Extending Markov Processes in Weak Duality by Poisson Point Processes of Excursions

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1 Extending Markov Processes in Weak Duality by Poisson Point Processes of Excursions Zhen-Qing Chen 1, Masatoshi Fukushima 2, and Jiangang Ying 3 1 Department of Mathematics, University of Washington, Seattle, WA 98195, USA, zchen@math.washington.edu.the research of this author is supported in part by NSF Grant DMS Department of Mathematics, Kansai University, Suita, Osaka , Japan, fuku2@mx5.canvas.ne.jp. The research of this author is supported in part by Grant-in-Aid for Scientific Research of MEXT No Department of Mathematics, Fudan University, Shanghai, China, jgying@fudan.edu.cn. The research of this author is supported in part by NSFC No Summary. Let a be a non-isolated point of a topological space E. Suppose we are given standard processes X and X b on E = E \ a} in weak duality with respect to a σ-finite measure m on E which are of no killings inside E but approachable to a. We first show that their extensions X and X b to E admitting no sojourn at a and keeping the weak duality are uniquely determined by the approaching probabilities of X, X b and m up to a non-negative constant δ representing the killing rate of X at a. We then construct, starting from X, such X by piecing together returning excursions around a and a possible non-returning excursion including the instant killing. This extends a recent result by M. Fukushima and H. Tanaka [16] which treats the case where X, X are m-symmetric diffusions and X admits no sojourn nor killing at a. Typical examples of jump type symmetric Markov processes and non-symmetric diffusions on Euclidean domains are given at the end of the paper. Dedicated to Professor Kiyosi Itô on the occasion of his 9th birthday 1 Introduction Let a be a non-isolated point of a topological space E and X = X t, ζ, P x} be a strong Markov process on E = E \ a} which admits no killings inside E and satisfies ϕ(x) := P x(ζ <, X ζ = a) > for every x E. We are concerned with a strong Markovian extension X of X from E to E such that X admits no sojourn at the one-point set a}. Natural questions

2 2 Zhen-Qing Chen, Masatoshi Fukushima, and Jiangang Ying arise: is X uniquely determined by X and how can it be constructed from X? When both X and X are required to be diffusions that are symmetric with respect to a σ-finite measure m on E with m(a}) =, affirmative answers to these questions were given quite recently in M. Fukushima and H. Tanaka [16]. It is shown in [16] that the entrance law and the absorption rate for the absorbed Poisson point processes of excursions attached to X away from a (due to K. Itô [24] and P.A. Meyer [27]) are uniquely determined by the approaching probability ϕ to a for X and the measure m, yielding the uniqueness of the extension X that admits no sojourn nor killing at the point a. Conversely such extension X can be constructed from X by piecing together the associated returning excursions and possibly a non-returning one away from a. The purpose of the present paper is to generalize the stated results of [16] to general standard processes X and X which are not necessarily symmetric but admitting weak dual standard processes X and X, respectively. We can no longer use the Dirichlet form theory which has played an important role in [16]. Nevertheless, the entrance law and the absorption rate for the absorbed Poisson point process of excursions of X at the point a can still be identified in 2 and 3 in terms of the approaching probabilities to a by X and X and m owing to the recent works on the exit system by P.J. Fitzsimmons and R.G. Getoor [12] and by the present authors [5]. It turns out that we must allow the killings of X and X at the point a in order to preserve the duality of X and X so that the uniqueness of extensions holds only up to a parameter δ that represents the killing rate of X at a (see Theorem 4.2). In 5, we shall construct such an extension X starting from X by piecing together the returning excursions around a and possibly a non-returning excursion from a including a killing at a. X and its dual X are assumed to be of no killings inside E. The sample path of the constructed process X is cadlag and is continuous at the times t when X t = a. If X is of continuous sample path, then so is X. In this construction, we can proceed along essentially the same line laid in [16] although some natural additional conditions on X and X including an off-diagonal finiteness of jumping measures will be required due to the lack of the symmetry and the path continuity. But we shall see that an integrability condition of the α-order approaching probability being imposed on X in [16] can be removed under a fairly general circumstance. As a typical example of a jump type Markov process, we consider in 6 the case where X is a censored symmetric α-stable process on an open set of R n studied by K. Bogdan, K. Burdzy and Z.-Q. Chen [3]. An example is also given on extending non-symmetric diffusions in Euclidean domains. Finally we remark at the end of 6 that the present results on the one point extensions can be applied to obtaining an extension to infinitely many points.

3 Extending Markov Processes in Weak Duality 3 2 Exit system and point process of excursions around a point Let E be a Lusin space (i.e. a space that is homeomorphic to a Borel sunset of a compact metric space), B(E) be the Borel σ-algebra on E and m be a σ-finite Borel measure on E. We consider a pair of Borel right processes X = (X t, ζ, P x ) and X = ( X t, ζ, P x ) on E that are in weak duality with respect to m: (C.1) Ĝ α f(x)g(x)m(dx) = f(x)g α g(x)m(dx) E E for every f, g B + (E) and α >, where G α, Ĝ α denote the resolvents of X, X respectively. We fix a point a E which is regular for a} with respect to X: (C.2) P a (σ a = ) = 1. Here σ a = inft > : X t = a} with the convention of inf :=. Under (C.1), we may and do assume that both X and X are of cadlag paths up to their life times (c. [21, 9]). Let E := E \ a}, m := m E, and [ ϕ(x) := P x (σ a < ), u α (x) := E x e ασ a ] for every x E. The corresponding functions for X will be denoted by ϕ and û α (x), respectively. For u, v B + (E ), (u, v) will denote the inner product of u and v in L 2 (E ; m ), that is, (u, v) := E u(x)v(x)m (dx). Denote by X = (Xt, ζ, P x) and X = ( X t, ζ, P x) the subprocesses of X and X killed upon leaving E, respectively. It is known that they are in weak duality with respect to m. The X -energy functional L () ( ϕ m, v) of the X -excessive measure ϕ m and an X -excessive function v is then well defined by L () 1 ( ϕ m, v) = lim t t ( ϕ P t ϕ, v), where P t is the transition semigroup of X (see [16, Lemma 2.1]). We shall now work with the exit system of X for the point a. To this end, it is convenient to take as the sample space Ω of the process X the space of all paths ω on E = E which are cadlag up to the life time ζ(ω) and stay at the cemetery after ζ. Thus, X t (ω) is just t-th coordinate of ω. Ω is equipped with the minimal completed admissible filtration F t, t } for X t, t }. The shift operator θ t is defined by X s (θ t ω) = X s+t (ω), s. We also introduce an operator k t, t, on Ω defined by Xs (ω) if s < t X s (k t ω) = if s t.

