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View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Saitama University Cyber Repository of Academic Resources J. Saitama Univ. Fac. Educ., 62(1)：249-266 （2013） On A Class of Historical Superprocesses in The Dynkin Sense DOKU,^ Isamu Faculty of Education, Saitama University Abstract We study a class of historical superprocesses in the Dynkin sense. This class is closely related to a group of superprocesses, namely, measure-valued branching Markov processes, which are associated with stable random measure. Then we prove the existence theorem for the class of historical superprocesses in Dynkin's sense. Key Words : Dynkin's historical superprocess, measure-valued Markov process, superprocess, stable random measure, existence. 1. Introduction The purpose of this paper is to investigate a class of historical superprocesses in Dynkin's sense. This class is closely related to another class of superprocesses, namely, measure-valued branching Markov processes, which are associated with stable random measure. Then it is proven that the class of historical superprocesses in Dynkin's sense exists under some suitable conditions. Now we shall begin with introducing a fundamental relationship between superprocess and its associated historical superprocess. 1.1 Superprocess with Branching Rate Functional In this subsection we shall introduce the so-called superprocess with branching rate functional, which forms a general class of measure-valued branching Markov processes with diffusion as a underlying spatial motion. WeR write the integral of measurable function f with respect to measure µ as hµ, fi = fdµ. For simplicity, d d MF = MF (R ) is the space of finite measures on R . Define a second order 1 r · r · r elliptic differential operator L = 2 a + b , and a = (aij) is positive definite 1;" 1;" d 1;" and we assume that aij; bi 2 C = C (R ). Here the space C indicates the totality of all H¨oldercontinuous functions with index " (0 < " 6 1), allowing their first order derivatives to be locally H¨oldercontinuous. Ξ = fξ; Πs;a; s ≥ 0; a 2 Rdg indicates a corresponding L-diffusion. Moreover, CAF stands for continuous additive functional in Probability Theory. -249- Definition 1. (A Locally Admissible Class of CAF; cf. Dynkin (1994), [17]) A continuous additive functional K is said to be in the Dynkin class with index q and we write K 2 Kq, (some q > 0) if (a) Z t sup Πs;a ϕp(ξr)K(dr) ! 0; (r0 ≥ 0) as s ! r0; t ! r0; (1) a2Rd s (b) each N, 9cN > 0 : Z t q d Πs;a ϕp(ξr)K(dr) 6 cN jt − sj ϕp(a); (for 0 6 s 6 t 6 N; a 2 R ): (2) s Rd + When we write Cb as the set of bounded continuous functions on , then Cb is the set of positive members g in Cb. The process X = fX; Ps,µ, s ≥ 0; µ 2 MF g is said to be a superprocess (or superdiffusion) with branching rate functional K or simply (L; K; µ)-superprocess if X = fXtg is a continous MF -valued time-inhomogeneous Markov process with Laplace transition functional −⟨ i −⟨ i P Xt;' µ,v(s;t) 6 6 2 2 + s,µe = e ; 0 s t; µ MF ;' Cb : (3) Here the function v is uniquely determined by the following log-Laplace equation Z t 2 d Πs;a'(ξt) = v(s; a) + Πs;a v (r; ξr)K(dr); 0 6 s 6 t; a 2 R : (4) s 1.2 Associated Historical Superprocess The historical superprocess was initially studied by E.B. Dynkin (1991) [15],[16], d see also Dawson-Perkins (1991) [5]. C = C(R+; R ) denotes the space of contin- uous paths on Rd with topology of uniform convergence on compact subsets of t R+. To each w 2 C and t > 0, we write w 2 C as the stopped path of w. We denote by Ct the totality of all these paths stopped at time t. To every w 2 C t we associate the corresponding stopped path trajectoryw ~ defined byw ~t = w for t ≥ 0. The image of L-diffusion w under the map : w 7! w~ is called the L-diffusion path process. Moreover, we define C× ≡ R ×^ C· f 2 R 2 Csg R + = (s; w): s +; w (5) C× ≡ R ×^ C· R ×^ C· and we denote by M( R) M( + ) the set of measures η on + which are finite, if restricted to a finite time interval. Let K be a positive CAF of ξ. X~ = ~ ~ s fX; Ps,µ; s ≥ 0, µ 2 MF (C )g is said to be a Dynkin's historical superprocess (cf. -250- ~ ~ Dynkin (1991), [16]) if X = fXtg is a time-inhomogeneous Markov process with ~ t state Xt 2 MF (C ), t ≥ s, with Laplace transition functional −⟨ ~ i −⟨ i P~ Xt;' µ,v(s;t) 6 6 2 Cs 2 + C s,µe = e ; 0 s t; µ MF ( );' Cb ( ); (6) where the function v is uniquely determined by the log-Laplace type equation Z t ~ ~ ~ 2 ~ ~ 6 6 2 Cs Πs;ws '(ξt) = v(s; ws) + Πs;ws v (r; ξr)K(dr); 0 s t; ws : (7) s We call this class of process an associted historical superprocess in Dynkin's sense in this article. 