Poisson point processes attached to symmetric diffusions. Masatoshi Fukushima Hiroshi Tanaka

Transcription

1 Research Report KSTS/RR-4/1 Jan. 13, 24 Poisson point processes attached to symmetric diffusions by Masatoshi Fukushima Hiroshi Tanaka Masatoshi Fukushima Kansai University Hiroshi Tanaka Department of Mathematics Faculty of Science and Technology Keio University 24 KSTS Hiyoshi, Kohoku-ku, Yokohama, Japan

2 KSTS/RR-4/1 January 13, 24 Poisson point processes attached to symmetric diffusions by Masatoshi Fukushima Department of Mathematics, Faculty of Engineering, Kansai University, Suita , Japan and Hiroshi Tanaka Miyamaedaira, Miyamae-ku, Kawasaki Japan - ABSTRACT. -Let a be a non-isolated point of a topological space S and X =(X t, t< ζ,p x ) be a symmetric diffusion on S = S \a such that P x (ζ <,X ζ = a) >,x S. By making use of Poisson point processes taking values in the spaces of excursions around a whose characteristic measures are uniquely determined by X, we construct a symmetric diffusion X on S with no killing inside S which extends X on S. We also prove that such a process X is unique in law and its resolvent and Dirichlet form admit explicit expressions in terms of X. Keywords: symmetric diffusion, Poisson point process, excursions, entrance law, energy functional, Dirichlet form 1 Introduction Let S be a locally compact separable metric space and a be a non-isolated point of S. We put S = S \a. The one point compactification of S is denoted by S. When S is compact already, is added as an isolated point. Let m be a positive Radon measure on S with Supp[m] =S.mis extended to S by setting m(a) =. We assume that we are given an m-symmetric diffusion X =(X t,p x )ons with life time ζ satisfying the following four conditions: A.1 P x (ζ <, X ζ a ) =P x (ζ < ), x S. We define the functions ϕ(x), u α (x), α >, of x S by ϕ(x) =P x (ζ <, X ζ = a), u α(x) =E x (e αζ ; X ζ = a). A.2 ϕ(x) >, x S, A.3 u α L 1 (S ; m), α >. A.4 u α C b (S ), G α (C b(s )) C b (S ), α >, where G α is the resolvent of X and C b (S ) is the space of all bounded continuous functions on S. By making use of excursion-valued Poisson point processes whose characteristic measures are uniquely determined by X, or to be a little more precise, by piecing together 1

3 KSTS/RR-4/1 January 13, 24 those excursions which start from a and return to a and then possibly by adding the last one that never returns to a, we shall construct in 4 of the present paper a process X on S satisfying (1) X is an m-symmetric diffusion process on S with no killing inside S, (2) X is an extension of X : the process on S obtained from X by killing upon the hitting time of a is identical in law with X. We call a process X on S satisfying (1),(2) a symmetric extension of X. We shall also prove in 5 that, under conditions A.1, A.2 for the given m-symmetric diffusion X on S, its symmetric extension is unique in law, satisfies condition A.3 automatically and admits the resolvent expressible as G α f(x) =G α f(x)+u α(x) G α f(a), x S, G α f(a) = (u α,f) α(u α,ϕ)+l(m,ψ), where (, ) denotes the inner product in L 2 (S ; m) and L(m,ψ) is the energy functional in Meyer s sense [21] of the X -excessive measure m = ϕ m and X -excessive function ψ =1 ϕ. Furthermore the associated Dirichlet form (E, F) onl 2 (S; m) will be seen in 5 to have the following simple expression; if we donote by F e its extended Dirichlet space, then F e = w = u + cϕ : u F,e, c constant, F = F e L 2 (S; m), E(w, w) =E(u,u )+c 2 E(ϕ, ϕ), E(ϕ, ϕ) =L(m,ψ), where (F,e, E) is the extended Dirichlet space for the given diffusion X. In 6, we shall present four examples. Example 6.1 concerns the uniqueness of the symmeric extension of the one dimensional absorbing Brownian motion. Example 6.2 treats the case where S is a bounded open subset of R d, (d 1), S = S a is the one point compactification of S and X is the absorbing Brownian motion on S. In this case, ϕ(x) =1, x S. The resulting Dirichlet form on L 2 (S; m) (m is the Lebsegue measure on S extended to S by m(a) = ) is given by F = w = u + c : u H 1 (S ), cconstant, E(w, w) = 1 u 2 (x)dx, 2 S which is easily seen to be regular, strongly local and irreducible recurrent. A more general Dirichlet form of this type will be presented in 3.2. This type of Dirichlet form first appeared in the paper [8] by the first author and it is recently utilized in a study of the asymptotics of the spectral gap for one parameter family of energy forms([17]). Our study is motivated by a wish to conceive a clearer picture of the sample path of the diffusion on S associated with such a Dirichlet form. Example 6.3 is essentially one-dimensional, where we shall see that the conditions A.2 and A.3 are satisfied if and only if the boundary is regular in Feller s sense. This example is reminiscent of an example by N. Ikeda and S. Watanabe[14]. Example 6.4 is higher dimensional, where the Dirichlet form associated with the constructed process X may not be regular. 2

4 KSTS/RR-4/1 January 13, 24 In order to identify right quantities to describe the excursion-valued Poisson point processes to be constructed in 4, we shall study in 2 and 3 a strongly local regular Dirichlet form on L 2 (S; m) for which the point a has a positive capacity. In particular, we shall find that the Dirichlet form and the associated resolvent admit exactly the above mentioned expressions. Furthermore, we shall see that the entrance law µ t governing the excursion law ought to be determined by m = µ t dt, an equation investigated by E.B.Dynkin, R.K.Getoor, P.J.Fitzsimmons and others ([11]). In a seminal work [15], K.Itô considered a standard process X on S for which a point a is regular for itself. A Poisson point process Y taking value in the space of excursions around a was then associated, and it was shown that the stopped process X obtained from X by the hitting time at a and the characteristic measure of Y together determine the law of X uniquely. It was implicitly assumed in [15] that the point a is recurrent in the sense that ϕ(x) =P x (σ a < ) =1,x S, σ a = inft >:X t = a. But, as was shown in P.A. Meyer [2], an absorbed Poisson point process can be still associated with X when a is non-recurrent. See Remark 4.2 in this regard. Since our present assumption on X requires ϕ only to be positive, we must handle not only returning excursions from the point a but also non-returning excursions. By restricting ourselves to the case that both X and X are symmetric diffusions however, we shall see that the characteristic measures on these different type of excursion spaces are uniquely determined by X so that, starting with X, we can give an explicit construction of X. The Dirichlet form (E, F) onl 2 (S; m) associated with a symmetric extension X of X may not be regular but it is quasi-regular in the sense of [19]. Accordingly we can make use of the quasi-homeomorphism in [3] to connect X with the regular Dirichlet form studied in 2, yielding the uniqueness of X and the explicit expression of (E, F). There are quite a few works [1], [24], [25], [26] dealing with generalizations of Itô s one [15]. See Remark 2.2 and Remark 4.1 in these regards. But construction and uniqueness of a symmetric extension X of a symmetric X as are formulated in the present paper have never been considered. 2 Strongly local Dirichlet form with a point of positive capacity 2.1 Description of the form and resolvent by absorbed process Let S be a locally compact separable metric space and a be a non-isolated point of S. We denote the complementary set S \a by S. Let m be a positive Radon measure on S with Supp[m] =S and with m(a) =. The inner product in each of the spaces L 2 (S; m), L 2 (S,m) will be designated by (, ). A Dirichlet form (E, F) onl 2 (S; m) is called regular if F C (S) ise 1 -dense in F and uniformly dense in C (S), where C (S) denotes the space of continuous functions on 3