4 4 Zhen-Qing Chen, Masatoshi Fukushima, and Jiangang Ying We adopt the usual convention that any numerical function of E is extended to E by setting its value at to be zero. Let us consider the random time set M(ω) M(ω) := t [, ) : X t (ω) = a}, (2.1) where indicates the closure in [, ). The random set M(ω) is closed and homogeneous on [, ). Define R t (ω) := t + σ a (θ t ω) for every t > and L(ω) := sups > : s M(ω)}, with the convention that sup :=. The connected components of the open set [, ) \ M(ω) are called the excursion intervals. The collection of the strictly positive left end points of excursion intervals will be denoted by G(ω). We can easily see that t G(ω) if and only if R t (ω) < R t (ω), and in this case R t (ω) = t. In particular, L(ω) G(ω) whenever L(ω) <. We further define the operator i t, t, on Ω by i t = k σa θ t. Then i s ω : s G} and i s ω : s G, R s < } are by definition the collection of excursions and the collection of returning excursions respectively of the path ω away from F, while i L(ω) (ω) = θ L(ω) (ω) is the non-returning excursion whenever L(ω) <. Note that those excursions belong to the excursion space W specified by which can be decomposed as with W = k σa ω : ω Ω, σ a (ω) > }, (2.2) W = W + W } (2.3) W + = w W : σ a < } and W = w W : σ a = and ζ > }. Here denotes the path identically equal to. The unit mass δ a} concentrated at the point a is smooth in the sense of [11] because a} is not semipolar by the assumption (C.2). Hence there is a unique positive continuous additive functional (PCAF in abbreviation) l = l t, t } of X with Revuz measure δ a}. Clearly l is supported by a} and any PCAF of X supported by a} is a constant multiple of l. We call l the local time of X at the point a. Since the point a is assumed to be regular for a}, t : X t = a} has no isolated points, and the equilibrium 1-potential E x [e[ σa ] is regular in the sense of [2, Definition IV.3.2] because E x ] [e σa ] = c E x e t dl t on E for some c >. Thus according to [26, 9] (see also [1], [8], [12] and [2])), there exists a unique σ-finite measure P on Ω carried by σ a > } and satisfying

5 Extending Markov Processes in Weak Duality 5 P [ 1 e σa] < (2.4) such that [ ] [ ] E x Z s Γ θ s = P (Γ ) E x Z s dl s s G for x E, (2.5) for every non-negative predictable process Z and every non-negative random variable Γ on Ω. Here E is the expectation under the law P. The pair (P, l) is the predictable version of the exist system for a originated in Maisonneuve [26, 9]. The measure P is Markovian with respect to the transition semigroup of X. We are particularly concerned with the σ-finite measure Q on the space of excursions W induced from P by Q (Γ ) = E (Γ k σa ). The measure Q is Markovian with respect to the semigroup Pt, t } of X and satisfies [ ] [ ] E x Z s Γ i s = Q [Γ ] E x Z s dl s, x E, (2.6) s G for every non-negative predictable process Z s and every non-negative random variable Γ on W. We define for f B + (E) ν t (f) := Q [f(x t )] = E [f(x t ); t < σ a ], t >. By the Markov property of Q, we readily see that ν t : t > } is an entrance law for X : ν t P s = ν t+s. Proposition 2.1 (i) ν t } t> is the unique X -entrance law characterized by ϕ m = Moreover ν t (E ) is finite for each t >. (ii) Q [W ] = L () ( ϕ m, 1 ϕ). Proof. (i). We put ˇν α (f) = ν t dt. (2.7) e αt ν t (f)dt. Then, for f B + b (E) and for v C b (E) vanishing at a, we have, using (C.1), (2.6) and the Revuz formula [21, (2.13)], (û α, v)ĝαf(a) = (Ĝαf Ĝ αf, v) = (f, G α v G αv) [ ] [ ] s+σa θ s = E f m e αt v(x t )1 M c(t)dt = E f m e αt v(x t )dt σ a s M s [ ] σa [ ] = E f m e αs e αt v(x t )dt θ s = ˇν α (v)e f m e αs dl s s M = ˇν α (v)ĝαf(a).

6 6 Zhen-Qing Chen, Masatoshi Fukushima, and Jiangang Ying Hence û α m = ˇν α, (2.8) from which (2.7) follows by letting α. Since ϕ m is a purely excessive measure of X, the uniqueness follows (cf. [2]). The finiteness of ν t follows from (2.4). (ii). By (i) and [5, Lemma 3.1], L () ( ϕ m, v) = lim ν t (v) for any X -excessive t function v. Hence L () ( ϕ m, 1 ϕ) = lim t Q [(1 ϕ)(x t )] = lim t Q [1 σa= θ t ; t < ζ σ a ] = Q [W ]. Remark 1. In the next section, we shall identify Q with the characteristic measure n of the absorbed Poisson point process of excursions associated with l. Proposition 2.1 was first proved by Fukushima-Tanaka [16] for n in the case that X is an m-symmetric diffusion by making use of the Dirichlet form of X. In a recent paper of Fitzsimmons-Getoor [12], various properties of some basic quantities for the exit system of a one point set including those in the above proposition have been obtained in the most general setting that X is just a Borel right process with an excessive measure m, in which case X can be taken to be a dual left continuous moderate Markov process. But the present proof, taken from a recent paper by Chen-Fukushima-Ying [5], is simpler under the condition (C.1) as far as Proposition 2.1 is concerned. The next proposition is taken from Fitzsimmons-Getoor [12, (2.1) and (2.17)]. Recall that L(ω) := sup M(ω). Proposition 2.2 Put δ = P (σ a = ). Then the followings are true: (i) P a (l > t) = exp( δt), t >. (ii) P a (L < ) = or 1 according to δ = or δ >. Let τ t, t } be the right continuous inverse of l = l t, t }, that is, τ t := infs : l s > t}, (2.9) with the convention that inf =. Since l is supported by a, we have (cf. [4, 5]) P a -a.s. τ lt = R t for every t. We see from the above that, after removing from Ω a P a -negligible set, and in this case, L(ω) < if and only if l (ω) <,

7 Hence, if we let then Extending Markov Processes in Weak Duality 7 l (ω) = l L (ω), τ l (ω) = L(ω) and τ l (ω) =. J l (ω) := s (, ) : τ s (ω) < τ s (ω)}, J l (ω) := l t : t G(ω)} (2.1) and s J l (ω) implies that s = l t (ω) for some t G(ω) with τ s (ω) = R t (ω) = t and τ s (ω) = R t (ω). In particular, l (ω) J l (ω) whenever it is finite. Finally the W -valued point process p = p(ω) associated with the local time l is introduced by D p(ω) = J l (ω) and p s (ω) = i τs ω for s D p(ω). (2.11) Note that p s (ω) : s D p(ω) } W and p s (ω) : s D p(ω), τ s < } W + is the collections of excursions and of the returning excursions away from a, respectively, while p l (ω)(= θ L (ω)) W } is the non-returning excursion whenever l (ω) < or, equivalently, L(ω) <. The counting measure of p is defined by n p ((s, t], Λ) = 1 Λ (p u ), Λ B(W ), (2.12) u D p (s,t] and n p (t, Λ) = n p ((, t], Λ) is then F τt -adapted as a process in t. Using (2.1), we now make the time substitute in the relation (2.6) to obtain [ ] [ ] l E a Z τs Γ i τs = Q [Γ ] E a Z τs ds. (2.13) s J l Inserting the predictable process Z u = 1 (,τt ](u), we arrive at the formula holding for the counting measure of the point process p associated with l: E a [n p (t, Λ)] = Q [Λ] E a [t l ] for every t and Λ B(W ). (2.14) This formula will be utilized in the next section. 3 Characteristic measure of absorbed Poisson point process In this section, we continue to work with the setting in 2 and investigate properties of the point process (p s, D p ) defined by (2.11) for the local time

