on satisfying gm P(, P gm -negligibility is the same as P m -negligibility. For a fixed filtration (M t on (Ω, M, we recall the notions of (M t -predictability, (M t - optionality and (M t -progressive measurability as follows (see [18] for more details: On [, [ Ω, the (M t -predictable (resp. (M t -optional σ-field P(M t (resp. O(M t is defined as the smallest σ-field over [, [ Ω containing all P ν (M-evanescent sets for all ν P( and with respect to which all M t -adapted lcrl (left-continuous, right-limited (resp. rcll processes are measurable. A process φ(s,ω on [, [ Ω is said to be (M t -progressively measurable provided [,t] Ω (s,ω φ(s,ω is B([,t] M t -measurable for all t >. It is well-known that (M t -predictability implies (M t -optionality, which in turn implies (M t -progressive measurability. For [, ]-valued functions S, T on Ω with S T, we employ the usual notation for stochastic intervals; for example, [S,T [ := {(t,ω [, [ Ω S(ω t < T(ω}, the other species of stochastic intervals being defined analogously. We write [S ] := [S, S ] for the graph of S. Note that these are all subsets of [, [ Ω. If S and T are (M t -stopping times, then [S,T ], [S,T [,, and [S ] are (M t -optional (see Theorem 3.16 in [1]. Definition 2.1 (AF An (F t -adapted (resp. (F m t -adapted process A = (A t t with values in [, ] is said to be an additive functional (AF in short (resp. AF admitting m-null set if there exist a defining set Ξ F and an -polar (resp. m-null set N satisfying the following conditions; (i P x (Ξ = 1 for all x \ N, (ii θ t Ξ Ξ for all t ; in particular, ω Ξ and P (Ξ = 1, because of ω = θ ζ(ω (ω for all ω Ξ, 8

9 (iii for all ω Ξ, A (ω is right continuous with left hand limits on [,ζ(ω[, A (ω =, A t (ω < for t < ζ(ω and A t+s (ω = A t (ω + A s (θ t ω for all t,s, (iv for all t, A t (ω = ; in particular, under the additivity in (iii, A t (ω = A ζ(ω (ω for all t ζ(ω and ω Ξ. An AF A (admitting m-null set is called right-continuous with left limits (rcll AF in brief if A ζ(ω exists for each ω Ξ. An AF A (admitting m-null set is said to be finite (resp. continuous additive functional (CAF in brief if A t (ω <, t [, [ (resp. t A t (ω is continuous on [, [ for each ω Ξ. A [, [-valued CAF is called a positive continuous additive functional (PCAF in short. Two AFs A and B are called equivalent if there exists a common defining set Ξ F and an -polar set N such that A t (ω = B t (ω for all t [, [ and ω Ξ. We call A = (A t t an AF on [,ζ [ or a local AF (admitting m-null set if A is (F t -adapted and satisfies (i, (ii, (iv and the property (iii in which (iii is modified so that the additivity condition is required only for t + s < ζ(ω. The notions of rcll AF, CAF and PCAF on [,ζ [ are similarly defined. Two AFs on [,ζ [, A and B, are called equivalent if there exists a common defining set Ξ F and an -polar set N such that A t (ω = B t (ω for all t [,ζ[ and ω Ξ. Remark 2.2 Any PCAF A on [,ζ [ can be extended to a PCAF by setting { limu ζ A u (ω, if t ζ(ω >, A t (ω :=, if t ζ(ω = for ω Ξ and setting A t (ω for ω Ξ c. The (F t -adaptedness of this extended A holds as follows: for a fixed T >, we know {A t T } {t < ζ} F t. From this, we have the F ζ - measurability of {A ζ T }. Indeed, {A ζ T } = t Q + {A t T,t < ζ} F ζ as {A t T,t < ζ} F ζ for any t. Thus {A t T } {t ζ} = {A ζ T } {t ζ} F t. Therefore, {A t T } F t for any T >, which gives the (F t -adaptedness of A. Noting ζ θ t = ζ t if t < ζ and ζ θ t = if t ζ, we conclude that A ζ = A t +A ζ θ t for any t [, [ on Ξ. Consequently, A t+s = A t +A s θ t holds for any t,s [, [ on Ξ. The following lemma is a special case of [17, Theorem 2.2]. Lemma 2.3 Let A,B be PCAFs such that for m-a.e. x, x [A t ] = x [B t ] for all t, and suppose that the Revuz measure µ A has finite total mass. Then A is equivalent to B. Remark 2.4 The above lemma may fail if the condition µ A ( < is not satisfied. For example, take = R d with d 2, and let X be Brownian motion on R d and µ A (dx = x d 1 dx. Then µ A is a smooth measure and it corresponds to a PCAF A of X. Let B t = A t + t, which is a PCAF of X with Revuz measure µ A (dx + dx. However ( x [A t ] = p(s,x,y y d 1 dy ds = = x [B t ] for every x R d \ {}. R d 9

