Measure-valued Diffusions and Stochastic Equations with Poisson Process 1

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1 Publihed in: Oaka Journal of Mahemaic 41 (24), 3: Meaure-valued Diffuion and Sochaic quaion wih Poion Proce 1 Zongfei FU and Zenghu LI 2 Running head: Meaure-valued Diffuion and Sochaic quaion 1 Inroducion Suppoe ha we are given a locally compac meric pace. Le C() denoe he e of bounded coninuou funcion on, and C () i ube of coninuou funcion vanihing a infiniy. The ube of non-negaive elemen of C() and C () are denoed repecively by C + () and C + (). Le (P ) be a rongly coninuou conervaive Feller emigroup on C () wih generaor (A, D(A)), where D (A) C (), and le D(A) = D (A) {1}. Suppoe in addiion ha b( ) C() and c( ) C + () have coninuou exenion o Ē, he one poin compacificaion of, and ha c( ) i bounded away from zero. Le M() be he pace of finie Borel meaure on equipped wih he opology of weak convergence. Le W = C([, ), M()) be he pace of all coninuou pah w : [, ) M(). Le τ (w) = inf{ > : w() = } for w W and le W be he e of pah w W aifying w() = w() = for all τ (w). We fix a meric on M() which i compaible wih i opology and endow W and W wih he opology of uniform convergence. Then for each µ M() here i a unique Borel probabiliy meaure Q µ on W uch ha for f D(A), M (f) = w (f) µ(f) under Q µ i a maringale wih quadraic variaion proce M(f) = w (Af bf)d,, (1.1) w (cf 2 )d,, (1.2) where µ(f) = fdµ. The yem {Q µ : µ M()} define a meaure-valued diffuion, which i he well-known Dawon-Waanabe uperproce. In he equel, we hall imply refer o i a a (A, b, c)-uperproce. We refer he reader o Dawon [1] and he reference 1 Reearch uppored by NSFC (No and No ). 2 Correponding auhor. -mail: lizh@ .bnu.edu.cn 1

2 herein for he conrucion and baic properie of he proce. A modificaion of he above model i o replace (1.1) by M (f) = w (f) µ(f) w (Af bf)d V (w, f)d,, (1.3) by uing a kernel V (µ, dx) from M() o, which can be regarded a a (A, b, c)- uperproce wih ineracive immigraion. Some inereing pecial cae of hi modificaion have been udied in he lieraure. Uing a Cameron-Marin-Giranov formula, Dawon [1, pp ] reaed he pecial cae where b( ) and V (µ, dx) = r(µ, x)µ(dx), µ M(), x, for a coninuou funcion r(, ) on M() and obained a uperproce wih non-linear birh-deah rae. The condiioned uperproce conruced by van and Perkin [5] and Roelly-Coppolea and Rouaul [16] correpond o he cae V (µ, dx) = µ(1) 1 µ(dx), µ M() \ {}, x. An inereing repreenaion of he condiioned uperproce wa given by van [4] in erm of an immoral paricle ha move around according o he underlying proce and hrow off piece of ma ino he pace. Le m be a σ-finie Borel meaure on and le q(, ) be a non-negaive Borel funcion on M(). We have anoher paricular form of (1.3) given by M (f) = w (f) µ(f) w (Af bf)d m(q(w, )f)d,, (1.4) where q(, ) can be inerpreed a an ineracive immigraion rae relaive o he reference meaure m. The proce defined by (1.4) and (1.2) i of inere ince i include a pecial cae (a lea formally) he uperproce wih non-linear birh-deah rae and he condiioned uperproce a hey are a.. aboluely coninuou wih repec o he reference meaure m, boh of which ha arien coniderable reearch inere. If q(ν, x) = q(x) only depend on x, he maringale problem ha a unique oluion and define a uperproce wih independen immigraion; ee e.g. Konno and Shiga [8] and Li and Shiga [12]. In he general cae, a oluion of he maringale problem could be conruced by an approximaion by paricle yem, bu he uniquene of oluion eem hard. Thi i imilar o he uperproce wih mean field ineracion udied by Méléard and Roelly [13, 14] for which he uniquene ill remain open. Inead of he maringale problem, Shiga [17] uggeed anoher approach o he ineracive immigraion uperproce, who gave he formulaion of a ochaic inegral equaion involving a uperproce and a yem of independen Poion procee on he pace of excurion of one-dimenional branching diffuion. For he paricular cae where A and µ m(q(µ, )) i bounded and Lipchiz relaive o he oal variaion meric, Shiga [17] conruced a oluion of he inegral equaion and howed ha hi oluion alo olve he maringale problem 2