8 8 Zhen-Qing Chen, Masatoshi Fukushima, and Jiangang Ying l = l t, t } at the point a. By the observation made after (2.11), it then holds that l = T where T = infs > : p s W }}. (3.1) In view of Proposition 2.2, T is exponentially distributed with parameter δ = P (σ a = ). Lemma 3.1 Under measure P a, p is an absorbed Poisson point process with absorption time T in Meyer s sense ([27]), that is, ( ) P a n p ((r + s 1, r + t 1 ], Λ 1 ) H 1,, n p ((r + s n, r + t n ], Λ n ) H n Fτr = 1 T >r} P a (n p ((s 1, t 1 ], Λ 1 ) H 1,, n p ((s n, t n ], Λ n ) H n ) +1 T r} 1 H1 () 1 Hn (), (3.2) for any s 1 < t 1,, s n < t n, H 1,, H n Z +, r >, Λ 1,, Λ n B(W ). Proof. The proof is the same as in [27, 2] although [27] considered only the conservative case. In fact, the identity τ r+u = τ r + τ u θ τr implies n p ((r + s, r + t], Λ) = n p ((s, t], Λ) θ τr and consequently we see from (3.1) and the strong Markov property of X that the left hand side of (3.2) (with n = 1) equals P Xτr (n p ((s, t], Λ) H) = 1 T >r} P a (n p ((s, t], Λ) H) +1 T r} P (n p ((s, t], Λ) H), whose last factor is equal to 1 H (). By virtue of [27, 1], there is on a certain probability space ( Ω, P) a W - valued Poisson point process p = p, s > } with domain D ep satisfying the following property. Let T = infs > : p s W }} and consider the stopped process p s, s > }: p s = p s for s D p = D ep (, T ]. (3.3) Then the point process p s, s > } under P a and p s, s > } under P are equivalent in law. Let us denote by n the characteristic measure of the W -valued Poisson point process p, s > }. Theorem 3.2 It holds that n = Q. (3.4) Therefore n is a σ-finite measure on W with n(σ a > t) < for every t >, and n is Markovian with respect to the transition semigroup P t, t } of X. The X -entrance law ν t, t > } of n defined by

9 is characterized by Define δ by Extending Markov Processes in Weak Duality 9 ν t (f) = n(f(x t ); t < σ a ), t >, f B + (E) ν t dt = ϕ m. (3.5) δ = n(ζ = ). (3.6) Then T is exponentially distributed with parameter L () ( ϕ m, 1 ϕ) + δ : P( T ( )) > t) = exp t (L () ( ϕ m, 1 ϕ) + δ for every t >. (3.7) Moreover, ν t (E ) < for each t > and L () ( ϕ m, 1 ϕ) <. Proof. Since p s : s D ep, p s W + } and T are independent, we have by (3.1) E a [n p (t, Λ)] = Ẽ u D ep (,t e T ] 1 Λ ( p u ) = n(λ) Ẽ[t T ] = n(λ) E a [t l ], which compared with (2.14) leads us to (3.4). Identities (3.5) and (3.7) are the consequences of Propositions 2.1 as Q (W }) = Q (W ) + Q ( }) = L () ( ϕ m, 1 ϕ) + δ. Then σ-finiteness of n and the last statement follow from (2.4). 4 Duality preserving one-point extension Let E be a locally compact separable metric space, a be a non-isolated point of E and m be a σ-finite measure on E := E \ a}. Contrarily to the preceding two sections, we shall start in this section with two given strong Markov processes X and X on E that are in weak duality with respect to m and have no killings inside E. We are concerned with their possible duality preserving extensions X and X to E that admit no sojourn at a. It turns out that we need to allow X and X have killings at a in order to guarantee their weak duality but they are unique up to a parameter δ that represents the killing rate of X at a. We shall assume that we are given two Borel standard processes X = (X t, P x, ζ ) and X = ( X t, P x, ζ ) on E satisfying the next three conditions. (A.1) X and X are in weak duality with respect to m ; that is, for every α > and f, g B + (E ),

10 1 Zhen-Qing Chen, Masatoshi Fukushima, and Jiangang Ying Ĝ αf(x)g(x)m (dx) = E f(x)g αg(x)m (dx), E where G α and Ĝ α are the resolvent of X and X, respectively. (A.2) X and X are approachable to a} but admit no killings inside E : for every x E, P x ( ζ <, X ζ = a) > and P x ( ζ <, X ζ E ) =, (4.1) P x ( ζ ) <, X bζ = a > and P ( ζ x <, X ) b E ζ =. (4.2) Here for a Borel set B E, the notation Xζ B means that the left limit of Xt at t = ζ exists under the topology of E and takes values in B E. We use the same convention for X. We shall use the same notations as in [16]: for x E and α >, ϕ(x) := P ( [ ] x ζ <, Xζ = a) and u α (x) := E x e αζ : Xζ = a. (4.3) As in 2, the X -energy functional of X -excessive measure µ and X - excessive function v is denoted by L () (µ, v). The corresponding notations for X will be designated by ϕ, û α, L(). We use (u, v) to denote the inner product between u and v in L 2 (E, m ), that is, (u, v) = E u(x)v(x)m (dx). We say that a strong Markov process X (resp. X) on E is an extension of X (resp. X ) if the subprocess on E of X (resp. X) killed upon hitting the point a is identical in law to X (resp. X ). Let us now consider two Borel right processes X = (X t, P x, ζ) and X = ( X t, P x, ζ) on E satisfying the next four conditions. (1) X and X are in weak duality with respect to a σ-finite measure m on E with m E = m. (2) X and X are extensions of X and X respectively. (3) The point a is regular for itself with respect to X: P a (σ a = ) = 1, where σ a = inft > : X t = a} is the hitting time of a by X. (4) X admits no sojourn at the point a, that is, ( ) P x 1 a} (X s )ds = = 1 for every x E. Under (1), we can and do assume that both X and X possess cadlag paths up to their life times.

11 Extending Markov Processes in Weak Duality 11 Proposition 4.1 Assume that the above conditions (1), (2), (3) and (4) hold. Then (i) The measure m does not charging on a}: m(a}) = (ii) X admits no jumping from E to the point a: for every x E, P x (X t E, X t = a for some t (, ζ)) =, (4.4) (iii) X admits no jump from the point a to E in the following sense: P x (X t = a, X t E for some t (, ζ)) = for q.e. x E. (4.5) Here q.e. means except on an m-polar set for X. (iv) The one point set a} is not m-polar for X. Let functions ϕ and u α be defined as in (4.3). Then ϕ(x) = P x (σ a < ) and u α (x) = E x [ e ασ a ] for x E. (4.6) (v) u α, û α L 1 (E, m ) for every α >. Proof. (i). This is immediate from (1), (4) for X as Ĝ α f(a)m(a}) = f(x)g α 1 a} (x)m(dx) = for every f B + (E). (ii). It follows from (4.1) and (2) that E P x (X σa E, σ a < ) = for every x E. (4.7) For any open set O that has a positive distance from a}, let σ n a, n }, η n, n } be the stopping times defined by η =, σ a = σ a, η n = σ n 1 a + σ O θ σ n 1, σ n a a = η n + σ a θ η n (4.8) with an obvious modification after one of them becomes infinity. Clearly the time set t (, ζ(ω)) : X t (ω) O, X t (ω) = a} σ n a (ω); n =, 1, 2, }. Thus it follows from the strong Markov property of X and (4.7) that for every x E, P x (there is some t > such that X t O, X t = a) =. Letting O increase to E establishes (4.4). (iii). Clearly, property (ii) also holds for X: ( P x Xt E, Xt = a for some t (, ζ) ) = for every x E. (4.9)