5 KSTS/RR-4/1 January 13, 24 S with compact support. It is called strongly local if E(u, v) vanishes whenever u, v F, Supp[u], Supp[v] are compact and v is constant on a negibourhood of Supp[u], where Supp[u] denotes the topological support of the measure u m. For the sake of a use in 3.2, we make here a remark: Remark 2.1. If a Dirichlet form (E, F) on L 2 (S; m) is regular and strongly local, then the strong locality stated above holds without assuming that Supp[v] is compact. Indeed, assuming the boundedness of v, take a function w F C (S) with w =1on a neighbourhood of K = Supp[u] and put v 1 = v w, v = v v 1. Then E(u, v 1 )=. Since v belongs to the part F G of (E, F) on the open set G = S \ K and (E, F G )isa regular Dirichlet form on L 2 (G; m) (cf.[9, Th.4.4.3]), we can find v n F C (G) which are E 1 -convergent to v. Hence E(u, v ) = lim n E(u, v n ) = and E(u, v) =. We consider a strongly local regular Dirichlet form (E, F) onl 2 (S; m) and an associated m-symmetric Hunt process X =(X t,p x )ons. In view of [9, The.4.5.3], X can then be taken to be a diffusion on S in the sense that all sample paths are continuous functions from [, ) tos, where S is the one-point compactification of S when S is non-compact and is an extra point isolated from S when S is compact. In either case will be the cemetery of the sample paths. Furthermore, X can be taken to be of no killing inside S in the sense that P x (X ζ =,ζ < ) =P x (ζ< ), x S, where ζ(ω) denotes the life time, namely, the hitting time of the cemetery of the sample path ω. In particular, when S is compact, P x (ζ = ) = 1 for all x S. We make the assumption that B.1 Cap(a) >. Here Cap(A) for A S is its 1-capacity relative to (E, F). In what follows, the quasi-continuity of functions on S will be understood with respect to this capacity. Each function u Fadmits its quasi-continuous version denoted by ũ. q.e. will means except for a set of zero capacity. The hitting probability and the α-order hitting probability of a are denoted by ϕ and u α respectively: ϕ(x) =P x (σ< ), u α (x) =E x (e ασ ), x S, (2.1) where σ is the hitting time of a by the process X defined by σ = inft >:X t = a. (2.2) The assumption B.1 implies that u α is a non-trivial element of F and it is the α-potential U α ν α of a positive measure ν α concentrated on a (cf. [9, 2.2]): Put E α (u α,v)=ṽ(a)ν α (a) v F. (2.3) F = u F:ũ(a) =. (2.4) Then (E, F ) is a regular strongly local Dirichlet form on L 2 (S ; m), which is associated with the part X = (X t,p x )ofx on the set S, namely, the diffusion process X 4

6 KSTS/RR-4/1 January 13, 24 obtained from X by killing upon the hitting time σ (cf. [9, 4.4]). X is of no killing inside S and, if we denote the life time of X by ζ, then ϕ, u α admit the expressions ϕ(x) =P x (ζ <, X ζ = a), u α(x) =E x(e αζ ; X ζ = a), x S, (2.5) in terms of the absorbed process X. We further consider the functions ψ (1) (x) =P x (ζ <, X ζ = ), ψ (2) (x) =P x (ζ = ), x S, (2.6) and put ψ = ψ (1) + ψ (2) so that ψ =1 ϕ. Denote by p t and G α the transition function and the resolvent of X respectively. The same notions for the absorbed process X will be denoted by p t and G α. The functions ϕ, ψ (1),ψ (2) on S are X -excessive. In particular, ψ (2) is X -invariant in the sense that ψ (2) = p t ψ(2), t >. Because of the m-symmetry of X, the measure m = ϕ m (2.7) is an X -excessive measure with m p t = p t ϕ m. Our first aim in this section is to show under the present setting that the form E as well as the resolvent G α are uniquely and explicitly determined by quantities depending only on the absorbed process X. We prepare a lemma. Lemma 2.1. For an X -excessive function v on S, 1 L(m,v) = lim t t m m p 1 t,v = lim t t (ϕ p t ϕ, v)( ). (2.8) is well defined as an increasing limit and it holds that If v is p t -invariant, then for each t> and α>, Proof. If we set e(t) =(ϕ p t ϕ, v), then L(m,v) = lim α α(u α,v). (2.9) L(m,v)= 1 t (ϕ p t ϕ, v) =α(u α,v). e(t + s) =e(t)+(p t ϕ p t+sϕ, v) =e(t)+(ϕ p sϕ, p t v) e(t)+e(s), and hence e(t)/t is increasing as t decreases and constant if v is p t -invariant.. We also see that α(u α,v)=α(ϕ αg αϕ, v) = increases to L(v) as α. e t (t/α) 1 (ϕ p t/αϕ, v) tdt We note that L(m,v) is nothing but the energy functional of the X -excessive measure m and the X -excessive function v in the sense of P.A. Meyer [21] when X is transient (cf.[4, 39], [11, p16]). In [4, 39], it is called the mass of v relative to m. 5

7 KSTS/RR-4/1 January 13, 24 Let F e (resp.f,e ) be the extended Dirichlet space of (F, E) (resp.(f, E)). Each element u F e admits its quasi continuous version denoted by ũ again. In view of [9, 4.6], it holds then that F,e = F e, = u F e :ũ(a) =, ϕ F e, E(ϕ, u) = u F e,, (2.1) F = F e L 2 (S; m) F = F,e L 2 (S,m). (2.11) Furthermore any w F e can be decomposed as w = u + cϕ, u F e,, c constant (2.12) and E(w, w) =E(u,u )+c 2 E(ϕ, ϕ). (2.13) Theorem 2.1. (i) It holds that E(ϕ, ϕ) =L(m,ψ)(= L(m,ψ (1) )+L(m,ψ (2) )). (2.14) (ii) u α is a non-trivial element of F L 1 (S ; m). (iii) For any f L 2 (S, m) and x S, G α f(x) =G αf(x)+ (u α,f) α(u α,ϕ)+l(m,ψ) u α(x), G α f(a) = (u α,f) α(u α,ϕ)+l(m,ψ). (2.15) (iv) Let δ a be a unit mass concentrated at a. Then it is of finite energy integral and its α-potential U α δ a is related to u α by Ũ α δ a = 1 α(u α,ϕ)+l(m,ψ) u α. (2.16) (v) The point a is regular for itself and also an instantaneous state with respect to X: P a (σ =, τ a =)=1, τ a = inft >:X t S. (2.17) Proof. We first give a proof of (ii). According to a general theorem ([9, Chap 4]), the formula obtained by the strong Markov property G α f(x) =G αf(x)+u α (x)g α f(a) x S, f L 2 (S; m), (2.18) represents the orthogonal decomposition of G α f Finto the space F and its orthogonal complement H α = c u α : c constant in the Hilbert space (F.E α ). We see that G α f(a) > for some f C + (S), because otherwise F = F from (2.18) contradicting to u α F. By (2.18), (u α, 1)G α f(a) (G α f,1) = (f,g α 1) 1 (f,1) <. α Next we prove (i) and (iii). For f C (S), the function w = G α f has two expressions: w = G α f + cu α = u + cϕ, c = G α f(a), u F e,. 6