3 (1.4) and (1.2). He proved ha he pahwie uniquene of oluion for he ochaic inegral equaion hold o hi oluion i a diffuion proce. Thi i a very inereing reul ince he uniquene of oluion of (1.4) and (1.2) i no known. A generalizaion of hi reul wa given in he recen work by Dawon and Li [2], where ome uperprocee wih dependen paial moion and ineracive immigraion were conruced from onedimenional excurion carried by ochaic flow. The main purpoe of hi paper i o eablih he reul of Shiga [17] when he paial migraion mechanim A i non-rivial. Since in hi cae he ma i mixed, i i no clear how o conruc he proce from one-dimenional excurion a in [17]. Forunaely, he echnique developed by Li and Shiga [12] can be combined wih hoe of Shiga [17] o olve he difficuly. The main idea of our approach i o formulae a ochaic equaion wih a Poion proce on he pace of meaure-valued excurion. Le {X : } be an (A, b, c)-uperproce wih deerminiic iniial ae X = µ and N(d, dx, du, dw) a Poion random meaure on [, ) [, ) W wih ineniy dm(dx)duq x (dw), where Q x i an excurion law of he (A, b, c)-uperproce carried by excurion growing up a x. We aume {X : } and N(d, dx, du, dw) are defined on a andard probabiliy pace and are independen of each oher. We hall prove ha he ochaic equaion Y = X + q(y,x) W w( )N(d, dx, du, dw),, (1.5) ha a pahwie unique coninuou oluion {Y : } and i diribuion on W olve he maringale problem given by (1.2) and (1.4); ee Theorem 4.1. The pahwie uniquene implie he rong Markov propery of {Y : }, o our reul give a parial oluion of he open problem on he Markov propery of he uperproce wih mean field ineracion; ee Méléard and Roelly [14, p.13]. In paricular, when = {a} i a ingleon, equaion (1.5) give a decompoiion of he one-dimenional diffuion proce {y() : } defined by dy() = cy()db() + β(y())y()d + γ(y())d,, (1.6) where c > i a conan, β( ) i a bounded Lipchiz funcion on [, ) and γ( ) i a non-negaive locally Lipchiz funcion on [, ) aifying he linear growh condiion. In he pecial cae where β( ) and γ( ) are conan, Piman and Yor [15] gave a conrucion of {y() : } by picking up excurion by a Poion poin proce, which erved a a preliminary o heir well-known reul on decompoiion of Beel bridge. See alo Le Gall and Yor [9]. In ecion 2 we recall ome baic fac on he (A, b, c)-uperproce and i immigraion procee wih deerminiic immigraion rae. In ecion 3, we dicu conrucion of immigraion procee wih predicable immigraion rae. The ochaic equaion wih a Poion proce of excurion i udied in ecion 4. 3

