!dp= fj f dv for any f e C* (S).

Transcription

1 252 [Vol. 42, 59. Fundamenal Equaions o f Branching Markov Processes By Nobuyuki IKEDA, Masao NAGASAWA, and Shinzo WATANABE Osaka Universiy, Tokyo Insiue of Technology, and Kyoo Universiy (Comm. by Kinjiro KUNUGI, M.J.A., March 12, 1966) We have given in he previous paper [2] a definiion of branching Markov processes and discussed some fundamenal properies of hem. Here we shall rea several fundamenal equaions which describe and characerize hese processes. 1. Fundamenal quaniies of branching Markov processes. In his paper we shall use consanly he noaion'' and he erminolog adoped in [2]. Definiion 1.1. Le X~ be a branching Markov process (abbreviaed as B.M.P.) on S. We denoe he killed process on S of X a he firs branching ime v by X~ ' and call i he non branching par on Sn of B.M.P. X. The non branching par on S' is called simply he non branching par of X, and is semi-group on B(S) is defined by (1.1) T f(x) = E. [f(x ); < v], Furher we denoe.f e B(S), x E S. (1.2) K(x, d, dy) = Px [v e d, Xz_ E dy], x e S, dy C S.2' Definiion 1.2. Assume ha here exiss a sysem {q(x); nn= 0, 2, 3,..., + } of non-negaives Borel measurable funcions on S and a sysem {ir(x, ndy); n= 0, 2,..., + o o} of non-negaives kernels3' on Sx S such ha (1.3) PX[XV E dy Xz-] =i'(x= dy), almos surely (P,) on {v.( oo }, x E S, dyes, where we pu (1.4) 7c(x, dy)= gn(x)irn(x, dyl Sn), n=0 and ~* denoes he sum over n= 0, 2,..., + oo and S = {d}. Then n= we shall call {q, n7r, n= 0, 2,..., + } he branching sysem of B.M.P. X. I is clear ha if a kernel ~r(x, dy) on S x S saisfing (1.3) is given, hen he sysem (1.5) gn(x) _ 2r(x, Sn), 7rn(x, dy) _ ir(x, dy)l gn(x), n= 0, 2,..., + 0, is he branching sysem of B.M.P. X. The above defined { T, K, q,, 7rn} are fundamenal quaniies of B. M. P, which compleely deermine he B.M.P. X. In his paper 1) In [2], branching Markov processes are denoed by x, bu in he following we wrie i as X. 2) We wrie as X0--=1im X, for any random ime a. 3),r(x, dy) is said o be a non-negaive kernel on Sx S, if for any Borel se B c S,,r(., B) is a Borel measurable funcion on S and for any x E S, 7r(x,.) is a non-negaive measure on S wih oal mass less han 1.

2 k No. 31 Fundamenal Equaions of Branching Markov Processes 253 we shall discuss some equaions defined hrough { T, K, qn, 7r}. More deailed sudy on hese equaions and he consrucion of a B.M.P. hrough hese equaions will be given in he forhcoming papers. Now, we assume in he following { T, K, qn, 7rn} are given a prio2 independen of B.M.P. X. Namely, Definiion 1.3. Le { T, K, qn, 2rn} be a sysem saisfying he following condiions: 1 ) Given a Markov process X = {X,, F,, x e S } on S and assume X0_ e S exiss, where ~ is he life ime of X, hen T and K are defined by (1.5) 5 K T f(x)=e [f(x ); x d d =P < e d j, X _ e x d e 5,f e B(S), x e S d and CS. 2 ) a) {q,,(x); n= 0, 2,.., + o o} is a sysem of non-negaive measurable funcions such as (1.7) q(x)=1, xe S, and n=0 b) {1rn(x, dy); n= 0, 2,.., --I-- oo } is a sysem of non-negaive kernels on S x Sn saisfying (1.8) wn(x, Sn) = 1, x e S, n= o, 2,, + oo. Then we call { T, K, qn, 2rn} (or { T, K, 7r} where i is given by (1.5)) a fundamenal sysem. If hese are given by Definiions 1.1 and 1.2., we call hem he fundamenal sysem of B.M.P. X. 2. Some preparaory resuls. We need he following Lemmas. Pu 21 C*(S)={f; f c C(S) ~I f <1}, and.( ) C * S = e C S and HfII1}. <_ The non-negaive par of C *(S) is denoed by C *(S)+. Lemma 2,1, i) The linear hull of {f I n; f e C*(S)+} is dense in C(Sn), ii) The linear hull of {f; f e C*(S)+} is dense in C (S).4' Lemma 2.2. Le vl, v2f, vk be signed measures on S-- {d} of bounded oal variaions. Then i) here exiss one and only one signed measure p on S--{d}, such as We denoe s-{} Then we have ii) and!dp= fj f dv for any f e C* (S). j=1 S-{d} =v~ov20... Ovk. =1 y110 y ~I vk 15~ p(s'-{d})= 1 1(S- {4}). j=1 Hence if v j are non-negaive, p is non-negaive, and if vj are probabiliy measures hen p is so. Definiion 2.1. For f e B*(S) and g e B(S)6' pu 4) C(S)={f; f is bounded coninuous on S}, C(S)= {f: f is bounded coninuous on Sn}, and Co(S) _ { f; f is bounded coninuous on S wih f(4)=0}. 5) I /1 and I denoe he oal variaions of p and v j, respecively. 6) B(S) = { f; f is bounded Borel measurable}, B*(S)(B*(S))= { f; f e B(S), I I f II < 1 (resp. I f <1)}.