12 12 Zhen-Qing Chen, Masatoshi Fukushima, and Jiangang Ying We combine the above with a time reversal argument based on the stationary Kuznetsov process (P, Z t, α < t < β) associated with X and X as was formulated in [21, 1] : the σ-finite measure P on a path space D((, ), E ) with a random birth time α and a random death time β is stationary under the time shift of the path, and furthermore, if we put Ẑ t = Z ( t) for t R, α = β and β = α, then Z t, t < β} (resp. Ẑt, t < β}) on Z E} (resp. Ẑ E}) is a copy of X t, t < ζ} (resp. X t, t < ζ}) under P m (resp. P m ). We shall use the formula (1.5) of [21, 1] which express a precise meaning of this property. Consider the set Λ = Z t = a and Z t E, for some t (α, β)}. Then Λ = Ẑt E and Ẑt = a, for some t ( α, β)}, and thus Λ = Λ r with r Q + Λ r = α < r < β, Ẑ t E, Ẑ t = a for some t > r}. According to (1.5) of [21, 1], P(Λ r ) is equal to the integral of the left hand side of (4.7) with respect to m for each rational r. Therefore P(Λ) =. Denote by h(x) the function of x E appearing in the left hand side of (4.5). By (1.5) of [21, 1] again, we have h(x)m(dx) = P (Z t = a and Z t E, for some t (, β), α < < β) E P(Λ) =. Consequently, h = m-a.e. and hence q.e. on E because h is X-excessive (cf. [5, 2]). (iv). On account of [2, p.59] (see also [21, Proposition 15.7] when E is a Lusin space), P x ( < σ a < σ a ) =, where σ a = inft : X t = a}, x E. On the other hand, (A.2) and (2) imply for ζ = σ a ζ that for x E, P x (σ a < σ a) P x (σ a <, X σa a) P x (ζ <, X ζ E ) =. Hence P x (σ a = σ a) = 1 and ϕ(x) = P x (ζ <, X ζ = a) = P x (σ a < ) for x E.

13 In particular, Extending Markov Processes in Weak Duality 13 P m (σ a < ) = ϕ(x)m(dx) > E by (A.2) and therefore a} is not m-polar for X. (v). By the strong Markov property of X, Ĝ α f(x) = Ĝ αf(x) + û α (x)ĝαf(a), x E. We can take a non-negative m-integrable function f on E such that Ĝαf(a) >. Then Ĝ α f(a)(û α, 1) (Ĝαf, 1) = (f, G α 1) 1 (f, 1) <, α yielding the m -integrability of û α. Similar, we have u α L 1 (E, m ). Theorem 4.2 Assume that X and X are two Borel right processes on E satisfying conditions (1), (2), (3) and (4) in this section. Let G α, α > } and Ĝα, α > } denote the resolvents of X and X, respectively. Then there exist constants δ, δ such that L () ( ϕ m, 1 ϕ) + δ = L () (ϕ m, 1 ϕ) + δ, (4.1) and for every f B + (E) and α >, G α f(a) = (û α, f) α(û α, ϕ) + L () ( ϕ m, 1 ϕ) + δ, (4.11) G α f(x) = G αf(x) + u α (x)g α f(a) for x E, (4.12) (u α, f) Ĝ α f(a) = α(u α, ϕ) + L () (ϕ m, 1 ϕ) + δ, (4.13) Ĝ α f(x) = Ĝ αf(x) + û α (x)ĝαf(a) for x E. (4.14) Corollary 4.3 Borel right processes X and X on E satisfying conditions (1)-(4) of this section are unique in law up to a parameter δ satisfying L() } δ max (ϕ m, 1 ϕ) L () ( ϕ m, 1 ϕ),. Proof of Theorem 4.2. In view of conditions (1)-(4) of this section and Proposition 4.1, X satisfies the conditions (C.1)-(C.2) of 2 so that Theorem 3.2 is applicable to X. The identity (4.12) is a simple consequence of the strong Markov property of X applied to the hitting time σ a. In order to show (4.11), we consider the local time l = l t, t } of X with Revuz measure δ a} and the W -valued point process p associated with l defined by (2.11). By Lemma 3.1, p under

14 14 Zhen-Qing Chen, Masatoshi Fukushima, and Jiangang Ying P a is an absorbed Poisson point process and admits the representation (3.3) in terms of a W -valued Poisson point process p defined on some probability space ( Ω, P) together with its hitting time T of W }. Let n be the characteristic measure of p. Then, for any non-negative predictable process a(t, w, ω), t, w W, ω Ω}, we have Ẽ [ ] a(s, p s, ω) = Ẽ a(s, w, ω)n(dw)ds, (4.15) s t W (,t] because the compensator of p equals t n( ) (cf. [23, II.3]). We now proceed along the same line as in [16, Remark 4.2]. The terminal time of w W is denoted by ζ(w): for w = k σa (ω) with ω Ω, ζ(w) = σ a (ω). We put for f B + (E ) ˇf α (w) = ζ(w) e αt f((w(t))dt, w W, α >. Note that t X t (ω) has only at most countably many discontinuous points. Thus by (2.2) and the condition (4), M(ω) has zero Lebesgue measure almost surely. So we have P a -a.s. e αt f(x t )dt = s<l = τs τ s e αt f(x t )dt + τ l e αt f(x t )dt s<l e ατs ˇfα (p s ) + e ατl ˇfα (p l ), (4.16) which is equivalent in law to e αs(s ) ˇfα ( p + s ) + e αs( T ) ˇfα ( p et ), under P, (4.17) s< e T where p + s, s > } is a Poisson point process defined by p + s = p s for s D ep + = s D ep : p s W + } and S(s) = r s ζ( p+ r ). The characteristic measure of p + s, s > } is the restriction n + of n on W +. First we claim that [ Ẽ e αs(s)] = exp( α(û α, ϕ)s). (4.18) Since e αs(s) 1 = r s e αs(r) e αs(r )} = } e αs(r ) e αζ(p+ r ) 1, r s it follows from (4.15) that [ Ẽ e αs(s)] 1 = c s [ Ẽ e αs(r)] dr,

15 with c = n + (1 e αζ ) = n(1 e αζ ; ζ < ) = n = α e αt n(t < ζ < )dt. Due to (3.5) (see also (2.8)), we have accordingly c = α Extending Markov Processes in Weak Duality 15 α ζ e αt ν t (ϕ)dt = α(û α, ϕ), e αt dt; ζ < which is finite by Proposition 4.1(v). The identity (4.18) then follows. On the other hand, we have from Theorem 3.2 and the basic properties of Poisson point processes, (i) T has an exponential distribution with exponent L () ( ϕ m, 1 ϕ) + δ, where δ is defined by (3.6). (ii) The three objects p + s, s > }, T and p et are independent. (iii) The law of p et is n (W }) 1 n = (L () ( ϕ m, 1 ϕ) + δ ) 1 n, where n is the restriction of n on W }. Taking these facts and formula (4.15) for p + into account, we get from (4.16),(4.17) and (4.18), G α f(a) = Ẽ e αs(s ) ˇfα ( p + s ) + e S( T e ) ˇfα ( p et ) s< et [ et ] = Ẽ e α(buα,ϕ)s ds n + ( ˇf α ) = = +Ẽ ( e α(buα,ϕ) e T ) (L () ( ϕ m, 1 ϕ) + δ ) 1 n ( ˇf α ) n + ( ˇf α ) α(û α, ϕ) + L () ( ϕ m, 1 ϕ) + δ + n( ˇf α ) α(û α, ϕ) + L () ( ϕ m, 1 ϕ) + δ, n ( ˇf α ) } α(û α, ϕ) + L () ( ϕ m, 1 ϕ) + δ which coincides with the right hand side of (4.11) because we have from Theorem 3.2 n( ˇf α ) = e αt ν t (f)dt = (û α, f). (4.13) can be obtained analogously. Under the weak duality assumption (1), the denominators of (4.11) and (4.13) must be equal. Since (û α, ϕ) = (u α, ϕ) (see the first two equations in the proof of Lemma 5.8), we must have the identity (4.1).