4 2 Deerminiic immigraion rae In hi ecion, we ummarize ome baic fac on he (A, b, c)-uperproce and i immigraion procee wih deerminiic immigraion rae. Le (Q ) denoe he raniion emigroup of he (A, b, c)-uperproce, which i deermined by e ν(f) Q (µ, dν) = exp{ µ(v f)}, f C + (), µ M(), (2.1) M() where V f i he unique poiive oluion of he evoluion equaion V f(x) + 1 d c(y)v f(y) 2 P b 2 (x, dy) = P b f(x),, x, (2.2) where (P b ) denoe he emigroup of kernel on generaed by A b := A b. By [1, pp ], here i a family of finie meaure L (x, dν) on M() := M() \ {} uch ha M() (1 e ν(f) )L (x, dν) = V f(x), >, x, f C + (). (2.3) Le (Q ) be he rericion of (Q ) o M(). I i eay o check ha (L (x, )) > i an enrance law for (Q ), ha i L r (x, )Q = L r+ (x, ) for r > and >. Then here i a unique σ-finie Borel meaure Q x on (W, B(W )) uch ha Q x (w( 1 ) dν 1,, w( n ) dν n ) = L 1 (x, dν 1 )Q 2 1 (ν 1, dν 2 ) Q n n 1 (ν n 1, dν n ) (2.4) for < 1 < 2 < < n and ν 1, ν 2,, ν n M(). Indeed, Q x i carried by he pah w W uch ha w (1) 1 w δ x a ; ee [11] and [12]. Moreover, i i eay o obain ha Q x {w()(f)} = P b f(x), >, x, f C + (). (2.5) Le B (W ) be he σ-algebra on W generaed by {w() : }. Roughly peaking, (W, B (W ), w()) under Q x i a Markov proce wih emigroup (Q ) > and onedimenional diribuion (L (x, )) >. The meaure Q x i known a an excurion law of (Q ). Now we fix a σ-finie reference meaure m on and uppoe ha q(, ) i a nonnegaive Borel funcion on [, ) uch ha m(q(, )) i a locally bounded funcion of. Then { } e ν(f) Q q r,(µ, dν) = exp µ(v r f) m(q(, )V f)d (2.6) define an inhomogeneou raniion emigroup (Q q r,) r. A diffuion proce wih raniion emigroup (Q q r,) r can be conruced a follow. Le {X : } be an (A, b, c)- uperproce wih deerminiic iniial ae X = µ and N(d, dx, du, dw) a Poion 4 r

5 random meaure on [, ) [, ) W wih ineniy dm(dx)duq x (dw). We aume {X : } and N(d, dx, du, dw) are defined on a andard probabiliy pace (Ω, A, P ) and are independen of each oher. For, le G be he σ-algebra generaed by he P -null e in A and he random variable {X, N(J A) : J B([, ] [, )), A B (W ), }. (2.7) We define he M()-valued proce {Y : } by q(,x) Y = X + w( )N(d, dx, du, dw), W, (2.8) where he inegraion area refer o {(, x, u, w) : <, x, < u q(, x), w W }. (We hall make he ame convenion in he equel.) Theorem 2.1 The proce {Y : } defined by (2.8) i an inhomogeneou diffuion proce relaive o (G ) wih raniion emigroup (Q q r,) r. Moreover, for each f D(A), M (f) = Y (f) Y (f) Y (Af bf)d m(q(, )f)d,, (2.9) i a maringale relaive o he filraion (G ) wih quadraic variaion proce M(f) = Y (cf 2 )d,. (2.1) Proof. Le N 1 (d, dx, du, dw) denoe he rericion of N(d, dx, du, dw) o {(, x, u, w) : >, x, < u q(, x), w W } and le N 1 (d, dw) be he image of N 1 (d, dx, du, dw) under he map (, x, u, w) (, w). Then N 1 (d, dw) i a Poion random meaure on [, ) W wih ineniy dq κ (dw), where Q κ (dw) = q(, x)q x (dw)m(dx), w W. Then he fir aerion follow by an obviou modificaion of he argumen of [12, Theorem 1.3] and [17, Theorem 3.6]; ee alo [1, Theorem 3.2]. The maringale characerizaion (2.9) and (2.1) can be proved by a calculaion of he generaor of (Q q r,) r. The conrucion (2.8) give clear inerpreaion for reference meaure m and immigraion rae q(, ) in he phenomenon. Since (2.9) i linear in f D(A), i define a maringale meaure M(d, dx) wih quadraic variaion meaure c(x)y (dx)d in he ene of Walh [18]. By a andard argumen one ge he following Theorem 2.2 For each and f C() we have a.. Y (f) = Y (P b f) + P f(x)m(d, b dx) + 5 m(q(, )P b f)d. (2.11)