10 Here p(s,x,y = (2πt d/2 exp ( x y 2 /(2t is the transition density function of X. As usual, if T is an (F t -stopping time and M a process, then M T is the stopped process defined by M T t := M t T. Following [1], we give the notion of local martingales of interval type: Definition 2.5 (Processes of interval type Let D be a class of (F t -adapted processes and denote by D loc its localization (resp. by D f-loc its localization by a nest of finely open Borel sets; that is, M D loc (resp. M D f-loc if and only if there exists a sequence M n D and an increasing sequence of stopping times T n with T n (resp. a nest {G n } of finely open Borel sets such that M Tn = (M n Tn (resp. M t = M n t for t < τ Gn for each n. Here a family {G n } of finely open Borel sets is called a nest if P x (lim n τ Gn = ζ = 1 for q.e. x. (But see Lemma 3.1. Clearly, D D loc (resp. D D f-loc and (D loc loc = D loc (resp. (D f-loc f-loc = D f-loc. If D is a subclass of AFs, then so is D loc (for if M D loc, then there exists M n and T n as above, and for each ω and t,s, there exists n N with s+t < T n (ω and s < T n (θ t ω, hence M t+s (ω = M t (ω+m s (θ t ω, while D f-loc is contained in the class of AFs on [,ζ [. (i B [, [ Ω is called a set of interval type if there exists a non-negative random variable S such that for each ω Ω the section B ω := {t [, [ (t,ω B} is [,S(ω] or [,S(ω[ and B ω. (ii Let B be an (F t -optional set of interval type. A real-valued stochastic process M on B (that is, M1 B = (M t (ω1 B (t,ω t is a real-valued stochastic process is said to be in D B if and only if there exists N D such that M1 B = N1 B, and is said to be locally in D on B (write M (D loc B if and only if S := D B c is the debut of B c and there exists an increasing sequence of (F t -stopping times {S n } with lim n S n = S and a sequence of M n D such that B ω n=1 [,S n(ω] P x -a.s. ω Ω and (M1 B Sn = (M n 1 B Sn for all n N and t, P x -a.s. ω Ω for q.e. x. Clearly, D B (D loc B. Moreover, D B 2 D B 1 and (D loc B 2 (D loc B 1 for any pair of (F t -optional sets B 1, B 2 of interval type with B 1 B 2. (iii Let B be an (F t -optional set of interval type. We set M 1 := {M M is a finite rcll AF, x [ M t ] <, x [M t ] = for -q.e. x and all t }, and speak of an element of (M 1 B (resp. (M 1 loc B as being an MAF on B (resp. a local MAF on B. Similarly, M := {M M is a finite rcll AF, x [M 2 t ] <, x[m t ] = for -q.e. x and all t }, and an element of M B (resp. (M loc B is a square-integrable MAF on B (resp. locally squareintegrable MAF on B. We further set M c : = {M M M is a CAF}, M d : = {M M M is a purely discontinuous AF}, 1

11 and an element of (M c loc B (resp. (M d loc B is called a locally square-integrable continuous MAF on B (resp. locally square-integrable purely discontinuous MAF on B. For M (M loc B, M admits a unique decomposition M = M c + M d with M c (M c loc B and M d (M d loc B (see Theorem 8.23 in [1]. In these definitions, we omit the usage on B when B = [, [ Ω. For a [, ]-valued function R on Ω and A Ω, R A := R 1 A + (+ 1 A c is called the restriction of R on A. Clearly, R R A. Remark 2.6 When B = [,R[ for a given (F t -stopping time R, there is another notion of locally in D on B, obtained by replacing (M1 B Sn = (M n 1 B Sn with M Sn 1 B = (M n Sn 1 B in our definition; this is a weaker notion than ours, because t 1 B (t,ω is decreasing and 1 B (t,ω1 B (s,ω = 1 B (t,ω for s t and ω Ω. This weaker notion is described in [18]. Definition 2.7 (MAF locally of finite energy Recall that M is the totality of MAFs of finite energy, that is, { } 1 M:= M M e(m := lim t 2t m[mt 2 ] <. We say that an AF M on [,ζ [ is locally in M (and write M Mf-loc if there exists a sequence {M n } in M and a nest {G n } of finely open Borel sets such that M t = M n t for t < τ Gn for each n N. In case X is a diffusion process with no killing inside, we can define the predictable quadratic variation M for M Mf-loc as follows: Note that M n t τ Gn = M m t τ Gn for n < m because of the continuity of M n. Owing to the uniqueness of Doob-Meyer decomposition, we see M n t τgn = M m t τgn. The predictable quadratic variation M of M Mf-loc as a PCAF is well-defined by setting M t = M n t, t < τ Gn, n N, with Remark 2.2 and by choosing an appropriate defining set and -polar set of M, where M n M and {G n } is a nest of finely open Borel sets such that M t = M n t, t < τ Gn. Proposition 2.8 (M loc [[,ζ[[ Mf-loc. More precisely, for each M (M loc [[,ζ[[, there exists a nest {G k } of finely open Borel sets such that 1 Gk M M for each k N, and the predictable quadratic variation process M can be constructed as a PCAF. Proof. Let M (M loc [[,ζ[[. Then there exists an increasing sequence {T n } of stopping times with lim n T n = ζ, (P x -a.s. ω Ω for q.e. x and M n M loc such that M t Tn 1 [,ζ[ (t T n = Mt T n n 1 [,ζ[ (t T n holds for all t P x -a.s. for q.e. x. We may assume that it holds for all ω Ω by changing the sample space. Note that [,ζ(ω[ n=1 [,T n(ω] for all ω Ω. Hence, Mt T m n 1 [,ζ[ (t T n = Mt T n n 1 [,ζ[ (t T n for n < m. As noted in Definition 2.5, we see that M is an AF on [, ζ [. Owing to the uniqueness of the Doob-Meyer decomposition for semimartingales on 11