8 KSTS/RR-4/1 January 13, 24 By [9, Cor.1.6.3, Th.2.1.7], We can find a sequence g n of uniformly bounded functions in F such that lim g n = ϕm a.e., n lim E(g n ϕ, g n ϕ) =. n Letting n in the equation we get Since the right hand side equals we arrive at E(w, g n )+α(w, g n )=(f,g n ), ce(ϕ, ϕ)+cα(u α,ϕ)=(f,ϕ) (αg αf,ϕ). G α f(a) = (f,ϕ αg αϕ) =(f,u α ), (u α,f) α(u α,ϕ)+e(ϕ, ϕ), f C (S). (2.19) (2.19) holds for any bounded Borel f. In particular, we have for any α>, and hence G α 1(a) = By letting α, we get from Lemma 2.1 (u α, 1) α(u α,ϕ)+e(ϕ, ϕ) 1 α, E(ϕ, ϕ) α(u α,ψ). E(ϕ, ϕ) L(m,ψ). In order to prove (2.14), notice that the assumption of the strong locality of E implies that the killing measure k in the Beurling-Deny representation of E vanishes (cf. [9, Th.4.5.3]). On account of [9, Lemma 4.5.2], f 2 dk = lim α f(x) S α S 2 (1 αg α 1(x))m(dx), f F C (S). From (2.18) and (2.19), we have 1 αg α 1(x) = 1 αg α(u α, 1) α1(x) α(u α,ϕ)+e(ϕ, ϕ) u α(x) u α (x) α(u α, 1) α(u α,ϕ)+e(ϕ, ϕ) u α(x) = E(ϕ, ϕ) α(u α,ψ) α(u α,ϕ)+e(ϕ, ϕ) u α(x). Take f F C (S) such that f(a). We have from (2.19) and the above inequality α f 2 (1 αg α 1)dm (E(ϕ, ϕ) α(u α,ψ))(αg α f 2 )(a). S 7

9 KSTS/RR-4/1 January 13, 24 By letting α, we get proving the desired identity (2.14). Proof of (iv). By (2.3), which combined with (2.15) gives (E(ϕ, ϕ) L(m,ψ))f(a) 2, (u α,f)=e α (u α,g α f)=g α f(a)ν α (a), ν α =(α(u α,ϕ)+l(m,ψ))δ a. Proof of (v). The regularity P a (σ = ) = 1 of the point a for itself follows from A.1 and a general fact that, for any Borel set B, the set of irregular points x B for B is of zero capacity ([9, Chap. 4]). If P a ( <τ a < ) >, then P a (X τa S ) = 1 contradicting the sample continuity and absence of the killing inside S for X. If a were a trap with respect to X, then G α f(a) =f(a)/α for any f L 2 (S; m) contradicting (2.15). Accordingly, a is an instantaneous state. Remark 2.2. (i) The present assumptions can be relaxed as follows: (a) The measure m on S is replaced by m = m + γδ a for a non-negative constant γ. (b) (E, F) is assumed to be a (not necessarily strongly) local regular Dirichlet form on L 2 (S; m), while its part (E, F )ons is assumed to be a strongly local Dirichlet form on L 2 (S ; m). Then, in view of the above proof of Theorem 2.1, we readily see that (2.14) and (2.15) remain true under the following modifications: G α f(x) =G αf(x)+ E(ϕ, ϕ) =L(m,ψ)+δ, (u α,f)+γf(a) α(u α,ϕ)+l(m,ψ)+δ + αγ u α(x), for a non-negative constant δ. Example 6.1 will indicate stochastic interpretations of the parameters γ and δ. (ii) The parameters γ, δ have appeared in Rogers description [24] of the most general extension of a general resolvent G α under a setting corresponding to ψ (1) =. Another parameter appearing in [24] is a family of measures n α,α >, on S, which is reduced to u α m under the present symmetry assumption. (iii) In the setting (i) in the above, G α is conservative if and only if ψ (1) = and δ =, and in this case the above expression is reduced to G α f(x) =G α f(x)+(1 αg α1,f)+γf(a) α(1 αg α1, 1) + αγ (1 αg α 1(x)). Such a formula was found by Y. Le Jan [18](see also [4, 78]) in a general setting to produce conservative resolvents out of a (not necessarily symmetric) submarkovian resolvent and its dual preserving the duality. 8

10 KSTS/RR-4/1 January 13, Description of the inverse local time In 4, we shall construct a diffusion on S with resolvent (2.15) by means of Poisson point processes of excursions, namely, by piecing together the excursions. In this subsection, let us study more about the roles of the measure m and the energy functional L(m,ψ) played in the present diffusion X on S. Let L(t) be the positive continuous additive functional (admitting exceptional set) associated with the smooth measure δ a (cf.[9, 5.1]): ( ) Ũ α δ a (x) =E x e αt dl(t) for q.e. x S. (2.2) In particular, (2.2) holds for x = a. L(t) is a local time at a in the sense that it increases only when X t = a: L(t) = t I a (X s )dl(s). We consider the right continuous inverse S(t) = infs : L(s) >t of L(t). It is well known that the increasing process (S(t),P a ) is a subordinator killed upon an exponential holding time (cf.[2]). Theorem 2.1 enables us to identify the Lévy measure of the subordinator and the killing rate. Indeed, according to [2, v (3.17)], (2.2) implies the identity E a (e αs(t) ) = exp( t/ũ α δ a (a)), which combined with (2.16) leads us to E a (e αs(t)) = e tl(m,ψ) exp[ tα(u α,ϕ)]. (2.21) We need a lemma which will play a basic role in 4 again. A family ν t t> of σ-finite measures on S is called an X -entrance law if ν t p s = ν s+t, s,t >. Then ν t (f), f B + (S ), is measurable in t and we may let Lemma 2.2. ˆν α (f) = e αt ν t (f)dt, α >, f B + (S ). (i) There exists a unique X -entrance law µ t such that m = µ t dt. (2.22) (ii) ˆµ α (f) =(u α,f), α >, f B + (S ). Consequently, t µ s (f)ds = Px (ζ t, X ζ = a)f(x)m(dx), t >, f B(S ). S (2.23) (iii) µ t (S ) <, t >. (iv) For any bounded X -excessive function v on S,µ t (v) is right continuous in t>. (v) For any X -excessive function v on S, the energy functional L(m,v) introduced in Lemma 2.1 admits an expression L(m,v) = lim t µ t (v). 9

11 KSTS/RR-4/1 January 13, 24 When v is p t -invariant, it holds for any t> that L(m,v)=µ t (v). (vi) L(m,ϕ)=. Proof. (i) Since p t ϕ(x) =P x (t<ζ <,Xζ = a), t, lim t m p t (f) =(p t ϕ, f) = for f L 1 (S,m), namely, m is purely excessive. Hence the desired assertion follows from a well known representation theorem provided that X is transient ([11, Th. 5.25]). But the present situation can be reduced to this case by observing that S 1 = x S : ϕ(x) > is a non-trivial X -invariant set q.e. and the restriction of X to S 1 is transient (cf. [9, 4.6]). (ii) For f C + (S ), we have and Hence t µ t (f)dt = ˆµ α (f) = µ t+s (f)dt = µ t (f) = d dt (ϕ, p t f), a.e. t. e αt d dt (ϕ, p t f)dt µ s (p t f)ds =(ϕ, p t f), = [ e αt (ϕ, p t f)] α e αt (ϕ, p t f)dt = (ϕ, f) α(ϕ, G α f)=(ϕ αg α ϕ, f) =(u α,f). (iii) By (ii) and Theorem 2.1 (ii), ˆµ α (1) = (u α, 1) <, from which the desired finiteness follows. (iv) On account of (iii), we have µ t+s (v) =µ t (p sv) µ t (v), s. (v) Since µ t,v is increasing as t (independent of t when v is p t -invariant), the assertions follow from m m p t,v = t µ s,v ds. (vi) Since S(t) is the right continuous inverse of an increasing continuous process L(t), P a (S(t) > ) = 1 and consequently we have by letting α in (2.21). L(m,ϕ) = lim α α(u α,ϕ)= We see by the above lemma that µ t (ϕ) is decreasing and right continuous in t> and so we can define a measure Θ on (, ) by Θ((s, t]) = µ s (ϕ) µ t (ϕ), <s<t. (2.24) 1