16 16 Zhen-Qing Chen, Masatoshi Fukushima, and Jiangang Ying In the above proof, we did not use the property of X having no jumps from the point a to E, which is proved in Proposition 4.1(iii). But this property reflects on the following property of the characteristic measure n of the absorbed Poisson point process p considered in the above proof. Proposition 4.4 n w() a} =. Proof. By (4.5), we have E a ( s G 1 Λ i s ) = for Λ = w() a} and we get n(λ) = Q (Λ) = from (2.6) and (3.4). Remark 2. In this section, we have assumed that E is a locally compact separable metric space. But all assertions in this section remain valid for a general Lusin space E except that the identities (4.6), (4.12), (4.14) hold only for q.e. x E rather than for every x E, because we need to replace the usage of [2, p59] by [21, (15.7)] in the proof of (4.6). The uniqueness statement in Corollary 4.3 should be modified accordingly in the Lusin space case. We also note that the expression (4.11) of the resolvent has been obtained in [12] by a different method for a general right process X and its excessive measure m, in which case X can be taken to be a dual moderate Markov process. But the present proof is more useful in the next section. 5 Extending Markov process via Poisson point processes of excursions As in 4, let E be a locally compact separable metric space and a be a fixed non-isolated point of E and m be a σ-finite measure on E := E \ a} with Supp[m ] = E. We extend m to a measure m on E by setting m(a}) =. Note that m could be infinity on a compact neighborhood of a in E. Let E = E } be the one point compactification of E. When E is compact, is added as an isolated point. 5.1 Excursion laws in duality We shall assume that we are given two Borel standard processes X = X t, P x, ζ } and X = X t, P x, ζ } on E satisfying the following conditions. (A.1) X and X are in weak duality with respect to m, that is, for every α >, and f, g B + (E ), Ĝ αf(x)g(x)m (dx) = f(x)g αg(x)m (dx), E E where G α and Ĝ α are the resolvents of X and X, respectively.

17 (A.2) X and X satisfy, for every x E, Extending Markov Processes in Weak Duality 17 P x ( ζ <, X ζ = a) >, P ( x ζ <, Xζ a, }) = P x(ζ < ), (5.1) P ( ζ ) x <, X bζ = a >, ( ζ ) <, X bζ a, } = P x( ζ < ).x (5.2) P x Here, as in 4, for a Borel set B E, the notation Xζ B means that the left limit of t Xt at t = ζ exists under the topology of E and takes values in B. The first condition in (5.1) (resp. (5.2)) means that X (resp. X ) is approachable to the point a, while the second condition in (5.1) (resp. (5.2)) implies that X (resp. X ) admits no killings inside E. As in 4, we put for x E and α >, ϕ(x) := P ( [ ] x ζ <, Xζ = a) and u α (x) := E x e αζ ; Xζ = a. (5.3) The corresponding notations for X will be designated by ϕ and û α. As in 2, the X -energy functional L () ( ϕ m, v) of the X -excessive measure ϕ m and an X -excessive function v is well defined. Similarly the X -energy functional L () (ϕ m, v) is well defined. The inner product of u, v in L 2 (E, m ) will be denoted by (u, v), that is, (u, v) = E u(x)v(x)m (dx). The space of all bounded continuous functions on E will be denoted by C b (E ). We impose some more assumptions: (A.3) u α, û α L 1 (E, m ) for every α >. (A.4) G f(x), Ĝ f(x), x E, are lower semi-continuous for any Borel f. Here G denotes the -order resolvent of X : [ ] G f(x) := E x f(x t )dt = lim G α αf(x) for x E and Borel function f on E. The -order resolvent Ĝ of X is similarly defined. We note that, if G α(c b (E )) C b (E ), Ĝ α(c b (E )) C b (E ), α >, then (A.4) is satisfied by the monotone class lemma. The next condition will be imposed only when X is non-symmetric, namely, when X X. (A.5) lim x a u α (x) = lim x a û α (x) = 1, for every α >.

18 18 Zhen-Qing Chen, Masatoshi Fukushima, and Jiangang Ying The next condition (A.6) will be imposed only when X is not a diffusion, namely, when P m ( X t X t for some t (, ζ ) ) >. Note that X then has the same property in view of [21, 1]. According to [31, (73.1), (47.1)], the standard process X on E has a Lévy system (N, H) on E. That is, N(x, dy) is a kernel on (E, B(E )) and H is a PCAF of X in the strict sense with bounded 1-potential such that for any nonnegative Borel function f on E (E }) that vanishes on the diagonal and is extended to be zero outside E E, E [ t ] x f(xs, Xs ) = E x f(xs, y)n(xs, dy)dh s (5.4) s t E for every x E and t. Similarly, the standard process X has a Lévy system ( N, Ĥ). Let µ H and µ bh be the Revuz measure of the PCAF H of X and the PCAF Ĥ of X with respect to the measure m on E, respectively. Define J (dx, dy) := N(x, dy)µ H (dx) and Ĵ (dx, dy) := N(x, dy)µ bh (dx). (5.5) The measures J and Ĵ are called the jumping measure of X and X, respectively. It is known (see [18]) that We now state the condition (A.6). J (dx, dy) = Ĵ(dy, dx) on E E. (5.6) (A.6) Either E \ U is compact for any neighborhood U of a in E, or for any open neighborhood U 1 of a in E, there exists an open neighborhood U 2 of a in E with U 2 U 1 such that J (U 2 \ a}, E \ U 1 ) < and Ĵ (U 2 \ a}, E \ U 1 ) <. Throughout this section, we assume that we are given a pair of Borel standard processes X and X on E satisfying conditions (A.1), (A.2), (A.3), (A.4), and additionally (A.5) in non-symmetric case and (A.6) in non-diffusion case. We aim at constructing (see Theorem 5.15) under these conditions their right process extensions X, X to E with resolvents (4.11), (4.13) respectively. Theorem 5.16 will then be concerned with some stronger conditions (A.1) and (A.4) to ensure the quasi-left continuity of the constructed processes so that they become standard. We note that, if X is an m -symmetric diffusion on E, then the present conditions (A.2), (A.3) are the same as the conditions A.1, A.2, A.3 assumed in [16, 4], while the present (A.4) is weaker than A.4 of [16, 4] as