6 3 Predicable immigraion rae In hi ecion, we fix a σ-finie reference meaure m on. Le (Ω, A, P ) be a andard probabiliy pace and N(d, dx, du, dw) and {X : } be a in he la ecion. Le G be he σ-algebra on Ω generaed by he P -null e in A and he random variable in (2.7). Le P be he σ-algebra on [, ) Ω generaed by funcion of he form g(, x, ω) = η (x, ω)1 {r }() + η i (x, ω)1 (ri,r i+1 ](), (3.1) i= where = r < r 1 < r 2 <... and η i (, ) i B() G ri -meaurable. We ay a funcion on [, ) Ω i predicable if i i P-meaurable. Theorem 3.1 Suppoe ha q(,, ) i a non-negaive predicable funcion on [, ) Ω uch ha {m(q(, )) 2 } i locally bounded in. Then he M()-valued proce Y = X + q(,x) W w( )N(d, dx, du, dw),, (3.2) ha a coninuou modificaion. Moreover, for hi modificaion and each f D(A), M (f) = Y (f) Y (f) Y (Af bf)d m(q(, )f)d,, (3.3) i a maringale relaive o he filraion (G ) wih quadraic variaion proce M(f) = Y (cf 2 )d,. (3.4) Le M(d, dx) denoe he ochaic inegral wih repec o he maringale meaure wih quadraic variaion meaure c(x)y (dx)d defined by (3.3) and (3.4). Then we have Theorem 3.2 For each and f C() we have a.. Y (f) = Y (P b f) + P b f(x)m(d, dx) + m(q(, )P b f)d. (3.5) The proce {Y : } conruced by (3.2) can be regarded a an (A, b, c)- uperproce allowing immigraion wih immigraion rae given by he predicable funcion q(,, ). To give he proof of he above heorem we need a e of lemma. Lemma 3.1 The reul of Theorem 3.1 and 3.2 hold if q(,, ) i of he form (3.1). 6

7 Proof. Oberve ha η i (x) i a deerminiic funcion on under he regular condiional probabiliy P { G ri }. Since G ri and he rericion of N(d, dx, du, dw) o (r i, ) [, ) W are independen, hi rericion under P { G ri } i ill a Poion random meaure wih ineniy dm(dx)duq x (dw). Noe ha {X : } i alo an a.. coninuou (A, b, c)-uperproce under P { G }. Then we conclude by Theorem 2.1 ha {Y : r 1 } under P { G } i an a.. coninuou (A, b, c)-uperproce allowing immigraion wih immigraion rae η ( ). Le r1 Y () = X + η (x) W w( )N(d, dx, du, dw), r 1. By Theorem 2.1, {Y () : r 1 } under P { G } i an a.. coninuou (A, b, c)-uperproce. Of coure, {Y () : r 1 } i ill an a.. coninuou (A, b, c)-uperproce under P { G r1 }. I i no difficul o ee ha Y = Y () + r 1 η1 (x) W w( )N(d, dx, du, dw), r 1 r 2. By Theorem 2.1 again, {Y : r 1 r 2 } under P { G r1 } i an a.. coninuou (A, b, c)- uperproce allowing immigraion wih immigraion rae η 1 ( ). Uing he above argumen inducively we can ee ha {Y : r i r i+1 } under P { G ri } i an a.. coninuou (A, b, c)-uperproce allowing immigraion wih immigraion rae η i ( ). By Theorem 2.1, {Y : } ha a coninuou modificaion. The maringale characerizaion of Theorem 3.1 and 3.2 follow from hoe of he immigraion proce wih deerminiic immigraion rae. Lemma 3.2 Suppoe ha here i a non-negaive deerminiic funcion q 1 ( ) L 1 (, m) uch ha q(, x, ω) q 1 (x) for a.a. (, x, ω) [, ) Ω. Le {g n } be a equence of non-negaive predicable funcion of he form (3.1) uch ha g n (, x, ω) q 1 (x) and g n (, x, ω) q(, x, ω) for a.a. (, x, ω) [, ) Ω. Le {Y (n) : } be defined by (3.2) in erm of g n (,, ). Then here i an M()-valued proce {Y : } uch ha lim n { Y (n) Y } = uniformly on each finie inerval of, where denoe he oal variaion meric. Proof. Since he reul of Theorem 3.2 hold for {Y (n) {Y (n) (f)} = µ(p b f) + : }, we have {m(g n (, )P b f)}d, f C(). (3.6) Oberve ha for any k n 1, boh g n g k and g n g k are predicable funcion of he form (3.1). Le Y (n,k) = X + gn(,x) g k (,x) 7 W w( )N(d, dx, du, dw)