12 [,ζ [ (see [1], we have M m t Tn 1 [,ζ[ (t T n = M n t Tn 1 [,ζ[ (t T n for n < m. Thus, we have M m t = M n t for t < T n and n < m. The predictable quadratic variation M of M is therefore well defined by setting M t := M n t for t < T n. Setting M t := M ζ := lim s ζ M s for all t ζ, we obtain a PCAF because of Remark 2.2. Let µ M be the Revuz measure corresponding to M and {F k } an -nest of closed sets such that µ M (F k < for each k, and let G k be the fine interior of F k. Then {G k } is a nest. In view of the proofs of Theorem and Lemma in [8], the stochastic integral 1 Gk M is of finite energy with e(1 Gk M = 1 2 µ M (G k, and its predictable quadratic variation 1 Gk M is a PCAF. Let µ k (resp. µ k be the Revuz measure corresponding to 1 Gk M (resp. 1 Gk M,M. By Lemma in [8], for M i M and f i L 2 (;µ Mi (i = 1,2, we have f 1 f 2 µ M1,M 2 = µ f1 M 1,f 2 M 2, hence (f 1f 2 (X s d M 1,M 2 s = f 1 M 1,f 2 M 2 t. From this, we see µ k,f 2 = µ k,f 2 = 1 Gk µ M,f 2 for any f L 2 (;µ M, consequently we have µ k = µ k = 1 Gk µ M by µ M (G k <. This yields 1 Gk M t = 1 Gk M,M t = 1 G k (X s d M s for t < ζ, hence M 1 Gk M t = for t < τ Gk. Therefore, M t = (1 Gk M t for t < τ Gk and 1 Gk M M. Corollary 2.9 Let M be a locally square-integrable MAF on [,ζ [ that is, M (M loc [[,ζ[[. Then there exists a Borel function ϕ on with ϕ(x,x = for all x such that M t M t = ϕ(x t,x t for every t ],ζ[, P m -a.e. Proof. By the proof of Proposition 2.8, there exists an -nest {F k } such that for each k N M k := 1 Fk M M and M t = Mt k, t < τ Fk. Let ϕ k be the jump function corresponding to M k. Then we have ϕ k (X t,x t = ϕ l (X t,x t, t < τ Fk P m -a.e. for k < l. From this, we see ϕ k = ϕ l J-a.e. on F k F k. We construct a Borel function ϕ on in the following manner. We set F :=, ϕ(x,y := ϕ k (x,y for (x,y F k F k \ (F k 1 F k 1, k N, ϕ(x,y := if (x,y \( k=1 F k k=1 F k. Then ϕ satisfies ϕ(x,x = for x. We also have ϕ = ϕ k J-a.e. on F k F k. Consequently, ϕ(x t,x t = ϕ k (X t,x t, t < τ Fk P m -a.e. This means that M t M t = ϕ(x t,x t, t < τ Fk P m -a.e. Therefore M t M t = ϕ(x t,x t, < t < ζ P m -a.e. We recall the definition of the shift operator θ s and the time-reversal operator r t on the path space Ω. For each s, the shift operator θ s is defined by θ s ω(t := ω(t + s for t [, [. Given a path ω {t < ζ}, the operator r t is defined by r t (ω(s := { ω((t s, if s t, ω(, if s t. (2.1 Here for r >, ω(r := lim s r ω(s is the left limit at r, and we use the convention that ω( := ω(. For a path ω {t ζ}, we set r t (ω := ω. We note that lim r t (ω(s = ω(t = r t (ω( and lim r t (ω(s = ω( = r t (ω(t. (2.2 s s t A key consequence of the m-symmetry assumption on the Hunt process X is that the measure P m, when restricted to {t < ζ}, is invariant under the time-reversal operator r t. 12

13 Clearly for t,s >, θ s : Ω Ω is F m t+s/f m t -measurable. The following lemma deals with the measurability issue of the time-reversal operator r t. Lemma 2.1 For each t >, r t : Ω Ω is F t /F -measurable and Fm t /Fm t -measurable. Proof. Let F i B( and s i [, [, i = 1,2,,n with s 1 < s 2 < < s k t < s k+1 < < s n for some k {1,2,,n}. Then rt 1 ( n i=1 X 1 s i (F i = n i=1 (X s i r t 1 (F i is equal to k i=1 ({X (t s i F i, t < ζ} { F i, t ζ} n i=k+1 ({X F i, t < ζ} { F i, t ζ} Ft. Next we show the Ft m /Ft m -measurability of r t. Take C Ft m. Then there exists D Ft and N F such that C D N and P m(n =. Since P m ({ω } =, by deleting {ω } = {ω Ω ζ(ω = } F F t, we may assume ω / C D N. Then, rt 1 (C rt 1 (D rt 1 (N, rt 1 (D,rt 1 (N Ft and P m (rt 1 (N = P m (rt 1 (N {t < ζ} + 1 N (ω P m (t ζ = P m (N {t < ζ} =. Definition 2.11 For any t >, we say two sample paths ω and ω are t-equivalent if ω(s = ω (s for all s [, t]. We say two sample paths ω and ω are pre-t-equivalent if ω(s = ω (s for all s [, t[. For an rcll AF A t adapted to (F t t, A t (ω = A t (ω if ω and ω are t-equivalent and A t (ω = A t (ω if ω and ω are pre-t-equivalent. These conclusions may fail to hold if the measurability conditions are not satisfied. We need the following notion: Definition 2.12 (PrAF A process A = (A t t with values in R := [, ] is said to be a progressively additive functional (PrAF in short (resp. PrAF admitting m-null set if A is (F t - adapted (resp. (F m t -adapted and there exist defining sets Ξ F, Ξ t F t (resp. Ξ F m, Ξ t F m t for each t > and an -polar (resp. m-null set N satisfying the following condition; (i P x (Ξ = 1 for all x \ N, and Ξ Ξ t Ξ s for every t > s >, and Ξ = t> Ξ t, (ii θ t Ξ Ξ for all t and θ t s (Ξ t Ξ s for all s ],t[; in particular, ω Ξ Ξ t and P (Ξ = P (Ξ t = 1 under (i, (iii for all ω Ξ t, A (ω is defined on [,t[ and it is right continuous on [,t ζ(ω[ and has left limit on ],t] ],ζ(ω[ such that A (ω =, A s (ω < for s [,t ζ(ω[ and A p+q (ω = A p (ω + A q (θ p ω for all p,q with p + q < t, (iv for all t, A t (ω =, (v for any t > and pre-t-equivalent paths ω,ω Ω, ω Ξ t implies ω Ξ t, A s (ω = A s (ω for any s [,t[ and A s (ω = A s (ω for any s ],t]. Furthermore, A is called an rcll PrAF (or an rcll PrAF admitting m-null set if for each t > and ω Ξ t, s A s (ω is right continuous on [,t[ and has left hand limits on ],t] and a PrAF 13