19 Extending Markov Processes in Weak Duality 19 is noted in the paragraph below (A.4). Therefore the results of this paper extend the construction problem treated in [16, 4] to a more general case. However we shall proceed along the same line as was laid in [16, 4]. In Theorem 5.17 at the end of this section, we shall present a stronger variant (A.2) of the condition (A.2) and prove using a time change argument that, under the conditions (A.1), (A.2), (A.4) and additionally (A.5) in non-symmetric case and (A.6) in non-diffusion case, the integrability condition (A.3) holds automatically and therefore can be dropped. As is shown in [5, Lemma 3.1], the measure ϕ m is X -purely excessive and accordingly there exists a unique entrance law µ t } t> for X characterized by ϕ m = µ t dt. (5.7) Analogously there exists a unique X -entrance law µ t } t> characterized by ϕ m = µ t dt. (5.8) Further by [5, Lemma 3.1], the Laplace transforms of µ t, µ t satisfy e αt µ t, f dt = (û α, f) and e αt µ t, f dt = (u α, f) (5.9) for every α > and f B + (E ). On account of the assumption (A.3), we then have that for every t >, µ t (E ) <, µ t (E ) <, and 1 µ s (E )ds <, We now introduce the spaces W and W of excursions by 1 µ s (E )ds <. (5.1) W = w : a cadlag function from (, ζ(w)) to E for some ζ(w) (, ]}, } W = w W : if ζ(w) < then w(ζ(w) ) := lim w(t) a, } (5.11). t ζ(ω) We call ζ(w) the terminal time of the excursion w. We are concerned with a measure n on the space W specified in terms of the entrance law µ t, t > } and the transition semigroup Pt, t } of X by [ n ] f 1 (w(t 1 ))f 2 (w(t 2 )) f n (w(t n ))n(dw) = E µt1 f k (Xt k t 1 ) W = µ t1 f 1 P t 2 t 1 f 2 P t n 1 t n 2 f n 1 P t n t n 1 f n, (5.12) for any < t 1 < t 2 < < t n, f 1, f 2,, f n B b (E ). Here, we use the convention that w W satisfies w(t) := for w W and t ζ(w), and any k=1

20 2 Zhen-Qing Chen, Masatoshi Fukushima, and Jiangang Ying function f on E is extended to E by setting f( ) =. Further, on the right hand side of (5.12), we employ an abbreviated notation for the repeated operations Proposition 5.1 (5.12). µ t1 ( f 1 P t 2 t 1 ( f 2 P t n 1 t n 2 ( f n 1 P t n t n 1 f n ) )). There exists a unique measure n on the space W satisfying Proof. Let n be the Kuznetsov measure on W uniquely associated with the transition semigroup Pt, t } and the entrance rule η u, u R} defined by η u = for u and η u = µ u for u >, as is constructed in [8, Chap XIX, 9] for a right semigroup. Because of the present choice of the entrance rule, it holds that the random birth time α for the Kuznetsov process is identically (cf. [2, p54]). On account of the assumption (A.2) for the standard process X on E, the same method of the construction of the Kuznetsov measure as in [8, Chap.XIX, 9] works in proving that n is carried on the space W and satisfies (5.12). We call n the excursion law associated with the entrance law µ t } for X. It is strong Markov with respect to the transition semigroup P t, t } of X. Analogously we can introduce the excursion law n on the space W associated with the entrance law µ t for X. We split the space W of excursions into two parts: W + := w W : ζ(w) < and w(ζ ) = a} and W := W \ W +. (5.13) For w W +, we define time-reversed path ŵ W by ŵ(t) := w((ζ t) ) = lim t t w(ζ t ), < t < ζ. (5.14) The next lemma asserts that the excursion laws n and n restricted to W + are interchangeable under this time reversion. Lemma 5.2 For any t k > and f k B b (S ), (1 k n), n } n f k (w(t t k )); W + = µ t1 f 1 Pt 2 f 2 Pt n 1 f n 1 Pt n f n ϕ, k=1 (5.15) n } n } n f k (w(t t k )); W + = n f k (ŵ(t t k )); W +. k=1 k=1 (5.16)

21 Extending Markov Processes in Weak Duality 21 Proof. (5.15) readily follows from (5.12) and the Markov property of n. As for (5.16), we observe that, for α 1,, α n >, e P n } )k=1 n α k n f k (w(t t k )); W + dt 1 dt n k=1 = nf (w); ζ <, w(ζ ) = a}, (5.17) where, with t + :=, F (w) = n! n <t 1< <t n<ζ k=1 } e α k(t k t k 1 ) f k (w(t k )) dt 1 dt n. Hence, for (5.16), it suffices to prove for f k C b (E ), 1 k n, nf (w); ζ <, w(ζ ) = a} = nf (ŵ); ζ <, w(ζ ) = a}. (5.18) Changing of variables ζ t k = s k for k n in the following expression F (ŵ) = n! n <t 1< <t n<ζ k=1 where t :=, and noting that } e α k(t k t k 1 ) f k (w((ζ t k ) )) dt 1 dt n, s = ζ and < t 1 < < t n < ζ if and only if < s n < < s 1 < ζ, we obtain F (ŵ) = n! where Γ s1 s n (w) = = n! n k=2 n <s n< <s 1<ζ k=1 <s n< <s 1< } e α k(s k 1 s k ) f k (w(s k )) ds 1 ds n Γ s1 s n (w)ds 1 ds n, } e α k(s k 1 s k ) f k (w(s k )) e α1(ζ s1) f 1 (w(s 1 ))1 (,ζ) (s 1 ). On the other hand, we get from (5.1) and the Markov property of n that n Γ s1s 2 s n (w); ζ <, w(ζ ) = a} = n f n (w(s n ))e αn(sn 1 sn) f 2 ((w(s 2 ))e α2(s1 s2) f 1 (w(s 1 ))u α1 (w(s 1 )); s 1 < ζ} = e P n k=2 α k(s k 1 sk µ sn f n P sn 1 s n f n 1 P sn 2 s n 1 f n 1 P s 2 s 3 f 2 P s1 s 2 f 1 û α1.

22 22 Zhen-Qing Chen, Masatoshi Fukushima, and Jiangang Ying Therefore, = n F (ŵ); ζ <, w(ζ ) = a} ds n µ sn f n Ĝ α n f n 1 Ĝ α n 1 f 3 Ĝ α 3 f 2 Ĝ α 2 f 1 û α1. In view of (5.8), the weak duality (A.1), (5.15) and (5.17), we arrive at n F (ŵ); ζ <, w(ζ ) = a} = ϕ m, f n Ĝ αn f n 1 Ĝ αn 1 f 3 Ĝ α3 f 2 Ĝ α2 f 1 û α1 ) = (f n ϕ, Ĝ αn f n 1 Ĝ αn 1 f 3 Ĝ α3 f 2 Ĝ α2 f 1 û α1 = ( f 1 G α 2 f 2 G α 3 f 3 G ) α n f n ϕ, û α1 = e α1t1 µ t1 f 1 G α 2 f 2 G α 3 f 3 G α n f n ϕdt 1 = n F (w); ζ < and w(ζ ) = a}, the desired identity (5.18). This establishes (5.16). Next we define W a := w W : w(+) := lim t w(t) = a}. (5.19) Lemma 5.3 n W \ W a } = and n W \ W a } =. Proof. The preceding lemma implies that We then have for each t > n W + \ W a } = n W + (w(+) = a) c} = n W + (ŵ(+) = a) c} = n W + (w(ζ ) = a) c} =. n ϕ(w(t)); (ζ > t) (w(+) = a) c } = n (W + \ W a ) (ζ > t) } =. As ϕ(x) > for every x E by the assumption (A.2). we conclude that n (W \ W a ) (ζ > t)} = for every t >, and therefore n (W \ W a )} = after letting t. The same property of n can be shown analogously.