12 KSTS/RR-4/1 January 13, 24 It then holds that Θ((s, t]) = µ s (ϕ p t s ϕ)= µ s,p (σ t s), and we get by letting t, Θ((s, )) = µ s (ϕ). (2.25) We note that Θ([δ, )) < for each δ> by virtue of Lemma 2.2 (iii). Lemma 2.3. It holds that α(u α,ϕ)= ( 1 e αu ) Θ(du). Proof. we have from Lemma 2.2 (ii) and (2.25) α(u α,ϕ) = αˆµ α (ϕ) =α = s αe αt dtθ(ds) = e αt Θ((t, ))dt (1 e αs )Θ(ds). On account of the formula (2.21), Lemma 2.3 and by noting that lim α α(u α,ϕ)=, we can get the next theorem from [2, Theorem 3.21]. Theorem 2.2. Define a measure Θ on (, ) by (2.24). On a certain probablity space (Ω, B,P), construct a subordinator Y t t with Lévy measure Θ and zero drift and a random variable Z, independent of Y t, with P (Z t) =e L(m,ψ)t, t. If we let S Y (t) t<z, (t) = t Z, then the process (S (t) t,p) is equivalent in law to (S(t) t,p a ). 3 Strongly local Dirichlet form with a recurrent point Let S and m be as in 2. In this section, we consider a special case of the Dirichlet form of 2 for which the point a is recurrent. 3.1 Description of associated Poisson point process and entrance law Let (E, F) be a strongly local regular Dirichlet form on L 2 (S; m) and X =(X t,p x )be an associated diffusion on S. In place of the assumption B.1 of 2, let us assume that B.2 ϕ(x) > m a.e. x S B.3 1 F e and E(1, 1) =. 11

13 KSTS/RR-4/1 January 13, 24 In the next subsection, we shall construct a typical example of a Dirichlet form (E, F) satisfying these conditions by a method of the one point compactification. The assumption B.2 implies that u 1 >, m-a.e. and Cap(a) = E 1 (u 1,u 1 ) (u 1,u 1 ) >, namely, the assumption B.1 of 1 (cf. [9, Lemma 4.2.1]). Further, the Dirichlet form (E, F) becomes irreducible because, from (2.15), we have for any Borel sets B 1,B 2 S of positive m-measures (I E,G α I F ) (u α,i E )(u α,i F )/α(u α,ϕ) >. Since (E, F) is recurrent by B.3, we have actually the property ϕ(x) =1, q.e. x S, (3.1) stronger than the assumption B.2 in view of [9, Th.4.6.6]. Thus the point a is not only regular for itself, instantaneous, but also recurrent. (2.15) is now reduced to G α f(x) =G α f(x)+ (u α,f) α(u α, 1) u α(x), x S, G α f(a) = (u α,f) α(u α, 1). (3.2) The positive continuous additive functional L(t) ofx associated with the unit mass δ a has the property that L( ) = and its right continuous inverse S(t) is a subordinator satisfying E a ( ) e αs(s) ds = 1 α(u α, 1) (3.3) on account of (2.16) and (2.2). Therefore we can follow directly the argument of [15, 6 case 2(b)] to conclude that D p = s : S(s) S(s ) >, (3.4) p s (t) = X S(s )+t, s D p, t<s(s) S(s ), (3.5) defines, under the law P a,aw a -valued Poisson point process p, where W a is the space of continous excursions in S from a to a: W a = w :[,ζ(ω)) S, continuous, <ζ(ω) <,w() = a, w(ζ ) =a. (3.6) Let n be the characteristic measure of the Poisson point process p. Then n is a σ- finite measure on the space W a and w(t), n is Markovian with respect to the transition function p t of X. The entrance law ν t associated with the characteristic measure n is defined by ν t (B) =nw : ζ(w) >t, w(t) B, B B(S), t>. (3.7) Recall that we have already considered an X -entrance law µ t specified by (2.22) which is now reduced to m = µ t dt. (3.8) The description (2.23) of µ t now reads t µ s (f)ds = Px (ζ t)f(x)m(dx), t>, f B(S ). S (3.9) 12

14 KSTS/RR-4/1 January 13, 24 Theorem 3.1. ν t = µ t, t >. Proof. By virtue of Lemma 2.2, it suffices to show that ˆν α (f) =(u α,f), f B b (S ). (3.1) We make use of the next general formula ) E a ( a(s, p s,ω) = E a a(s, w, ω)n(dw)ds s t W a (,t] (3.11) holding for any non-negative predictable function a(s, w, ω) on[, ) W a Ω, Ω being a filtered sample space on which the diffusion process X is defined (cf. [14, p62].) Since m(a) is assumed to be zero, I a (X t )dt =, P a -almost surely. By (3.4) and (3.5), we have for f B b (S), ( ) ( ) S(s) G α f(a) = E a e αt f(x t )dt = E a e αt f(x t )dt s> S(s ) ( ) ζ(ps) = E a e αs(s ) e αt f(p s (t))dt. s> We let Γ(w) = ζ(w) e αt f(w(t))dt. a(s, w, ω) =Γ(w) e αs(s,ω) is then predictable and we get by (3.11) ( ) G α f(a) = E a e αs(s ) Γ(p s ) s> = Γ(w)n(dw) E a (e αs(s)) ds. W a Since Γ(w)n(dw) =ˆν α (f), W a (3.2) and (3.3) lead us to the desired identity (3.1). By Theorem 3.1 and [15, Th. 6.3], the finite dimensional distribution of W a, n can be described as follows: f 1 (w(t 1 ))f 2 (w(t 2 )) f n (w(t n ))n(dw) =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, W a (3.12) for any <t 1 <t 2 < <t n 1,t n, f 1,f 2,,f n B b (S ). Here we use the convention that w W satisfies w(t) =, t ζ(w), and any function f on S is extended to S by setting f( ) =. In 4, we shall start with an m-symmetric diffusion X on S and an expression like the above with µ t being specified by (2.22). See 4 for the abbrevaited notation appearing on the right hand side of (3.12). 13

15 KSTS/RR-4/1 January 13, 24 Actually Theorem 3.1 can be extended to a general case where condition B.3 of the recurrence is not assumed as we shall see in Remark 4.2 at the end of 4. We note that the excursion law around a regular point of a general Markov process can be also formulated in terms of Maisonneuve s exit system[5]. Some property of the integral in t of the associated entrance law was investigated by R.K. Getoor [1]. 3.2 Construction of form by one-point compactification In this subsection, we start with a Dirichlet form with underlying space S and extend it by the one-point compactification to a Dirichlet form with underlying space S = S a satisfying B.2 and B.3 (and consequently B.1). Let S be a locally compact separable metric space and m be a bounded positive measure on S with Supp[m] =S. We consider a regular strongly local Dirichlet form (E, F )onl 2 (S ; m) satisfying the Poincaré inequailty: (u, u) A E(u, u) u F A>. (3.13) Denote by S = S a the one-point compactification of S and by L 2 (S; m)(= L 2 (S ; m)) the space of square integrable functions on S with respect to I S m. Let us introduce a space (E, F) by F = F + constant functions on S, (3.14) E(w 1,w 2 ) = E(f 1,f 2 ),w 1 = f 1 + c 1, w 2 = f 2 + c 2, f i F, c i constant. (3.15) Theorem 3.2. (i) (E, F) is a regular strongly local Dirichlet form on L 2 (S; m) possessing as its core the space C = C + constant functions on S, where C = F C (S ). (ii) (E, F) and the associated diffusion on S satisfy B.2, B.3. Proof. (i) Suppose f F is a constant. By the regularity of (E, F ), there exist f n F C (S ) which are E 1 -convergent to f. We have then E(f,f) = lim n E(f,f n )= on account of the strong locality of (E, F ) and Remark 2.1 stated in the beginning of 2.1. (3.13) then implies f = and the definition (3.14) and (3.15) makes sense. If w n = f n + c n Fis an E 1 -Cauchy sequence then f n is E 1 -convergent to some f F by (3.13) and hence w n is E 1 -convergent to f + c for some constant c. Clearly C is dense both in F and C(S), namely, (E, F) is regular. Suppose, for w i = f i + c i C, that w 1 is constant on a neighbourhood of Supp(w 2 ). When c 2 =, E(w 1,w 2 ) = by the strong locality of (E, F ). When c 2, the set U = S \ Supp(w 2 ) is either empty or a non-empty relatively compact open subset of S. In the former case, f 1 = and E(w 1,w 2 )=. In the latter case, f 2 = c 2 on U, while Supp(f 1 ) U and E(w 1,w 2 )=E(f 1,f 2 ) = again. Hence (E, F) is strongly local on account of [9, Th.3.1.2]. The Markov property w F v =( w) 1 F, E(v, v) E(w, w) is evident, because, for w = f +c, w F,cconstant, we have v =[( c) f] (1 c) +c. 14