2. Examples of Measure-Valued Processes 2.1 Dawson-Watanabe Superprocess X = fXt; t ≥ 0g is said to be the Dawson-Watanabe superprocess (cf. Watan- abe (1968), [29], Dawson (1975), [1]) if fXtg is a Markov process taking values d d in the space MF (R ) of finite measures on R , satisfying the following martingale d problem (MP): i.e., there exists a probability measure P 2 P(MF (R )) on the d sapce MF (R ) such that for all ' 2 Dom(∆) Z t 1 Mt(') ≡ hXt;'i − hX0;'i − hXs; ∆φids (8) 0 2 is a P-martingale and its quadratic variation process is given by Z t 2 hM(')it = γ hXs;' ids: (9) 0 Or equivalently, the Laplace transition functional of fXtg is given by −⟨ i −⟨ i E Xt;' X0;u(t) 2 + Rd \ e = e ; for ' Cb ( ) Dom(∆); (10) where u(t; x) is the unique positive solution of evolution equation @u 1 1 = ∆u − γu2; u(0; x) = '(x): (11) @t 2 2 The Dawson-Watanabe superprocess Xt inherits the branching property from the approximating branching Brownian motions (cf. Dawson (1993), [2]). Namely, Pt(·; ν1 + ν2) = Pt(·; ν1) ∗ Pt(·; ν2); (12) where Pt(·; µ) denotes the transition probability with the initial data µ. The branching property extends in the obvious way to initial measures of the form ν1 + ··· + νn. Conversely, for any integer n, the distribution of the superprocess -251- started from initial measure µ is written as that of the sum of n independent copies of the superprocesses each started from µ/n. This implies that the Dawson- Watanabe superprocess is infinitely divisible. The following proposition is well known. The statement is the analogue of the classical L´evy-Khintchine formula (cf. Sato (1999), [28]: Theorem 8.1, p.37), that characterizes the possible characteristic functions of infinitely divisible distri- butions on Rd, in the measure-valued setting. Theorem 2. (Canonical Representation Theorem) Let (E; E) be a Polish space, E and X be infinitely divisible Random measure on (E; ). Then there exist measuresR Xd 2 MF (E) and m 2 M(MF (E)), m =6 0 such that for 8' 2 Cb(E), f1 − e−⟨ν;'igm(dν) < 1 and Z −⟨X;'i −⟨ν;'i − log E[e ] = hXd;'i + f1 − e gm(dν): (13) If m(f0g) = 0, then Xd and m are unique. (cf. Etheridge (2000), [20]: Theorem 1.28, p.18) d It is natural to ask whether we can construct other processes in MF (R ) (i.e. superprocesses) with infinitely divisible distributions. As a matter of fact, using the above canonical representation formula as a starting point, we can construct a more general superprocess. To keep the notation as simple as possible, we restrict our plan to time homogeneous Markov processes satisfying two conditions: (i) branching property; (ii) infinite divisibility. Let us denote by Y = fYtg a time homogeneous Markov process. We have −⟨Yt;'i −⟨µ,Vt'i Eµ[e ] = e : (14) This operator Vt satisfies the property Vt+s = Vt ◦ Vs. Comparing the formula (13) and Eq.(14), we can write with the uniqueness of the canonical representation, Z Z −⟨ν;'i hµ, Vt'i = '(y)Yd(µ, t; dy) + (1 − e )m(µ, t; dν); (15) d d R MF (R ) where we put for simplicity Z Z Yd(µ, t; dy) = Yd(x; t; dy)µ(dx); m(µ.t, dν) = m(x; t; dν)µ(dx): Moreover, we can derive an important relation from (15) Z E h i − −⟨ν;'i − h i Vt'(x) = δx [ Yt;' ] + (1 e ν; ' )m(x; t; dν): (16) -252- When we denote by Pt the linear semigroup associated with Vt, then we have E h i @Pt' Pt'(x) = δx [ Yt;' ]; (x) = A'(x): (17) @t x=0 Under the integrable condition on m Z 1 hν; 1i ^ hν; 1i2 m(x; t; dν) 6 C; (9C > 0); 8t < 1 (18) t 1 and someR proper measurability on the kernel n(x; dθ), a measure on (0; ) satis- 1 ^ 2 1 fying 0 θ θ n(x; dθ) < , the compactness argument (e.g. Le Gall (1999) [31]) allows us to obtain v(t; x) = Vt'(x) satisfying @v (t; x) = Av − b(x)v − c(x)v2 @t Z 1 + (1 − e−θv(t;x) − θv(t; x))n(x; dθ): (19) 0 Here A is the generator of a Feller semigroup, c ≥ 0 and b are bounded measurable functions, and n : (0; 1) ! (0; 1) is a kernel satisfying the integrability condition, where required is uniformity with respect to the parameter x.
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