14 (or a PrAF admitting m-null set is said to be finite (resp. continuous if A s (ω <, s [,t[ (resp. continuous on [,t[ for every ω Ξ t. We say that an AF A on [,ζ [ (resp. AF A on [,ζ [ admitting m-null set is a PrAF on [,ζ [ (resp. PrAF on [,ζ [ admitting m-null set if A is (F t -adapted (resp. (Ft m -adapted, and there exist Ξ F, Ξ t F t (resp. Ξ F, m Ξ t Ft m for each t > and an -polar (resp. m-null set N such that (i, (ii, (iii, (iv and (v hold: (i : P x (Ξ = 1 for all x \ N, Ξ Ξ t for all t >, Ξ = t> Ξ t, and Ξ t {t < ζ} Ξ s {s < ζ} for s < t. (iii : For each ω Ξ t {t < ζ}, the same conclusion as in (iii holds. (v : For any t > and pre-t-equivalent paths ω,ω Ω {t < ζ}, the same conclusion as in (v holds. The notion of rcll PrAF on [,ζ [ (or rcll PrAF admitting m-null set is similarly defined. Remark 2.13 (i Our notion of PrAF is different from what is found in Walsh [19]. (ii very PrAF (resp. PrAF on [,ζ [ is an AF (resp. AF on [,ζ [. (iii The MAF M u and the CAF N u of -energy appearing in Fukushima s decomposition (1.2 can be regarded as finite rcll PrAFs in view of the proof of Theorem in [8]. In this case, the defining sets for M u as PrAF are given by Ξ := {ω Ω Ms un (ω converges uniformly on ],t] for t for some subsequence n k } F Ξ t for every t >, where M un t := {ω Ω Ms un (ω converges uniformly on ],t] for some subsequence n k } F t := u n (X t u n (X (u n(x s f n (X s ds with f n := n(u nr n+1 u and u n := R 1 f n = nr n+1 u. Hence a MAF of stochastic integral type g(x s dm u s (g,u F with g L 2 (;µ u can be regarded as a finite rcll PrAF. Consequently, any MAF of finite energy also can be regarded as an rcll PrAF, in view of the assertion of Lemma in [8] and Lemma 2.14 below. (iv very M Mf-loc can be regarded as a PrAF on [,ζ [, hence every M M [[,ζ[[ loc is so. Since every local martingale can be written as the sum of a local martingale with bounded jumps (and hence a locally square-integrable martingale and a local martingale of finite variation, we conclude that every local MAF is a PrAF. Lemma 2.14 Let (A n be a sequence of finite rcll PrAFs with defining sets Ξ n F, Ξ n t F t. For each t >, set { Ξ t := ω n NΞ } n t A n converges uniformly on [,t[ F t and { Ξ := ω n NΞ } n A n converges uniformly on [,t[ for every t [, [ F. 14

15 Suppose that there exists an -polar set N such that P x (Ξ = 1 for x \ N. If we define A t := lim n A n t on Ω, then A is a finite rcll PrAF with its defining sets Ξ, Ξ t. Proof. We only show that for any t > and pre-t-equivalent paths ω,ω, ω Ξ t implies ω Ξ t. Suppose that ω Ξ t and ω is pre-t-equivalent to ω. It easy to see that ω n N Ξn t. We then see the uniform convergence of A n s (ω = A n s (ω for s ],t]. Therefore ω Ξ t, A s (ω = A s (ω for s [,t[ and A s (ω = A s (ω for s ],t]. Recall that {θ t,t > } denotes the time shift operators on the path space for the process X. Lemma 2.15 For t,s >, (i θ t r t+s ω is s-equivalent to r s ω if t + s < ζ(ω or s ζ(ω; (ii r t θ s ω is pre-t-equivalent to r t+s ω. Moreover, if ω is continuous at s, then r t θ s ω is t-equivalent to r t+s ω. Proof. (i: We may assume t + s < ζ(ω. For v [,s], θ t r t+s ω(v = ω((s v = r s ω(v and so θ t r t+s ω is s-equivalent to r s ω. (ii: Note that t + s < ζ(ω is equivalent to t < ζ(θ s ω. It follows from the definition, if t + s < ζ(ω, { ω((t + s v, if v < t, (r t θ s ω(v = (2.3 ω(s, if v = t, while r t+s ω(v = ω((t + s v for v t. Hence typically r t θ s ω is only pre-t-equivalent to r t+s ω. Fix t >. Set H t s := F t for s [,t]; and H t s := F s for s ]t, [. Then (H t s s is a filtration over (Ω, F, and F s H t s for all s. Lemma 2.16 The following assertions hold for any fixed t > : (i Let ϕ be a Borel function on and set X := X. Then [, [ Ω (s,ω 1 [[,ζ[[ (s,ω1 Γt (ωϕ(x s (ω,x s (ω is (Hs t-optional for any Γ t F t. (ii Let A be an rcll PrAF with defining sets Ξ F, Ξ t F t. If we set A (ω := and A t s (ω := 1 Ξ t (ω(1 [,t] (sa s (ω + 1 ]t, [ (sa t (ω for ω Ω, then [, [ Ω (s,ω 1 [[,ζ[[ (s,ω(a t s(ω A t s (ω is (H t s-optional. 15