3 254 N. IKEDA, M. NAGASAWA, and S. WATANABE [Vol. 42, (2.2) n <f g>(x)= {l(xk)l'kf(xi) g, if x c Sn and (x1,..., xn) E x, 0, if x = a or d. If f e C*(S) and g e C(S), hen clearly < f g> E C (S). Definiion 2.2. For f e B * (S) pu (2.3) F[x, f ] _ rr(x, dy)f(y). S Then F defines a non-linear operaor on B*(S) ino B(S). Theorem 2.1. Le { T, K, 2r} be a fundamenal sysem of Definiion 1.3. Then: 1 ) For n= 0, 2, 3,..., + oo 7~ here exiss a unique non-negaive kernel T (, x, dy) on S'>< Sn such as S (2.4) T (, x, dy)f (y) = T f (x), for any f e C *(S) and x E Se', Sn and wih T (, x, Sn) 1. 2 ) There exiss a unique non-negaive kernel!p'(x, ds, dy) on Sx ([0, oo) x S) such as (2.5) (x, ds, dy) f (y) = T f K(., d s, dz)f[z, f ] (x), s s for any f e C*(S), and (2.6)?'(x, [0, ] x S) =1-- T (, x, S n), x E Sn, n=0, 2,..., ) For f e B(S), we pu T f(x)= T (, x, dy)f(y), x E Sn, n=0, 2, 3,..., +00. Then {T,?P'} saisfies and T 1 +?P'(x, [0, ] x S) =1,!P'(x, [0, ] x dy)=?p'(x, [0,r] x dy)+ T {?P'(., [0, --r] x dy)}(x), 0_<<r<.8' 3. Fundamenal equaions of B.M.P. In his secion we assume ha we are given a fundamenal sysem { K, 7r} of Definiion 1.3., and le T and P' be hose of Theorem 2, M-equaion. Definiion 3.1. For f e C(S), consider he following equaion (3.1) u(x) = T f (x) + P(x, dr, dy)u-r(y), x e S, s which we call Moyal equaion (M-equaion) corresponding o { T, K, 2r}, A soluion of (3,1) is called a soluion of M-equaion for he iniial value f. Theorem 3.1. Suppose ha a B.M,P. X has he branching sysem and saisfies he condiion (c. 3) of Theorem 1 in [21. Le T be he semi-group of B,M,P. X. Then u(x)= Tf(x) is a soluion of M-equaion corresponding o he sysem {T00, K, 2r} of X for he iniial value f e C (S). Proof is easily performed using Theorem 1 in [2] and so-called Dynkin's formula [1]. 7) For n=+ oo, pu T (, d, {4})=1. 8) Moyal [3] called his (P,?lT)-condiion.

4 No. 31 Fundamenal Equaions of Branching Markov Processes S-equaion. Definiion 3.2. Consider for f e C(S), he following equaion (3.2) u(x)= T (x)+ LK(x, ds, dy)f[y, u_si, x E S, and we call i Skorohod equaion (S-equaion) corresponding o { T, K, 7r}.9) Theorem 3.2. (Skorohod [5]) Suppose a B.M.P. X has he branching sysem, hen u(x)- Tf(x), x e S, is a soluion of S-equaion corresponding o he sysem { T, K, 7r} of X for iniial value f E C(S), where T is he semi-group of B.M.P. X Semi-linear parabolic equaion (backward equaion). We now se an assumpion. Assumpion 1. {T, K, 2} of Definiion 1.3 saisfies he following condiions; 1 ) The Markov process X in 1 ) of Def. 1.3, is obained as follows; Given a srongly coninuous semi-group U on C(S) saisfying U1= 1, and a funcion k e C(S)+, le X be he Hun process corresponding o U. Then X is he exp - k(x.)ds -subprocess of X. 2 ) The kernel 7r(x, dry) defines F[., f ] E C(S) for any f e C*(S)+.1 ) Now le ( (resp. C ) be he generaor (Hille-Yosida sense [6] ) of U (resp. T ) and ~(C) (resp.(( )) be is domain, hen we have and CM =-k, C and K is given by (cf. [4] ) K(x, ds, dy)f(y)= \ T (kf)(x)ds, f e C(S), x e S. s Definiion 3.3. Consider he following equaion au (3.3) a _ u + kf[., u01, =C u+k(f[., uj --u), and we call i he semi-linear parabolic equaion (backward equaion) corresponding o { T, K, ~c}. Theorem 3.3. Suppose a B.M.P. X has he branching sysem and saisfies he Assumpion 1. Then i) he semi-group T of X is srongly coninuous on C (S). Le G be is generaor. I f o <f< 1 and f e ~(( ), hen f e i(g) and (3.4) Gf(x)=<f Yf+kF[., f IJ>(x), x E S. ii) For 0 f < 1 and f e (( ), u0(x) = Tf (x), x e S, is a soluion of (3.3) corresponding o he sysem { T, K, 2r} o f X, which saisfies u-f ~--*0 ( 0). 9) 10) Noice ha M -equaion is an equaion on S, while S-equaion is defined on S. If T is srongly Feller, we need no assume 2 ).