23 Extending Markov Processes in Weak Duality 23 Lemma 5.4 For any neighborhood U of a in E, define τ U (w) = inft > : w(t) / U} for w W. Then n τ U < ζ} < and n τ U < ζ} <. Proof. We only give a proof for n. Let V be any neighborhood of a in E. It suffices to show n(τ U < ζ) < for some neighborhood U of a with U V. We choose such U as follows. Let us fix a relatively compact open neighborhood U 1 of a in E. When X is a diffusion, we put U = V U 1. When X is not a diffusion and the second condition of (A.5) is fulfilled, we take U 2 in the condition for U 1 and put U = V U 2. By virtue of the relation ϕ u 1 = G 1ϕ = G u 1 and the assumption (A.4), the function G 1ϕ is lower semi-continuous on E. Furthermore, since ϕ is X -excessive and strictly positive by assumption (A.2), G 1ϕ is moreover strictly positive on E. As U 1 is compact in E, Since G 1ϕ(x) = δ := 1 2 inf G 1ϕ(x) >. (5.2) x U 1\U e t P x ( t < ζ <, X ζ = a) dt, we have P x ( δ < ζ <, X ζ = a) > δ for every x U 1 \ U. (5.21) We shall use the notation τ U not only for w W but also for the sample path of the Markov process X. Using the preceding lemma, we have n τ U < ζ } = lim n ɛ < τ U < ζ } = lim µ ɛ (dx)p x τu < ζ } = I + II, ɛ ɛ where I : = lim ɛ II : = lim ɛ From (5.21) and (5.1), it follows that U U U µ ɛ (dx)p x ( τu < ζ, X τ U U 1 \ U ), µ ɛ (dx)p x ( τu < ζ, X τ U E \ U 1 ).

24 24 Zhen-Qing Chen, Masatoshi Fukushima, and Jiangang Ying [ I lim ɛ µ ɛ (dx)e x δ 1 ( P X τu δ < ζ <, Xζ = a) ; U τ U < ζ, Xτ U U 1 \ U ] δ 1 lim µ ɛ (dx)p ɛ x(δ < ζ <, Xζ = a) E δ 1 lim µ ɛ (dx)p ɛ x(δ < ζ ) E = δ 1 lim ɛ µ ɛ+δ (E ) δ 1 µ δ (E ) <. II may not vanish when X is not a diffusion. In this case, let (N(x, dy), H) be the Lévy system of X appearing in the condition (A.5). Note that [ τu ] II = lim µ ɛ (dx)e x 1 U (Xs )N(Xs, E \ U 1 )dh s ɛ U [ ] lim µ ɛ (dx)e x 1 U (Xs )N(Xs, E \ U 1 )dh s ɛ E = lim µ ε (dx) G µ K (x) ɛ E where µ K (dx) := 1 U (x)n(x, E \ U 1 )µ H (dx) is the Revuz measure of the PCAF of X K t := t 1 U (X s )N(X s, E \ U 1 )dh s, t, and G µ K (x) := E x [K ]. Note that µ K is a finite measure on E by assumption (A.5). For α > and x E, we define [ ] G αµ K (x) := E x e αt dk t. Observe that αg αg µ K increases to G µ K as α. We have, by (5.7), the identity G αg µ K = G G αµ K and [21, (9.3)], µ ε (dx) G µ K (x) = lim α µ ε (dx) G G αµ K (x) E α E = lim α lim α µ ε P t, αg αµ K dt = lim α αĝ α ϕ, µ K = µ t, αg αµ K dt = lim ϕ m, αg αµ K α E ϕ(x)µ K (dx) µ K (E ) <.

25 Extending Markov Processes in Weak Duality 25 Hence we get the desired finiteness of II. When the first condition of (A.5) is fulfilled, the first half of the preceding proof is enough if we replace U, U 1 with V, E respectively. Lemma 5.5 n(w ) = L ( ϕ m, 1 ϕ) < and n(w ) = L (ϕ m, 1 ϕ) <. Proof. Since n ( ζ > t; W ) = µ t, 1 ϕ, the first identity follows from [5, Lemma 3.1] by letting t. Take a relatively compact neighborhood U of a in E. Since a E and is a one-point compactification of E, we have Hence for any t >, ζ < and w(ζ ) = } τ U < ζ}. (5.22) n ( W ) = n ζ <, w(ζ ) = } + n ζ = } n τ U < ζ} + n ζ > t} = n τ U < ζ} + µ t (E ), which is finite by Lemma 5.4 and (5.1). The second assertion can be shown similarly. 5.2 Poisson point processes on W a } and a new process X a By Lemma 5.3, the excursion law n is concentrated on the space W a defined by (5.19). In correspondence to (5.13), we define W + a := w W + : lim t w(t) = a} and W a := w W : lim t w(t) = a}, so that W a = W a + + Wa. In the sequel however, we shall employ slightly modified but equivalent definitions of those spaces by extending each w from an E -valued excursion to E-valued one as follows: W a = w : a cadlag function from [, ζ(w)) to E for some ζ(w) (, ] withw() = a, w(t) E for t (, ζ(w)) and w(ζ(w) ) a, } if ζ(w) < }. (5.23) Any w W a with the properties ζ(w) < and w(ζ(w) ) = a will be regarded to be a cadlag function from [, ζ(w)] to E by setting w(ζ(w)) = a. We further define W + a := w : a cadlag function from [, ζ(w)] to E for some ζ(w) (, ) with w(t) E for t (, ζ(w)) and w() = w(ζ(w)) = w(ζ(w) ) = a}, W a := W a \ W + a.

26 26 Zhen-Qing Chen, Masatoshi Fukushima, and Jiangang Ying The excursion law n will be considered to be a measure on W a defined by (5.23). Let us add an extra point to W a which represents a specific path constantly equal to. Fix a non-negative constant δ and we assign a point mass δ to } and extend the measure n on W a to a measure n on W a } by n(λ) if Λ Wa n(λ) = (5.24) n(λ W a ) + δ if Λ for Λ W a }. The restrictions of n to W + a and W a } are denoted by n + and n, respectively. Let p = p s : s D p } be a Poisson point process on W a } with characteristic measure n defined on an appropriate probability space (Ω a, P). We then let p + and p be the point processes obtained from p by restricting to W + a and W a } respectively, that is, D p + = s D p : p s W a + } and D p = s D p : p s Wa }}. (5.25) Then p + s, s > }, p s, s > } are mutually independent Poisson point processes on W a + and Wa } with characteristic measures n + and n, respectively. Clearly, p s = p + s + p s. Recall that ζ(p + r ) denotes the terminal time of the excursion p + r. We define J(s) := r s ζ(p + r ) for s > and J() :=. (5.26) Lemma 5.6 (i) J(s) < a.s. for s >. (ii) J(s)} s is a subordinator with [ E e αj(s)] = exp ( α(û α, ϕ)s). (5.27) Proof. (i) We write J(s) as J(s) = I + II with I := ζ(p + r ) and II := ζ(p + r ). r s, ζ(p + r ) 1 r s, ζ(p + r )>1 Since n + (ζ > 1) µ 1 (E ) < by (5.1), r in the sum II is finite a.s. and hence II < a.s. On the other hand, E(I) = s n + (ζ; ζ 1) s n + (ζ 1) 1 } = s n + 1 (,ζ) (t)dt = s 1 n + (ζ > t)dt s 1 µ t (E )dt, which is finite by (5.1). Hence I < a.s. (ii) This can be shown exactly in the same way as that for (4.18) in the proof of Theorem 4.2 by using the identity (5.9).