16 Proof. (i: Note that 1 [[,ζ[[ is (Hs-predictable. t The assertion is clear if ϕ = f g for bounded Borel functions f,g on. The monotone class theorem for functions tells us the desired result. (ii: Since A t is (Hs-adapted t and rcll on Ω and A t is (Hs-adapted t and lcrl on Ω, (s,ω A s (ω is (Hs t-optional and (s,ω At s (ω is (Ht s -predictable. Consequently, (s,ω At s (ω At s (ω is (Hs-optional. t By Lemma 3.2 of [3], for a finite rcll AF A = (A t t, there is a Borel function ϕ : R with ϕ(x,x = for all x such that A t A t = ϕ(x t,x t, for every t ], ζ[, P m -a.e. (2.4 Moreover, if ϕ is another such function, then J(ϕ ϕ =. As before, we refer to such a function ϕ as a jump function for A. Recall that if M M [[,ζ[[ loc, then there exists a jump function ϕ (unique in the above sense so that M t M t = ϕ(x t,x t for t ],ζ[, P m -a.e. Lemma 2.17 Let A be a finite rcll PrAF with defining sets {Ξ, Ξ t, t }. Then there exists a real valued Borel function ϕ on with ϕ(x,x = for x such that A with defining sets Ξ := Ξ t := { } ω Ξ A s (ω A s (ω = ϕ(x s (ω,x s (ω for s ],ζ(ω[, { ω Ξ t A s (ω A s (ω = ϕ(x s (ω,x s (ω for s ],t[ ],ζ(ω[ is again an rcll PrAF admitting m-null set. The analogous assertion holds for PrAFs on [,ζ [, and in particular for elements of (M loc [[,ζ[[ Proof. Let ϕ : R be a Borel function vanishing on the diagonal and define Ξ, Ξ t in terms of ϕ as above. Clearly, Ξ = t> Ξ t, Ξ t Ξ s for s < t. Moreover, we see that θ t Ξ Ξ for t, θ t s ( Ξ t Ξ s for s < t. For two pre-t-equivalent paths ω,ω, we see that ω Ξ t implies ω Ξ t. By the previous lemma, Γ := {(s,ω 1 [[,ζ[[ (s,ω1 Ξt (ω(a t s(ω A t s (ω ϕ(x s (ω,x s (ω } is (Hs t -progressively measurable for any fixed t > and the debut of Γ is D Γ (ω := inf{s 1 [[,ζ[[ (s,ω1 Ξt (ω(a t s (ω At s (ω ϕ(x s (ω,x s (ω }, which is an (Hs t -stopping time by (A5.1 in [18]. In particular, {ω Ω 1 [[,ζ[[ (s,ω1 Ξt (ω(a s (ω A s (ω ϕ(x s (ω,x s (ω = for s [,t[} = {ω Ω t < D Γ (ω} H t t = F t. }, 16