16 KSTS/RR-4/1 January 13, 24 (ii) B.2 follows from the Poincaré inequality (3.13). Denote by X and X =(X t,p x,ζ ) the diffusions associated with (E, F) and (E, F ) respectively. Then X is the part of X on S and hence ϕ(x) =P x (ζ < ), x S, Denote by G the -order resolvent operator of X. Since m(s ) <, (3.13) implies that G 1 F and Ex (ζ )=G 1(x) < q.e. proving (3.1). It is obvious from (3.14),(3.15) that 1 F and E(1, 1) =. (E, F ) is not necessarily irreducible on S, but (E, F) defined by (3.14),(3.15) is irreducible recurrent on S in view of the observation made in the preceding subsection. See Example Construction of a symmetric extension via excursion valued Poisson point processes In this section, we start with an m-symmetric diffusion X on S and construct first an excursion law with which Poisson point processes of two different kinds of excusions around the point a are associated. We then construct an m-symmetric diffusion X on S = S a by piecing together those excursions. The resolvent of the resulting diffusion X turns out to be identical with (2.15). 4.1 An excursion law and its basic properties Let S be a locally compact separable metric space and a be a non-isolated point of S. We put S = S \a. The one point compactification of S is denoted by S. When S is compact already, is added as an isolated point. Let m be a positive Radon measure on S with Supp[m] =S.mis extended to S by setting m(a) =. We assume that we are given an m-symmetric diffusion X =(X t,p x )ons with life time ζ satisfying the following: A.1 P x (ζ <, X ζ a ) =P x (ζ < ), x S. We define the functions ϕ, u α,ψ (1),ψ (2),ψ by (2.5) and (2.6), namely, for x S, ϕ(x) =P x (ζ <, X ζ = a), u α(x) =E x(e αζ ; X ζ = a), ψ =1 ϕ = ψ (1) + ψ (2), ψ (1) (x) =P x (ζ <, X ζ = ), ψ (2) (x) =P x (ζ = ). Let us assume that A.2 ϕ(x) >, x S, and A.3 u α L 1 (S ; m), α >. Denote by p t,g α the transition function and the resolvent of X respectively. Our last assumption concerns the regularity: 15

17 KSTS/RR-4/1 January 13, 24 A.4 u α C b (S ), G α(c b (S )) C b (S ), α >, where C b (S ) is the space of all bounded continuous functions on S. The measure m could be infinite on a compact neighbourhood of a in S, but it is finite on each level set of u α due to the condition A.3. We also note here the next relation which will be utilized in the sequel: u α (x) =ϕ(x) αg α ϕ(x) 1 αg α 1(x), x S. Define m by m = ϕ m, which is an X -excessive measure with m p t = p t ϕ m. In view of Lemma 2.2, there exists a unique X -entrance law µ t related to the measure m by (2.22), namely, and it satisfies that m = µ t dt. On account of the assumption (A.3), we then have that ˆµ α (f) =(u α,f), f B + (S ). (4.1) µ t (S ) <, t >, 1 µ t (S )dt <. (4.2) We now introduce the spaces W, W of excursions by W = w : ζ(w) (, ], w is a continuous function from (,ζ(w)) to S, W = w W :if ζ(w) <, then w(ζ(w) ) a. (4.3) ζ(w) will be called 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 and the transition function p t by f 1 (w(t 1 ))f 2 (w(t 2 )) f n (w(t n ))n(dw) =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, W (4.4) for any <t 1 <t 2 < <t n,f 1,f 2,,f n B b (S ). Here, we use the convention that w W satisfies w(t) =, t ζ(w), and any function f on S is extended to S by setting f( ) =. Further, on the right hand side of (4.4), we employ an abbreviated notation for the repeated operations 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 )]. Proposition 4.1. There exists a unique measure n on the space W satisfiying (4.4). 16

18 KSTS/RR-4/1 January 13, 24 Proof. Let n be the Kuznetsov measure on W uniquely associated with the transition semigroup p t and the entrance rule η u defined by η u = for u, η u = µ u for u> as is constructed in [5, Chap XIX, 9] for a right semigroup. Because of the present choice of the entrance rule, it holds that α = where α is the birth time which is random in general(cf. [11, p54].) On account of the assumption A.1 for the diffusion X on S, the same method of the construction of the Kuznetsov measure as in [5, Chap.XIX, 9] works in proving that n is supported by the space W and satisfies (4.4). We call n the excursion law associated with the entrance law µ t. We split the space W of excursions into two parts: W + = w W : ζ(w) <, w(ζ ) =a, W = W \ W +. (4.5) Note that W = W 1 W 2 with W 1 = w W : ζ(w) <, w(ζ ) =, W 2 = w W : ζ(w) =. For w W +, we define ŵ W by ŵ(t) =w(ζ t), <t<ζ. (4.6) The next lemma says that the restriction of the excursion law to W + is invariant under time reversion. This is a present variant of the time reversal arguments that have been formulated in general contexts ([23], [12], [6], [7]). Lemma 4.1. 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 p t 2 f 2 p t n 1 f n 1 p t n f n ϕ, (4.7) k=1 n n n f k (w(t t k )); W + = n f k (ŵ(t t k )); W +. (4.8) k=1 k=1 Proof. (4.7) readily follows from (4.4) and the Markov property of n. As for (4.8) we observe that, for α 1,,α n >, n e α 1t 1 α nt n n f k (w(t t k )); W + dt 1 dt n (4.9) k=1 equals with F (w) = <t 1 < <t n<ζ nf (w); ζ <, w(ζ ) =a n k=1 e α k(t k t k 1 ) f k (w(t k )) dt 1 dt n, (t =). 17