27 Extending Markov Processes in Weak Duality 27 In view of Lemma 5.4 and Lemma 5.6, by subtracting a P-negligible set from Ω a if necessary, we may and do assume that the next three properties hold for every ω Ω a : J(s) < for every s >, (5.28) lim J(s) =, (5.29) s and, for any finite interval I (, ) and any neighborhood U of a in E, s I : τu (p + s ) < ζ(p + s ) } is a finite set. (5.3) Let T be the first time of occurrence of the point process p s, s > }, namely, T = infs > : s D p }. (5.31) Since by Lemma 5.5 n (W a }) = n(w a ) + δ = L ( ϕ m, 1 ϕ) + δ <, we see that T and p T are independent and P(T > t) = e (L( bϕ m,1 ϕ)+δ)t and p dist T = (L( ϕ m, 1 ϕ) + δ ) 1 n. (5.32) We are now in a position to produce a new process X = X t, t } out of the point processes of excursions p ±. (i) For t < J(T ), there is an s such that We define J(s ) t J(s). X a t := p + s (t J(s )) if J(s) J(s ) >, a if J(s) J(s ) =. (5.33) It is easy to see that X a is well-defined. (ii) If p T W a, then we define ζ ω := J(T ) + ζ(p T ) and Xa t := p T (t J(T )) for J(T ) t < ζ ω. (5.34) (iii) If p T =, then we define In this way, the E-valued path ζ ω := J(T ). (5.35) X a t, t < ζ ω } is well-defined and enjoys the following properties:

28 28 Zhen-Qing Chen, Masatoshi Fukushima, and Jiangang Ying X a = a, is cadlag in t [, ζ ω ) and continuous when X a t = a, and X a ζ ω a, } whenever ζ ω <. (5.36) The second property is a consequence of (5.3). If p T W a and ζ ω <, then Xζ a =. If T <, ω p T =, then T / D p + and hence by (5.35), we have Xζ a = ω Xa J(T ) = a. Thus the third property holds. For this process X a = Xt a, t < ζ ω, P}, let us put [ ] ζω G α f(a) = E e αt f(xt a )dt, α >, f B(E). (5.37) Similarly we assign a non-negative mass δ to the death path and extend the measure n on W a to a measure ˆn on W a }. By making use of the Poisson point process p on W a } with the characteristic measure ˆn on a certain probability space ( Ω a, P), we can construct a cadlag process X a t, t < ζˆω, P) on E quite analogously. The corresponding quantity to (5.37) is denoted by Ĝαf(a). We can then obtain the first identity of the next proposition exactly in the same way as in the proof of Theorem 4.2 using (5.9), Lemma 5.6 and (5.32). An analogous consideration gives the second identity. Proposition 5.7 For α > and f B(E), it holds that G α f(a) = Ĝ α f(a) = For α > and f B(E), define (û α, f). α(û α, ϕ) + L () ( ϕ m, 1 ϕ) + δ (5.38) (u α, f) α(u α, ϕ) + L (ϕ m, 1 ϕ) + δ. (5.39) G α f(x) := G αf(x) + G α f(a)u α (x) for x E, (5.4) Ĝ α f(x) := Ĝ αf(x) + Ĝαf(a)û α (x) for x E. (5.41) Lemma 5.8 G α, α > } and Ĝα, α > } are sub-markovian resolvents on E. They are in weak duality with respect to m if and only if L () ( ϕ m, 1 ϕ) + δ = L () (ϕ m, 1 ϕ) + δ. (5.42) Proof. By making use of the resolvent equations for G α, Ĝ α, their weak duality with respect to m and the equations u α (x) u β (x) + (α β)g αu β (x) =, α, β >, x E, (5.43) û α (x) û β (x) + (α β)ĝ αû β (x) =, α, β >, x E, (5.44) we can easily check the resolvent equations

29 Extending Markov Processes in Weak Duality 29 G α f(x) G β f(x) + (α β)g α G β f(x) =, x E, Ĝ α f(x) Ĝβf(x) + (α β)ĝαĝβf(x) =, x E. Moreover we get as in [16, Lemma 2.1] that αg α 1(x) = αg α(û α, ϕ) + α(û α, 1 ϕ) α1(x) + u α (x) α(û α, ϕ) + L( ϕ m, 1 ϕ) + δ 1 u α (x) + u α (x) = 1, x E, and similarly, αg α 1(a) 1. The m-weak duality Ĝ α f(x)g(x)m(dx) = E E f(x)g α g(x)m(dx), f, g B + (E), holds if and only if the denominators of the right hand sides of (5.38) and (5.39) coincide. Since (û α, ϕ) = (u α, ϕ) by the above equations for u α, û α, we get the last conclusion. 5.3 Regularity of resolvent along the path of X a Let U n } be a decreasing sequence of open neighborhoods of the point a in E such that U n U n+1 and U n = a}. For α > and < ρ < 1, let n=1 A = A α,ρ := x E : u α (x) < ρ}. We then define σ n := inft > : X t U n E }, τ n := inft > : X t U n A}, and σ := lim n σ n, with the convention that inf =. The stopping time σ may be called the approaching time to a of X. The next lemma can be proved exactly in the same way as the proof of [16, Lemma 4.7]. Lemma 5.9 For any α >, ρ (, 1) and x E, Lemma 5.1 lim n P x τ n < σ < } =. (5.45) The following are ture. (i) For any x E, P x-a.s. on σ < }, lim t σ u α (X t ) = 1 for every α >. (5.46)

30 3 Zhen-Qing Chen, Masatoshi Fukushima, and Jiangang Ying (ii) n(λ W a + ) = where } Λ = w W a : α >, lim inf u α (w(t)) < 1. t ζ (iii) n( Λ) = where } Λ = w W a : α >, lim inf û α (w(t)) < 1. t Proof. Let < ρ < 1. If σ < and if lim t σ u α (Xt ) < ρ, then for any small ɛ > there exists t (σ ɛ, σ) such that u α (Xt ) < ρ, and so τ n < σ for all n. Therefore by the preceding lemma ( ) P x lim inf u α (Xt ) < ρ, σ < =. t σ Since u α is decreasing in α and ρ can be taken arbitrarily close to 1, we obtain (5.46). (ii) follows from (i) as n(λ W a + ) = lim n(λ W a + ɛ < ζ}) ɛ ( = lim ɛ E µ ɛ (dx)p x lim inf u α (Xt ) < 1, σ < for every α > t σ ) =. (iii). Part (ii) combined with Lemma 5.2 and a similar reasoning as in the proof of Lemma 5.3 leads us to n( Λ W + a ) = n(ŵ Λ} W + a ) =, and also n( Λ) =. Denote by Q + the set of all positive rational number and by C b (E) the space of all bounded continuous functions on E. Let us fix an arbitrary countable subfamily L of C b (E). We extend functions u α (x) and G αf(x) for f C b (E) to be functions on E by setting u α (a) = 1 and G αf(a) = respectively. Functions û α and Ĝ αf are similarly extended to E. As u α and G αf for a non-negative f C b (E) are α-excessive with respect to the process X, it is well-known (cf. [2]) that u α (Xt ), G αf(xt ) are right continuous in t [, ζ) P x a.s. x E. (5.47) Suppose that X is m -symmetric: X = X. Then u α = û α and hence by Lemma 5.1 ( n lim inf u α (w(t)) < 1 t ) =.