17 Hence, {ω Ξ t A s (ω A s (ω ϕ(x s (ω,x s (ω = for s ],t[ ],ζ(ω[} = {ω Ξ t A s (ω A s (ω ϕ(x s (ω,x s (ω = for s [,t[ [,ζ(ω[} = {ω Ξ t 1 [[,ζ[[ (s,ω(a s (ω A s (ω ϕ(x s (ω,x s (ω = for s [,t[} F t. Therefore, Ξ t F t and Ξ F. The proof for PrAFs on [,ζ [ is similar, so we omit it. The following theorem is a key to our extension of Nakao s operator Γ. Its proof is complicated by measurability issues but the idea behind it is fairly transparent. We will use the convention X (ω := X (ω. Theorem 2.18 (Dual PrAF Let A be a finite rcll PrAF on [,ζ [ with defining sets Ξ, Ξ t admitting m-null set. Suppose there is a Borel function ϕ on with ϕ(x, x = for x such that ϕ(x s (ω,x s (ω = A s (ω A s (ω, s ],t[ ],ζ[ and all ω Ξ t. Set  t (ω := A t (r t (ω + ϕ(x t (ω,x t (ω for t [,ζ(ω[ and Ât(ω := for t [ζ(ω, [. (2.5 Then  is an rcll PrAF on [,ζ [ admitting m-null set such that for all t ],ζ[, P m -a.e.  t = A t r t + ϕ(x t,x t and  t Ât = ϕ(x t,x t Proof. Let Ξ F, Ξ t Ft m, t > be the defining sets of A admitting m-null set. We easily see (Ξ t {t < ζ} rs 1 (Ξ s {s < ζ} for s ],t[ by use of Lemma 2.15(i and θ t s Ξ t Ξ s. Set Ξ t := rt 1 (Ξ t for t > and Ξ := Ξ t> t. Then, we see Ξ = Ξ t>,t Q t by use of (Ξ t {t ζ} = {t ζ} and the monotonicity of rt 1 (Ξ t {t < ζ}. Indeed, we have Ξ Ξ t>,t Q t ( Ξs {s < ζ} {t ζ} for any < s < t with t Q. Taking the intersection r 1 t r 1 t over t ]s, [ Q, we have Ξ t>,t Q Ξ t Ξ s for all s >, which yields the assertion. We prove θ t Ξ Ξ for each t, in particular, θt Ξ θs Ξ, equivalently θ 1 s Ξ θt 1 Ξ if s [,t]. Suppose ω Ξ. Then r t+s ω Ξ t+s. If t + s < ζ(ω, then r t+s ω Ξ s, otherwise r t+s ω = ω Ξ s. Hence we have r s θ t ω Ξ s by Lemma 2.15(ii. Therefore r s θ t ω Ξ s for all s >, which implies θ t ω Ξ. Next we prove θ t s ( Ξ t Ξ s for s ],t[. Take ω Ξ t. Then r s θ t s ω is pre-s-equivalent to r t ω Ξ t Ξ s by Lemma 2.15(ii, hence r s θ t s ω Ξ s. Therefore θ t s ω Ξ s for all s ],t[. From Ξ t Ft m, we get Ξ t Ft m by Lemma 2.1. Since ( Ξ t c = rt 1 ((Ξ t c = rt 1 ((Ξ t c {t < ζ} holds by noting ω Ξ t, we have P m (( Ξ t c = P m (( Ξ c =. By (2.2, v r s (ω(v is continuous at v = s. Hence, on Ξ t {t < ζ}, we have ϕ(x s,x s r s = ϕ(x s,x s r s =, in particular, A s r s = A s r s for s ],t[. The remainder of the proof is devoted to showing that  is an rcll PrAF on [, ζ[ with defining sets Ξ, Ξ t such that on Ξ t {t < ζ}, Âs = A s r s +ϕ(x s,x s, s ],t[. First note that for ω Ξ, 17

18 Ât(ω < for any t ],ζ(ω[, because by taking T ]t,ζ(ω[, r T ω Ξ T implies r t ω Ξ t, hence A t (r t ω <. Moreover, for ω Ξ t {t < ζ}, we see r s ω Ξ s {s < ζ} and A s (r s ω < for all < s < t. For two pre-t-equivalent paths ω,ω Ω {t < ζ} with t >, we show ω Ξ t implies ω Ξ t and Âs(ω = Âs(ω for s [,t[. Recall ω Ξ t {t < ζ} Ξ s {s < ζ} for s [,t] and note that ω and ω are s-equivalent for any s [,t[. On the other hand, s < ζ(ω is equivalent to s < ζ(ω for any s [,t[. Then we see r s ω Ξ s is s-equivalent to r s ω for any s [,t], which implies r s ω Ξ s for any ],t] and A s (r s ω = A s (r s ω for any s [,t]. Fix t >. On Ξ t {t < ζ} and for any p,q > with p + q < t, by Lemma 2.15,  p+q = A (p+q r p+q + ϕ(x p+q,x (p+q = (A p + A q θ p r p+q + ϕ(x p+q,x (p+q = A p r p+q + A q θ p r p+q + ϕ(x p+q,x (p+q = (A p r p+q + ϕ(x p,x p r p+q + A q r q + ϕ(x p+q,x (p+q = (A p r p θ q + ϕ(x q,x q + (Âq ϕ(x q,x q + ϕ(x p+q,x (p+q = (Âp ϕ(x p,x p θ q + Âq + ϕ(x p+q,x (p+q = Âp θ q + Âq. On Ξ t {t < ζ}, again by Lemma 2.15 and (2.2, for any s > and u ], s[,  s Âs u = Âu θ s u = (A u r u + ϕ(x u,x u θ s u = A u r u θ s u + ϕ(x s,x s = A u r s + ϕ(x s,x s. So lim(âs Âs u = ϕ(x s,x s. u This shows that  has left limit at s ],t[ and Âs Âs = ϕ(x s,x s. To show the right continuity of  on Ξ t {t < ζ} at any s ],t[, note for any u ], t s[, by Lemma 2.15 and (2.2,  s+u Âs = Âu θ s = (A u r u + ϕ(x u,x u θ s = A u r u θ s + ϕ(x s+u,x (s+u = A u r s+u + ϕ(x s+u,x (s+u. Since (A v A v r s+v = ϕ(x v,x v r s+v = ϕ(x s,x s, while by Lemma 2.15 and (2.2, (A v A v r s+v = lim u (A v A v u r s+v = lim u A u θ v u r s+v = lim u A u r v+u + ϕ(x s,x s. 18