19 KSTS/RR-4/1 January 13, 24 Hence, for (4.8), it suffices to prove nf (w); ζ <, w(ζ ) = a = nf (ŵ); ζ <, w(ζ ) = a. (4.1) Performing the change of variables ζ t k = s k, 1 k n, in the expression of F (ŵ) and by noting that we obtain with t k = ζ s k, t k t k 1 = s k 1 s k, 1 k n, s = ζ, F (ŵ) = = Γ s1 s n (w) = <t 1 < <t n <ζ <s n < <s 1 <ζ, <s n< <s 1 <ζ n k=1 Γ s1 s n (w)ds 1 ds n <s n< <s 1 < n k=2 e α k(s k 1 s k ) f k (w(s k )) ds 1 ds n e α k(s k 1 s k ) f k (w(s k )) e α 1(ζ s 1 ) f 1 (w(s 1 ))I (,ζ) (s 1 ). On the other hand, we get from (4.4) and the Markov property of n that Therefore, n Γ s1 s 2 s n (w); ζ<, w(ζ ) =a = n f n (w(s n ))e αn(s n 1 s n) f 2 ((w(s 2 ))e α 3(s 1 s 2 ) f 1 (w(s 1 )u α1 (w(s 1 )); s 1 <ζ = e αn(s n 1 s n) α n 1 (s n 2 s n 1 ) α 2 (s 1 s 2) µ sn f n p s n 1 s n f n 1 p s n 2 s n 1 f n 1 p s 2 s 3 f 2 p s 1 s 2 f 1 u α1. n F (ŵ); ζ<,w(ζ ) =a = ds n µ sn f n G α n f n 1 G α n 1 f 3 G α 3 f 2 G α 2 f 1 u α1. In view of (2.7), the symmetry of G α, (4.7) and (4.9), we arrive at n F (ŵ); ζ<,w(ζ ) =a = m,f n G α n f n 1 G α n 1 f 3 G α 3 f 2 G α2 f 1 u α1 = (f n ϕ, G α n f n 1 G α n 1 f 3 G α 3 f 2 G α2 f 1 u α1 )=(f 1 G α 2 f 2 G α 3 f 3 G αn f n ϕ, u α1 ) = e α 1t 1 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<,w(ζ ) =a the desired identity (4.1). Next we put W a = w W : lim t w(t) =a. (4.11) 18

20 KSTS/RR-4/1 January 13, 24 Lemma 4.2. n W \ W a =. Proof. The preceding lemma implies that n W + \ W a = n W + (w(+) = a) c We then have for each t> = 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) =, which combined with the assumption A.2 leads us to It then suffices to let t. Lemma 4.3. It holds then that n (W \ W a ) (ζ >t) =. For any neighbourhood U of a in S, we let τ U c = inft >:w(t) U c, w W. n τ U c <ζ <. Proof. We may assume that the closure U in S is compact. Let f(x) =ϕ(x) u 1 (x), x S. Then f(x) =Ex 1 e ζ ; ζ <,X ζ = a >, x S. Since u α (x) u 1 (x) f(x), α, the assumption A.3 implies that f is lower semicontinuous on S and hence c = inf f(x) x U is positive. We then have, for each δ> and x U, Px (δ<ζ <,X ζ = a) Ex c Ex 1 e ζ ; ζ δ, X ζ = a 1 e ζ ; δ<ζ <,X ζ = a c (1 e δ ). Choose δ> so small that is positive. For such δ, r = c (1 e δ ) P x (δ<ζ <,X ζ = a) r, x U. (4.12) We shall use the notation τ U c not only for w W but also for the sample path of the Markov process X. Using the preceding lemma, (4.12) and (4.2), we are led to n τ U c <ζ = lim n ɛ <τ U c <ζ = lim µ ɛ (dx)p x τu c <ζ ɛ ɛ U lim ɛ µ ɛ (dx)ex r 1 PX (δ<ζ τuc <,X ζ = a); τ U c <ζ U r 1 lim µ ɛ (dx)px (δ <ζ <,X ζ = a) r 1 lim µ ɛ (dx)px (δ <ζ ) ɛ S ɛ S = r 1 lim µ ɛ+δ (S ) r 1 µ δ (S ) <. ɛ 19

21 KSTS/RR-4/1 January 13, 24 The next lemma states a relation of the excursion law n to energy functionals L(m,v) introduced in Lemma 2.1. Lemma 4.4. (i) n(w + )=L(m,ϕ), n(w )=L(m,ψ), n(wi )=L(m,ψ (i) ), i =1, 2. (ii) n(w1 ) <, n(w 2 )=µ t(ψ (2) )=αˆµ α (ψ (2) )=α(u α,ψ (2) ) <, t >,α >. Proof. (i) Since n(ζ >t; W + )= µ t,ϕ, the first identity follows from Lemma 2.2 (v) by letting t. The proof of the other indentities is the same. (ii) Take a neighbourhood U of a in S with compact U. We have then by the preceding lemma n(w1 )=n(ζ <,w(ζ ) = ) n τ U c <ζ <. Since ψ (2) is p t -invariant, the second assetion follows from (i), Lemma 2.1, Lemma 2.2 and assumption A.3. In particular, n(w )=n(w 1 )+n(w 2 ) is finite. We shall see that n(w + )=. 4.2 Poisson point processes on W a and a new process X By Lemma 4.2, the excursion law n is concentrated on the space W a defined by (4.11). Accordingly, we consider the spaces W + a = w W + : lim t w(t) =a, 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 S -valued excursion to S-valued continuous one as follows: W a = w : ζ(w) (, ], w is a continuous function from [,ζ(w)) to S, w() = a. w(t) S, t (,ζ(w)), w(ζ(w) ) a if ζ(w) <, (4.13) Any w W a for which ζ(w) <,w(ζ(w) ) =a will be regarded to be a continuous function from [,ζ(w)] to S by setting w(ζ(w)) = a. We further let W + a = w : ζ(w) (, ), w is a continuous function from [,ζ(w)] to S, w(t) S,t (,ζ(w)), w() = w(ζ(w)) = a, (4.14) W a = W a \ W + a. (4.15) The excursion law n will be considered to be a measure on W a defined by (4.13) and we denote by n +, n, the restrictions of n to W + a, W a defined by (4.14) and (4.15) respectively. Let p s,s > be a Poisson point process on W a with characteristic measure n defined on an appropriate probability space (Ω,P). We then let p + s = ps if p s W a +, otherwise, (4.16) 2

22 KSTS/RR-4/1 January 13, 24 p s = ps if p s Wa, otherwise, (4.17) where is an extra point disjoint of W a. Then p + s,s >, p s,s > are mutually independent Poisson point processes on W a +, W a with characteristic measures n+, n respectively. Furthermore p s = p + s + p s. (4.18) By means of the terminal time ζ(p + r ) of the excursion p + r, we let J(s) = ζ(p + r ), s >. (4.19) r s We put J() =. Lemma 4.5. (i) J(s) < a.s. for s>. (ii) J(s) s is a subordinator with E e αj(s) = exp α(u α,ϕ)s. (4.2) Proof. (i) We write J(s) asj(s) = I + II with I = ζ(p + r ), II = ζ(p + r ). r s,ζ(p + r ) 1 r s,ζ(p + r )>1 Since n + (ζ>1) µ 1 (S ) < by (4.2), r in the sum II is finite a.s. and hence II < a.s. On the other hand, E(I) = sn + (ζ; ζ 1) sn + (ζ 1) 1 = sn + I (,ζ) (t)dt = s 1 n + (ζ>t)dt s 1 µ t (S )dt, which is finite by (4.2). Hence I< a.s. (ii) Clearly J(s) s is increasing and of stationary independent increment. Since e αj(s) 1= e αj(r) e αj(r ) = e αj(r ) e αζ(p+ r ) 1, r s r s we have with E e αj(s) 1= c s E e αj(r) dr, c = n + (1 e αζ )=n(1 e αζ ; ζ<,w(ζ) =a) ζ = n α e αt dt; ζ<,w(ζ) =a = α = α e αt n(t <ζ<,w(ζ) =a)dt e αt µ t (ϕ)dt = αˆµ α (ϕ) =α(u α,ϕ) <. 21