8 and Z (n,k) = X + gn(,x) g k (,x) W w( )N(d, dx, du, dw). Since Y (n) {Z (n,k) Y (k) : } o ha Y (n,k) (1) Z (n,k) (1), we may apply (3.6) o {Y (n,k) : } and { Y (n) Y (k) } e b ( ) {m( g n (, ) g k (, ) )}d. By dominaed convergence, he righ hand ide goe o zero uniformly on each finie inerval of a n. Then here i an M()-valued proce {Y : } uch ha {Y (f)} = µ(p b f) + {m(q(, )P b f)}d, f C(), (3.7) and lim n { Y (n) Y } = uniformly on each finie inerval of. Lemma 3.3 Suppoe ha he condiion of Lemma 3.2 hold. Then he proce {Y : } obained here i independen of he choice of {g n } in he ene ha if {Z : } obained from anoher equence wih he ame properie, hen Y = Z a.. for each. Moreover, (3.2) hold a.. for each. Proof. Le {q n } be anoher equence having he properie of {g n }. Then {g n q n } and {g n q n } have he ame properie. Le {Y : } and {Y : } be he procee obained repecively from {g n q n } and {g n q n }. Clearly, Y Y Y a.. for each. Bu, {Y (1)} = {Y (1)} = {Y (1)} by (3.7), o we have Y = Y = Y a.. for each. Thu {Y : } i independen of he choice of {g n }. To how (3.2), le Z denoe he value of i righ hand ide. We fir aume in addiion here i a ricly poiive deerminiic funcion q 2 ( ) L 1 (, m) uch ha q 2 (x) q(, x, ω) for all (, x, ω) [, ) Ω. For k 1, le {Y k, : } and {Z k, : } be he proce obained by Lemma 3.2 from he non-negaive predicable funcion q(, x, ω) + q 2 (x)/k and q(, x, ω) q 2 (x)/k, repecively. Since q(, x, ω) q 2 (x)/k < q(, x, ω) < q(, x, ω) + q 2 (x)/k, we have Z k, Z, Y Y k, a.. for each. Bu, by (3.7) i i eay o how ha {Y k, (1) Z k, (1)} 2e b m(q 2 )/k, o we mu have Y = Z a.. for each. In he general cae, we may apply he above reaoning o q(, x, ω) + q 2 (x)/k and {Y k, : } o ge Y k, = X + q(,x)+q2 (x)/k 8 W w( )N(d, dx, du, dw)

9 and {Y k, (f)} = µ(p b f) + {m([q(, ) + q 2 /k]p b f)}d. Clearly, Y k, decreae o Z a k. A in he proof of Lemma 3.2, i i eay o how ha lim k { Y k, Y } = uniformly on each finie inerval of, o he deired reul hold. Lemma 3.4 Under he aumpion of Theorem 3.1, chooe a ricly poiive funcion q 1 ( ) L 1 (, m) and le q n (, x, ω) = q(, x, ω) (nq 1 (x)). Le {Z (n) : } be defined by (3.2) in erm of q n (,, ). Then we have lim n { Z (n) Y } = uniformly on each finie inerval of, where {Y : } i defined by (3.2). Proof. A in he proof of Lemma 3.2 one can how ha here i an M()-valued proce {Z : } uch ha lim n { Z (n) Z } = uniformly on each finie inerval of. A in he proof of Lemma 3.3 we have Y = Z a.. for each. Lemma 3.5 The reul of Theorem 3.1 and 3.2 hold if i compac. Proof. We fir aume he condiion of Lemma 3.2 hold. Le {Y (n) approximaing equence given by Lemma 3.2 and define {M (n) of {Y (n) : } and g n (,, ). By (3.6) we have {Y (n) (1)}d Then for T > and ε >, here i η > uch ha e b [µ(1) + m(q 1 )]d. : } be he : } by (3.3) in erm { T } T P Y (n) ( b )d > η/2 2η 1 b e b [µ(1) + m(q 1 )]d ε. Moreover, from (3.4) we obain T {M (n) T (1)2 } = T {Y (n) (c)}d c e b [µ(1) + m(q 1 )]d. (3.8) In view of he maringale characerizaion (3.3) and (3.4) for {Y (n) η > 2µ(1) + 2m(q 1 )T we have { P up T { ε + P } Y (n) (1) > η up T T } Y (n) (1) > η, Y (n) ( b )d η/2 : }, chooing 9