19 we conclude that lim A u r s+u =. u On the other hand, for any s lim ϕ(x s+u,x (s+u = lim ϕ(x (v u,x v u r s+v = lim(a v u A (v u r s+v u u u = (A v A v r s+v =. Hence we have for s > lim(âs+u Âs =. u In other words,  is right continuous at any s ],t[ on Ξ t {t < ζ}. We also see lim u<s,s,u (Âs+u Âs =. Thus we can define the limit Â(ω := lim s  s (ω for ω Ξ t {t < ζ} for any t >. We also see  (ω = lim s  s (ω for ω Ξ t {t < ζ} for any t >, because lim s ϕ(x s,x s =. Next we prove Â(ω = for ω Ξ t {t < ζ} for any t >. Take ω Ξ t {t < ζ} for some fixed t >. It suffices to show that lim u  s u (θ u ω = Âs(ω for s [,t[. Owing to Lemma 2.15(ii, we have  s u (θ u ω = A (s u (r s u θ u ω + ϕ(x s (ω,x s (ω = A (s u (r s ω + ϕ(x s (ω,x s (ω = A s u (r s ω ϕ(x u (ω,x u (ω + ϕ(x s (ω,x s (ω = A s u (r s ω Âu(ω + Âu (ω + ϕ(x s (ω,x s (ω A s (r s ω + ϕ(x s (ω,x s (ω as u = Âs(ω. The F m t -measurability of Ât is clear from (2.5. This proves the theorem. 3 Stochastic integral for Dirichlet processes The following fact will be used repeatedly in this section. Since a Hunt process is quasi-left continuous, for each fixed t >, we have X t = X t, P x -a.s. for every x. Before embarking on the definition of our stochastic integral, we prepare the following lemma for later use. Lemma 3.1 The following assertions hold. (i Let {G n } be an increasing sequence of finely open Borel sets. Then the following are equivalent. 19

20 (a {G n } is a nest, that is, P x (lim n σ \Gn ζ = ζ = 1 for q.e. x. (b = n=1 G n q.e. (c P x (lim n σ \Gn = = 1 for m-a.e. x. (d P x (lim n σ \Gn = = 1 for q.e. x. In particular, for an increasing sequence {F n } of closed sets, {F n } is an -nest if and only if P x (lim n σ \Fn = = 1 for m-a.e. x. (ii For a function f on, f F loc if and only if there exist an -nest {F k } of closed sets and {f k k N} F b such that f = f k q.e. on F k. Proof. (i: For the implications (ia (ib, see Theorem 4.6 in [14]. The implication (id= (ia is clear. Next we show (ib= (ic. Since each G n is finely open, it is quasi-open by Theorem 4.6.1(i in [8]. So there exists a common nest {A l } of closed sets such that ( \ G n A l is closed for all n,l N. Set σ := lim n σ \Gn. We then have that for all n N X σ\gn \ G n P x -a.s. on {σ < } for q.e. x. We have P x (lim l σ \Al = = 1 q.e. x. Since σ(ω < and lim l σ \Al (ω = together imply σ(ω < σ \Al (ω for some l = l (ω N, we have that there exists l N such that σ \Gn < σ \Al for all n > l l, P x -a.s. on {σ < } for q.e. x. This means ( { } P x (σ < P x lim X σ\gn ( \ G n A l for all n > l, σ < l lim P x (X σ\gn ( \ G l A l for all n > l, σ < l lim P x (X σ ( \ G l A l, σ < l lim P x (X σ \ G l, σ < l = P x (X σ \ G l, σ < l=1 for m-a.e. x, because of the -polarity of \ l=1 G l, where we use the quasi-left continuity of X up to and the closedness of ( \ G l A l. The implication (ic (id follows from the fact that x P x (σ < is the limit of a decreasing sequence of excessive functions and Lemma in [8]. (ii: The if part is clear by (i because τ Fk = τ Gk, where G k is the fine interior of F k. We only prove the only if part. Take f F loc. Then there exist {f k k N} F and an increasing sequence {G k } of finely open sets with = k=1 G k q.e. such that f = f k m-a.e. on G k. We may take f k F b for each k N, by replacing f k with ( k f k k, and G k with G k { f < k}. Note that f and f k are quasi-continuous, so f = f k q.e. on G k. Taking an -quasi-closure G k of Gk, we have f = f k q.e. on G k (see [13] for the definition of -quasi-closure. Let {An } be a common -nest of closed sets such that for each k,n N, G k An is closed. Set F k := G k Ak. By (i, {G k } is a nest, hence G k is a nest of q.e. finely closed sets, because of τgk τ Gk. Here we = 2

21 recognize G k as a finely closed Borel sets by deleting an -polar set. Since {An } is a nest of closed sets, {F k } is so, that is, P m (lim k τ Fk ζ =. Therefore {F k } is an -nest of closed sets. We easily see that for each k N, f = f k q.e. on F k. Recall that any locally square-integrable MAF M on [,ζ [ admits a jump function ϕ on with ϕ(x,x = for x such that M t = ϕ(x t,x t for t ],ζ[ P m -a.e. When M M, we can strengthen this statement by replacing ],ζ[ with ], [ in view of Fukushima s decomposition and the combination of Theorem and Lemma in [8]. Lemma 3.2 Let φ be a Borel function on satisfying φ(x,x = for all x. (i Suppose that N(1 ( φ 2 φ µ H S. Then there exists a unique purely discontinuous local MAF K on [,ζ [ (i.e. K (M 1 loc [[,ζ[[ such that K t K t = φ(x t,x t for all t [,ζ[, P x -a.s. for q.e. x. (ii If N(1 ( φ 2 φ µ H S, then K can be taken to be a local MAF (i.e. K M 1 loc and K t K t = φ(x t,x t for all t [, [, P x -a.s. for q.e. x. Proof. The proof of (ii is similar to that of (i, so we only prove (i. By martingale theory (see, e.g. [1], the hypothesis implies that the compensated process K (2 t := <s t φ(x s,x s 1 { φ(xs,x s >1}1 {s<ζ} is a local MAF of X of finite variation on [,ζ [ and K (1 t ( := lim φ(x s,x s 1 {ε< φ(xs,x ε s 1} 1 {s<ζ} <s t N(X s,dyφ(x s,y1 { φ(xs,y >1} dh s N(X s,dyφ(x s,y1 {ε< φ(xs,y 1} dh s is a purely discontinuous locally square-integrable MAF of X on [,ζ [. Thus K := K (1 + K (2 is a purely discontinuous MAF on [,ζ [ with jump function φ. The uniqueness is clear from the martingale theory. Definition 3.3 Let M be a local MAF on [,ζ [ (that is, M (M 1 loc [[,ζ[[ with jump function ϕ. Assume that for q.e. x, P x -a.s. ( ϕ 2 1 { bϕ 1} + ϕ 1 { bϕ >1} (Xs,yN(X s,dydh s < for every t < ζ, (3.1 21