23 KSTS/RR-4/1 January 13, 24 In virtue of Lemma 4.3 and Lemma 4.5, we may assume that the next three properties hold for any ω Ω by subtracting a P -negligible set from Ω if necessary: J(s) < s>, (4.21) lim J(s) =, (4.22) s and, for any finite interval I (, ) and any neighbourhood U of a in S, s I : τu c(p + s ) <ζ(p+ s ) is a finite set. (4.23) Let T be the time of occurrence of the first excursion of the point process p s,s>, namely, T = mins >:p s. (4.24) Since n(wa )=L(m,ψ) < by Lemma 4.4, we can see that T and p T are independent and P (T >t)=e L(m,ψ)t, the distribution of p T = L(m,ψ) 1 n. (4.25) 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 ), we determine s by J(s ) t J(s), (4.26) and let X t = p + s (t J(s )) if J(s) J(s ) >, a if J(s) J(s ) =. (ii) For J(T ) t<ζ ω J(T )+ζ(p T ), we let (4.27) In this way, the S-valued continuous path is defined and X t = p T (t J(T )). (4.28) X t, t<ζ ω, X ζω = if ζ ω <. Continuity of the path is a consequaence of (4.23). For this process X t, t<ζ ω,p, let us put ( ζω ) G α f(a) =E e αt f(x t )dt, α >, f B(S). (4.29) 22

24 KSTS/RR-4/1 January 13, 24 Proposition 4.2. It holds that Proof. We use the notation G α f(a) = (u α,f) α(u α,ϕ)+l(m,ψ). (4.3) ˆf α (w) = ζ(w) e αt f(w(t))dt, w W a. We have then ζω e αt f(x t )dt = J(s) J(T )+ζ(p e αt T ) f(x t )dt + e αt f(x t )dt s<t J(s ) J(T ) = e αj(s ) ˆfα (p + s )+e αj(t ) ˆfα (p T ), s<t and consequently ( ) G α f(a) = E e αj(s ) ˆfα (p + s )+e αj(t ) ˆfα (p T ) s<t ( T = E = = ) e αˆµα(ϕ)s ds n + ( ˆf ( ) α )+E e αˆµα(ϕ)t L(m,ψ) 1 n ( ˆf α ) n + ( ˆf α ) αˆµ α (ϕ)+l(m,ψ) + n ( ˆf α ) αˆµ α (ϕ)+l(m,ψ) n( ˆf α ) αˆµ α (ϕ)+l(m,ψ) = ˆµ α (f) αˆµ α (ϕ)+l(m,ψ). It then suffices to substitute (4.1) in the last expression. 4.3 Continuity of resolvent along X Lemma 4.6. For α> and f B(S), define G α f(a) by the right hand side of (4.3) and extend it to a function on S by setting G α f(x) =G αf(x)+g α f(a)u α (x), x S. (4.31) Then G α α> is an m-symmetric (sub)markovian resolvent on S. Proof. By making use of the resolvent equation for G α, the m-symmetry of G α and the equation u α (x) u β (x)+(α β)g α u β(x) =, α,β >, x S, we can easily check the resolvent equation The m-symmetry of G α G α f(x) G β f(x)+(α β)g α G β f(x) =, x S. S G α f(x)g(x)m(dx) = 23 S f(x)g α g(x)m(dx)

25 KSTS/RR-4/1 January 13, 24 holding for any non-negative Borel functions f,g is clear. Moreover we get by Lemma 2.1 that and similarly, αg α 1(a) 1. αg α 1(x) = αg α(u α,ϕ+ ψ) α1(x)+u α (x) α(u α,ϕ)+l(m,ψ) 1 u α (x)+u α (x) =1, x S, Let U n be a decreasing sequence of open neighbourhoods of the point a in S such that U n U n+1 and n=1u n = a. Let We then set A = A α,ρ = x S : u α (x) <ρ for α>, <ρ<1. σ n = inft >:X t U n S,σ a = lim n σ n, τ n = inft >:X t U n A, with the convention that inf =. Lemma 4.7. For any α>, ρ (, 1) and x S, Proof. Since lim P n x τ n <σ a < =. (4.32) σ a < = ζ <, X ζ = a and σ a = ζ on the set σ a <, we have for x S and m<n u α (x) = Ex e ασ a ; τ n <σ a + E x e ασ a ; τ n σ a = Ex e ατ n ( ) u α X τn ; τn <σ a + E x e ασ a ; τ n σ a ρex e ατ n ; τ n <σ a + E x e ασ a ; τ n σ a e α(τn σa) ; τ m <σ a + Ex e ασ a ; τ n σ a. ρe x By letting first n and then m, we obtain u α (x) ρ lim m E x e ασ a ; τ m <σ a + lim n E x e ασ a ; τ n σ a = Ex e ασ a (1 ρ) lim n E x e ασ a ; τ n <σ a = u α (x) (1 ρ) lim n E x e ασ a ; τ n <σ a, which implies and so (4.32) must hold. lim n E x e ασ a ; τ n <σ a = Lemma 4.8. Let α>. (i) For any x S, lim u α (Xt )=1 Px -a.s. on σ a <. (4.33) t σ a (ii) n(λ) = where Λ= w W + a : α >, lim u α(w(t)) 1. t ζ 24

26 KSTS/RR-4/1 January 13, 24 Proof. If σ a < and if lim t σa u α (Xt ) < ρ, then for any small ɛ > there exists t (σ a ɛ, σ a ) such that u α (Xt ) < ρ, and so τ n < σ a for all n. Therefore by the preceding lemma limt σa u α (Xt ) <ρ, σ a < =. P x Since u α is decreasing in α and ρ can be taken arbitrarily close to 1, we obtain (4.33). (ii) follows from (i) as n(λ) = lim n(λ ɛ<ζ) ɛ = lim µ ɛ (dx)px (lim u α (Xt ) 1)=. ɛ S t σa We extend u α to a function on S by setting u α (a) =1. By Lemma 4.8 (ii) combined with Lemma 4.1 and a similar reasoning as in the proof of Lemma 4.2, we may assume, subtracting a suitable n-negligible set from W a + (resp. Wa ), that u 1 (w(t)) is continuous in t [,ζ] (resp. t [,ζ).) Lemma 4.9. Let <ρ<1 and set W ρ = w W a + : max 1 u 1(w(t)) >ρ. t ζ Then n + ( W ρ ) <. Proof. The proof is similar to that of Lemma 4.3. For any x such that 1 u 1 (x) =ρ and for δ = log(1 ρ 2 ) >, we have Px (σ a >δ) Ex 1 e σ a ; σ a >δ = Ex 1 e σ a E x 1 e σ a ; σ a δ Therefore if we set 1 u 1 (x) (1 e δ )=ρ (1 e δ )= ρ 2. A = x S :1 u 1 (x) ρ, τ = inft >:w(t) S \ A, then n + ( W ρ )=n + (τ<ζ) = lim n + (ɛ<τ<ζ ) = lim µ ɛ (dx)px (τ <ζ ) ɛ ɛ A ( ) 2 lim ɛ µ ɛ (dx)ex P Xτ A ρ (σ a >δ); τ<ζ 2 ρ lim ɛ µ ɛ (dx)px (σ a >δ) S 2 ρ lim µ ɛ (dx)px (ζ >δ)+ 2 ɛ S ρ lim µ ɛ (dx)px (ζ <σ a = ) ɛ S = 2 ρ lim ɛ µ ɛ+δ(1) + 2 ρ lim ɛ µ ɛ(ψ (1) ), 25