5 n 256 N. IKEDA, M. NAGASAWA, and S. WATANABE [Vol. 42, 3.4. Forward equaion. Assumpion 1 is se. Le += {f, f E C(S), 0<f<1}, and le A(f) be a funcional defined on +. A funcional derivaive of A(f) a f e ~+ owards g e C(S) is defined by lim A(f +~g)-a(f ) if his limi exiss. We denoe i by DgA(f ). Definiion 3.4. Consider, for f e ~+ f ~(Ci ), he following equaion (3.5) aa(,f) a = D~(f)A(, f), where c(f)=c3 f+k( )F[, f]. We call (3.5) forward equaion corresponding o {T, K, 7r}. Theorem 3.4. Suppose a B.M.P. X has he branching sysem and saisfies he Assumpion 1, hen if we pu for x e S-{d}, Ax(, f) - Tf (x), f e ~+ f ~(~ ), i is a soluion of he forward equaion corresponding o he sysem { T, K, 2c} of X wih he iniial condiion Ax(0+, f)=f(x) 3.5. Equaion of he mean number of paricles. Definiion 3.5. For f e B(S), pu 3.6 v x = ~, f (x), if x e Sn, (x1,..., xn) e x, 0, if x = a or d. v Then f (x) is a measurable funcion on S. Definiion 3.6. Pu for non-negaive f e B(S), G(x,f)= and consider an equaion Js 7r(x, dy)f(y)e(y),11) (3.7) u(x) = e T (ef)(x) + e s K5(x, ds, dy)g(y, u_,), where e(x)= Px[ea= + oo ] (e4 is he explosion ime). We call (3.7) he (generalized) equaion of he mean number of paricles corresponding o {T, K, i}. We inroduce Assumpion 2. 1 ) For f E C(S), G(., f) e C(S), and 2 ) II G(, f) I EMI f II, (M is a posiive e(x) e C(S) and e > 0 on S. consan). Definiion 3.7. Under Assumpions 1 and 2, we pu (3.8) a e 1 1e 11) For f =f +-f we is definie. pu G(x,f)=G(x,f+)-G(x,f-) if he righ hand side

6 No. 3] Fundamenal Equaions of Branching Markov Processes 257 and we call (3.8) he (parabolic) equaion of he mean number of paricles. Theorem 3.5. Suppose a B.M.P. X has he branching sysem and 0 < e(x) _< 1, x e S. Pu, for f e B(S) and f >_ 0 (3.9) Hf(x)=,.1 T(e f)(x). e(x) Then i saisfies H+sf(x) - HHsf (x), x e S, and, i f f e C(S), hen u(x)=hf(x) is a soluion of (3.7) corresponding o he sysem {T, K, 2r} of X wih iniial value f. Moreover if B.M.P. X saisfies Assumpions 1 and 2, hen H is a srongly coninuous semi-group on C(S) wih II H I e, (c is a posiive consan), and if of e ~(( ), hen eu e ~(5~ ) and u is a soluion of (3.8) saisfying u -- f I ---0 ( 0), Remark. If e m 1 and if we pu f m 1, hen we have (3.10) H1(x) = Ex [he number of paricles a ], which represens he mean number of paricles a. References [1] [2] [3] [4] [5] [6] Dynkin, E. B.: Markov Processes. Springer (1965). Ikeda, N., M. Nagasawa, and S. Waanabe: On branching Markov processes. Proc. Japan Acad., 41, (1965). Moyal, J. E.: Disconinuous Markov processes. Aca Mah., 98, (1957). Nagasawa, M., and K. Sao: Some heorems on ime change and killing of Markov processes. Kodai Mah. Sem. Rep., 15, (1963). Skorohod, A. V.: Branching diffusion processes. Theory of Prob. Appl., 9, (1964). Yosida, K.: On he differeniabiliy and he represenaion of one-parameer semi-group of linear operaors. Jour. Mah. Soc. Japan, 1, (1948).