22 where ϕ(x,y := ϕ(x,y + ϕ(y,x. Define, P m -a.e. on [,ζ [, Λ(M t := 1 2 (M t + M t r t + ϕ(x t,x t + K t for t [, ζ[, (3.2 where K t is the purely discontinuous local MAF on [,ζ [ with for q.e. x. K t K t = ϕ(x t,x t for every t < ζ, P x -a.s. (3.3 Remark 3.4 (i The condition (3.1 is nothing but N(1 ( ϕ 2 ϕ µ H S. In particular, condition (3.1 is satisfied by the jump function of any element of (M loc [[,ζ[[. (ii It follows from Remark 2.13(iv and Theorem 2.18 that Λ(M is a continuous PrAF admitting m-null set on [,ζ [. (This is because Remark 2.13(iv and Theorem 2.18 imply that the process defined on [,ζ[ by B t := M t r t + ϕ(x t,x t is a rcll PrAF, with left-limit process B t = B t ϕ(x t,x t. It follows that Λ(M is rcll on [,ζ[ and that Λ(M t = Λ(M t for all t ],ζ[, P m -a.e. Note that K t := s t ϕ(x s,x s 1 {s<ζ} ϕ(x s,yn(x s,dydh s, t < ζ, satisfies K t = K t r t P m -a.e. on {t < ζ} for fixed t >. In view of Theorem 2.18, it is then clear from the definition that Λ is a linear operator that maps local MAFs on [,ζ [ satisfying condition (3.1 into even CAFs on [,ζ [ admitting m-null set. (iii If {M n,n 1} is a sequence of MAFs having finite energy and converging in probability to M, then it is easy to see that M n t r t, ϕ n (X t,x t = M n t M n t and ϕn (X t,x t converge to M t r t, ϕ(x t,x t = M t M t and ϕ(x t,x t in probability, respectively, under P m. Hence we have Λ(M n t converges to Λ(M t in measure for each t >. (iv For u F, Λ(M u t = 1 2 (Mu t + M u t r t + u(x t u(x t = N u t P m -a.e. on {t < ζ} for each fixed t. The first equality above is just the definition of Λ(M u while the second follows by applying r t to both sides of (1.2, because X t = X t and N u r t = N u t, P m -a.e. on {t < ζ}. (The last property is proved in [5, Theorem 2.1] when X is a diffusion, but the same proof works for general symmetric Markov process X. Since both Λ(M u t and N u t are continuous in t, we even have, P m -a.e., Λ(M u t = N u t for all t < ζ. We are going to show that Λ(M defined above coincides on [, ζ[ with Γ(M defined in (1.5 by Nakao when M is an MAF of finite energy. An AF Z is called even (resp. odd if and only if 22

23 Z t r t = Z t (resp. Z t r t = Z t P m -a.e. on {t < ζ} for each t >. For a rcll process Z with Z = and T >, we define R T Z t := (R T Z t := Z T Z (T t for t T, with the convention Z = Z =. Note that R T Z t so defined is an rcll process in t [, T]. Lemma 3.5 Suppose that Z is an rcll PrAF. Then P m -a.e. on {T < ζ}, { Zt r T, if Z is even R T Z t = Z t r T, if Z is odd for every t [, T]. (3.4 Proof. Let Z be an rcll PrAF and let T >. By Lemma 2.15, When Z is even, Z t r T = (Z T Z T t θ t r T = Z T r T Z T t r T t for all t < T. (3.5 Z t r T = Z T Z T t = Z T Z (T t = R T Z t P m -a.e. on {T < ζ} for each fixed t < T. Since both sides are right continuous in t [,T[, we have P m -a.e. R T Z t = Z t r T for every t [,T]. When Z is an odd AF of Z, (3.4 can be proved similarly. Theorem 3.6 For an MAF M of finite energy, Λ(M defined above coincides on [, ζ [ with Γ(M defined in (1.5 P m -a.e.. Proof. For u F and < t < T, since N u is an even CAF, by Lemma 3.5, (M u t + 2N u t r T = (u(x t u(x + N u t r T = u(x (T t u(x T + N u T N u (T t = M u (T t Mu T = R T M u t. Since both (M u t + 2N u t r T and R T M u t are right continuous in t, we have P m -a.e. on {T < ζ}, R T M u t = (M u t + 2N u t r T for every t [, T]. (3.6 For u D(L F and v F b, define M t = v(x s dms u, which is a MAF of finite energy. Note that, since u D(L, Nt u = Lu(X sds is a continuous process of finite variation. For each fixed < t < T and n 1, define t i = it/n and s i = T t + t i. Using the standard Riemann-sum 23