10 { ε + P up T [ µ(1) + M (n) (1) + ] } m(g n (, ))d > η/2 { } ε + P up M (n) (1) > η/2 µ(1) m(q 1 )T T { } 4(η/2 µ(1) m(q 1 )T ) 2 M (n) T (1)2 4(η/2 µ(1) m(q 1 )T ) 2 c T e b [µ(1) + m(q 1 )]d by a maringale inequaliy; ee e.g. [6, p.34]. Conequenly, lim up η n 1 { P up T } Y (n) (1) > η =. Thu {Y (n) : } viewed a procee in C([, ), M()) aify he compac conainmen condiion of [3, p.142]. (Noe ha C([, ), M()) i a cloed ubpace of D([, ), M()).) By Iô formula, for G C 2 (IR m ) and {f 1,, f m } D(A), G(Y (n) 1 2 (f 1 ),, Y (n) (f m )) G(Y (n) (f 1 ),, Y (n) (f m )) m G i(y (n) (f 1 ),, Y (n) (f m ))[m(g n (, )f i ) + Y (n) (Af i bf i )]d i=1 m G i,j=1 ij(y (n) (f 1 ),, Y (n) (f m ))Y (n) (cfi 2 )d i a coninuou maringale. From (3.8) and he maringale characerizaion of Lemma 3.1 we ee ha {Y (n) (1) 2 } i dominaed by a locally bounded poiive funcion independen of n 1. By [3, pp ] we conclude ha {Y (n) : } i a igh equence in C([, ), M()). Conequenly, {Y : } ha a coninuou modificaion and {Y (n) : } converge a.. o hi modificaion in he opology of C([, ), M()). Noe alo ha m(g n (, )f i )d m(q(, )f i )d,, in he opology of C([, ), IR). Then he maringale characerizaion (3.3) and (3.4) for {Y : } follow from Lemma 3.1 and [7, p.342]. If he condiion of Lemma 3.2 doe no hold, we may conider he addiional approximaing equence {Z (n) : } given by Lemma 3.4. Then a modificaion of he above argumen how ha {Z (n) : } i a igh equence, o we alo have (3.3) and (3.4). The equaliy (3.5) follow in he ame way a Theorem 2.2. Proof of Theorem 3.1 and 3.2. Noe ha (P ) can be exended o a Feller raniion emigroup ( P ) on Ē, he one poin compacificaion of. Since m can be viewed a 1

11 a σ-finie meaure on Ē and ince b( ) and c( ) have coninuou exenion b( ) and c( ) on Ē, we can alo regard {X : } and {Y : } a objec aociaed wih ( P ). Applying Lemma 3.5 in hi way we ee ha {Y : } ha a M(Ē)-valued coninuou modificaion {Ȳ : } which aifie he correponding maringale characerizaion (3.3) and (3.4). Then he wo heorem will follow from Lemma 3.5 once i i proved ha Oberve ha for any f C(Ē), P {Ȳ({ }) = for all [, T ]} = 1, T >. (3.9) M T ( f) := Ȳ( P T b f) Ȳ( P T b f) = P T b f(x) M(d, dx) Ē m( P b T fq(, ))d i a coninuou maringale in [, T ] wih quadraic variaion proce M T ( f) = Ȳ ( c( P b T f) 2 )d, where ( P b ) i defined from ( P ) and b. By a maringale inequaliy we have 4 { P up T T Ȳ( P b T f) Ȳ( P b T f) {Ȳ( c( P b T f) 2 )}d. m( P T b fq(, ))d 2 } Chooe a equence { f k } C(Ē) uch ha f k 1 { } boundedly a k. Since each Ȳ i a.. uppored by, replacing f by f k in he above and leing k we obain (3.9). 4 A ochaic equaion wih Poion proce We fix a σ-finie reference meaure m on. Le (Ω, A, P ) be a andard probabiliy pace on which N(d, dx, du, dw) and {X : } are given a in ecion 2. Le G be he σ-algebra on Ω generaed by he P -null e in A and he random variable in (2.7). Suppoe ha q(, ) i a Borel funcion on M() uch ha here i a conan K uch ha m(q(ν, )) K(1 + ν ), ν M(), (4.1) and for each R > here i a conan K R > uch ha m( q(ν, ) q(γ, ) ) K R ν γ (4.2) 11