27 KSTS/RR-4/1 January 13, 24 which is finite in view of (4.2) and Lemma 4.4. For α>, f B(S), we defined the resolvent G α f by G α f(x) =G αf(x)+g α f(a)u α (x), x S with G α f(a) of Proposition 4.2. We now extend G αf(x) tos by setting G αf(a) =. In the last subsection, we have constructed a process X t t [,ζω) out of the Poisson point processes p +, p on W a +,W a defined on a probability space (Ω,P). Proposition 4.3. [,ζ ω ), P-a.s. Let u = G α f with f C b (S). Then u(x t ) is continuous in t Proof. As was remarked immediately after the proof of Lemma 4.8, u 1 is continuous along any sample point functions of p + = p + s,s > and p = p s,s >. Moreover, by Lemma 4.9, we can subtract a suitable P -negligible set from Ω so that, in addition to the properties (4.21),(4.22) and (4.23), p + satisfies the following property for every sample point ω Ω: for any finite interval I (, ) and for any ρ (, 1), s I : max (1 u 1 (p + s (t))) >ρ is a finite set. (4.34) t ζ(p + s ) Then it is not hard to see that not only X t but also u 1 (X t ) are continuous in t [,ζ ω ). From the inequality G 1 1(x) 1 u 1(x), x S, we see that lim G t t 1 1(X t)= if X t = a. Hence G 1 f(x t) has the same property as the above for f C b (S). Since G 1 f(x t)is clearly continuous on t [,ζ ω ):X t a by the assumption A.4, it is continuous on [,ζ ω ). We have thus proved the continuity of G 1 f(x t ). The continuity of G α f(x t ) follows from the resolvent equation proved in Lemma Markov property of X Let us define p t f(x) for t>,x S, f B(S), as follows: Evidently p t f(a) =E (f(x t ); ζ ω >t), (4.35) p t f(x) =p t f(x)+e x p t σ a f(a); σ a t, x S. (4.36) Lemma 4.1. p t+s = p t p s, t, s >. e αt p t fdt = G α f, α >. (4.37) Proof. Takeanyf C b (S). By (4.36) and the resolvent equation in Lemma 4.6, we have for any x S e αt e βs p t+s f(x)ds dt = e αt p t (G β f)(x) dt, (4.38) 26

28 KSTS/RR-4/1 January 13, 24 1 because the left hand side equals α β (G βf(x) G α f(x)) = G α G β f(x). We first consider the case where x = a. Then the functions inside of the both hand sides of (4.38) are continuous in t> in virtue of the continuity of X and Proposition 4.3. Hence we have for any t> e βs p t+s f(a)ds = p s (G β f)(a) = e βs p t (p s f)(a)ds. Since both p t+s f(a), p t (p s f)(a) are right continuous in s>, we get p t+s f(a) =p t (p s f)(a), t >, s>. (4.39) We next consdier the case where x S. Using (4.37), we obtain p t+s f(x) = p t+sf(x)+ex p t+s σa f(a); σ a t + s = p t+s f(x)+e x p t σ a (p s f)(a); σ a t + Ex p t+s σ a f(a) :t<σ a t + s. On the other hand, p t (p s f)(x) =p t (p sf)(x)+ex p t σ a (p s f)(a); σ a t. Hence it suffices to prove that p t+sf(x)+ex p t+s σa f(a); t<σ a t + s = p t (p s f)(x). (4.4) Put g(x) =E x p s σa f(a); σ a s, then, we are led from p s f(x) =p sf(x)+g(x) to and consequently, (4.4) is reduced to p t (p sf)(x) =p t+s f(x)+p t g(x), E x p t+s σa f(a); t<σ a t + s = E x(g(x t ); ζ >t). (4.41) With the notation θ t to denote the usual shift, the left hand side of (4.41) equals pt+s σa f(a); ζ >t,σ a >t,σ a θ t s E x = Ex ps σa θ t f(a); ζ >t,σ a θ t s [ ] = Ex E X p t s σa f(a); σ a s ; ζ >t, which coincides with the right hand side of (4.41) as was to be proved. Lemma C b (S), t >, Suppose g B(S) and lim ɛ p ɛ g(x) =g(x), x S. Then, for any f lim p ɛ (fp t g)(x) =f(x)p t g(x), x S. (4.42) ɛ 27

29 KSTS/RR-4/1 January 13, 24 Proof. Fix x S. Clearly, for any neighbourhood U of x, and hence lim p ɛ I U (x) =1, ɛ p ɛ fp t g (x) =p ɛ fi U p t g (x)+o(ɛ). For any δ>, take a neighbourhood U of x such that Then f(y) f(x) <δ, y U. p ɛ (fp t g)(x) f(x)p ɛ (p t g)(x) p ɛ ( f f(x) p t g )(x) p ɛ ( f f(x) I U p t g )(x)+o(ɛ) δ g + o(ɛ). On the other hand, we have from the preceding lemma that Consequently lim f(x)p ɛ (p t g)(x) = lim f(x)p t (p ɛ g)(x) =f(x)p t g(x). ɛ ɛ lim ɛ p ɛ (fp t g)(x) f(x)p t g(x) δ g, which means (4.42) because δ> can be taken arbitrarily small. Proposition 4.4. (i) For α 1,,α n >, n ( ) E e α k(t k t k 1 ) f k (X tk ) dt 1 dt n = G α 1 f 1 G α2 f 2 G αn f n (a), <t 1 < <t k=1 n<ζ ω (4.43) where we set t =by convention. (ii). X = X t, t<ζ ω,p is a Markov process on S with transition function p t and initial distribution concentrated at a. Proof. We shall employ the following notations: F (X; t; α 1,f 1,,α n,f n )= and, for w W a, F (w; t; α 1,f 1,,α n,f n )= n t<t 1 < <t k=1 n<ζ ω n t<t 1 < <t k=1 n<ζ(w) e α k(t k t k 1 ) f k (X tk ) dt 1 dt n, e α k(t k t k 1 ) f k (w(t k )) dt 1 dt n. (i). The left hand side of (4.43) will be denoted by G(α 1,f 1,,α n,f n ), namely, E F (X;;α 1,f 1,,α n,f n ) = G(α 1,f 1,,α n,f n ). (4.44) 28

30 KSTS/RR-4/1 January 13, 24 For <s<t,we denote by I(s) the expression n ( ) e α 1t 1 f 1 (X t1 ) e α k(t k t k 1 ) f k (X tk ) dt 2 dt n J(s )<t 1 <J(s) dt 1. Then F (X;;α 1,f 1,,α n,f n )= Further, if we put for 1 m n then I m (s) = t 1 <t 2 < <t k=2 n<ζ ω <s<t m J(s )<t 1 < <t k=1 m<j(s) J(s)<t m+1 < <t l=m+1 n<ζ ω I(s)+F (X; J(T ); α 1,f 1,,α n,f n ). e α k(t k t k 1 ) f k (X tk ) dt 1 dt m n I(s) = Moreover, each I m (s) can be written as with F m (s) = G m (s) = Therefore m J(s )<t 1 < <t k=1 m<j(s) e α l(t l t l 1 ) f l (X tl ) dt m+1 dt n, n I m (s). m=1 I m (s) =F m (s)g m (s) J(s)<t m+1 < <t n<ζ ω e αm+1(tm+1 J(s)) F (X;;α 1,f 1,,α n,f n )= e α k(t k t k 1 ) f k (X tk ) e α m+1(j(s) t m) dt 1 dt m, <s<t m=1 n l=m+2 Next, let us put (with the convention that α n+1 =) F (w; α 1,f 1,,α m,f m ; α m+1 ) = m <t 1 < <t k=1 m<ζ(w) e α l(t l t l 1 ) f l (X tl ) dt m+1 dt n. n F m (s)g m (s)+f(x; J(T ); α 1,f 1,,α n,f n ). (4.45) e α k(t k t k 1 ) f k (w(t k )) e α m+1(ζ(w) t m) dt 1 dt m, (4.46) so that F m (s) =e α 1J(s ) F (p + s ; α 1,f 1,,α m,f m ; α m+1 ). (4.47) 29