12 for ν and γ M() aifying ν(1) R and γ(1) R. inegral equaion: Y = X + q(y,x) We conider he ochaic W w( )N(d, dx, du, dw),. (4.3) By a (rong) oluion of (4.3) we mean a coninuou M()-valued proce {Y : } which i adaped o he filraion (G ) and aifie (4.3) wih probabiliy one. A oluion of hi equaion can be regarded a an immigraion (A, b, c)-uperproce wih ineracive immigraion rae given by q(, ). Lemma 4.1 Le R and le q 1 (, ) and q 2 (, ) be Borel funcion on M() aifying q 1 (ν, ) q 2 (ν, ) q(ν, ) for ν(1) R. Suppoe ha {Y (1) : } and {Y (2) : } are oluion of (4.3) wih q(, ) replaced by q 1 (, ) and q 2 (, ) repecively. Le τ = inf{ : Y (1) (1) R or Y (2) (1) R}. Then {Y (1) τ : } and {Y (2) τ : } are indiinguihable. Proof. Since each {Y (i) ha m(q(y (i) and, )I {τ} ) i bounded. Le τ (1) q(y Y = Z = τ : } i coninuou, q(y (i), x)i {τ} i predicable. Noe alo,x) q(y (2),x) (1) q(y,x) q(y (2),x) Applying Theorem 3.1 o he predicable funcion we ee ha M (1) = Y (1) + W w( )N(d, dx, du, dw) W w( )N(d, dx, du, dw). (, x, ω) q(y (1), x) q(y (2), x)i {τ} Y (b)d i a coninuou maringale. By Doob opping heorem, {Y τ(1)} = Similarly, we have {Z τ(1)} = {m(q(y (1), ) q(y (2), ))I {τ} }d {m(q(y (1), ) q(y (2), ))I {τ} }d m(q(y (1), ) q(y (2), ))I {τ} d 12 {Y (b)i {τ} }d. {Z (b)i {τ} }d.

13 By (4.2) we obain {[Y τ(1) Z τ(1)]} = {m( q(y (1), ) q(y (2), ) I {τ} )}d + {[Y (b) Z (b)]i {τ} }d K R { Y (1) Y (2) I {τ} }d + b K R + b {[Y (1) Z (1)]I {τ} }d { Y (1) τ Y τ }d (2) (K R + b ) {[Y τ(1) Z τ(1)]}d {[Y τ(1) Z τ(1)]}d. Oberve ha Y (1) τ Y (2) τ Y τ(1) Z τ(1). Then Gronwall inequaliy yield { Y (1) τ Y (2) τ } {[Y τ(1) Z τ(1)]} = for all. Since {Y (1) τ : } and {Y (2) τ : } are coninuou, hey are indiinguihable. Lemma 4.2 There i a mo one oluion of (4.3). Proof. Suppoe {Y : } and {Y : } are wo oluion of (4.3). Le τ n = inf{ : Y (1) n or Y (1) n}. By Lemma 4.1, {Y τn : } and {Y τ n : } are indiinguihable for each n 1. Thu τ n = inf{ : Y (1) n} = inf{ : Y (1) n}. By coninuiy of pah, τ n a.. a n and hence {Y : } and {Y : } are indiinguihable, ha i, (4.3) ha a mo one oluion. Lemma 4.3 Suppoe here i a conan L uch ha m(q(ν, )) L for all ν M() and (4.2) hold for all ν and γ M() wih K R replaced by L. Then here i a oluion {Y : } of (4.3). Moreover, for hi oluion and each f D(A), M (f) = Y (f) Y (f) Y (Af bf)d m(q(y, )f)d,, i a coninuou maringale relaive o he filraion (G ) wih quadraic variaion proce M(f) = Y (cf 2 )d,. 13

14 Proof. Since {X : } i a.. coninuou, he funcion (, x, ω) q(x (ω), x) i predicable. We define an approximaing equence {Y (n) : } inducively by Y () = X and for n 1. Le Y (n) = X + Y (n) = q(y (n 1) q(y (n 1),x),x) q(y (n 2),x) W w( )N(d, dx, du, dw) W w( )N(d, dx, du, dw) and Z (n) = q(y (n 1),x) q(y (n 2),x) W w( )N(d, dx, du, dw). From Theorem 3.1 and 3.2 i follow ha and {Y (n) (1)} = {m([q(y (n 1), ) q(y (n 2), )]P 1)}d b {Z (n) (1)} = Then we ue (4.2) and he fac Y (n) {m([q(y (n 1) Y (n 1), ) q(y (n 2), )]P 1)}d. b Y (n) (1) Z (n) (1) o ee and { Y (n) Y (n 1) } {m( q(y (n 1) Le b T { Y (n 1), ) q(y (n 2), ) P 1)}d b Y (n 2) }d, { Y (1) Y () } = {m(q(x, )P b 1)}d LT e b T for T. By a andard argumen, one how ha here i an M()-valued proce {Y : } uch ha lim n { Y (n) Y } = uniformly on each finie inerval of. Le Y = X + q(y,x) W w( )N(d, dx, du, dw). 14

15 By a calculaion imilar a he above we ge { Y (n) Y } Le b T { Y (n 1) Y }d for T. Then we alo have lim n { Y (n) Y } = uniformly on each finie inerval of, o ha a.. Y = Y and (4.3) i aified. By Theorem 3.1, {Y : } ha a coninuou modificaion and we have he maringale characerizaion. Lemma 4.4 For each n 1 define a coninuouly differeniable funcion a n ( ) on [, ) uch ha { 1 if z n 1, a n (z) = n/z if z n + 1, and a n(z) 1/z for all z. condiion of Lemma 4.3. Then q n (ν, x) := q(a n (ν(1))ν, x) aifie he Proof. Oberve ha he aumpion on a( ) imply ha a n (z)z n for every z. By (4.1) and he definiion of q n (, ) we have m(q n (ν, )) K(1 + a n (ν(1))ν(1)) K(1 + n). On he oher hand, for ν and γ M() le η = ν +γ and le g ν and g γ denoe repecively he deniie of ν and γ wih repec o η. Wihou lo of generaliy, we may aume ν(1) γ(1). By he mean-value heorem we have ha ν(1) a n (ν(1)) a n (γ(1)) ν(1) a n(z) ν(1) γ(1) ν γ, where ν(1) z γ(1). I follow ha m(q n (ν, ) q n (γ, )) = m(q(a n (ν(1))ν, ) q(a n (γ(1))γ, )) K n a n (ν(1))ν a n (γ(1))γ K n η( a n (ν(1))g ν a n (γ(1))g γ ) K n [ a n (ν(1)) a n (γ(1)) η(g ν ) + a n (γ(1))η( g ν g γ )] K n [ a n (ν(1)) a n (γ(1)) ν(1) + ν γ ] 2K n ν γ. Tha i, q n (, ) aifie he condiion of Lemma 4.3. The following heorem generalize he reul of [17, Corollary 5.5]: Theorem 4.1 Under he condiion (4.1) and (4.2), here i a unique oluion {Y : } of (4.3). Moreover, {Y : } i a meaure-valued diffuion and for each f D(A), M (f) = Y (f) Y (f) Y (Af bf)d 15 m(q(y, )f)d,, (4.4)

16 i a coninuou maringale relaive o he filraion (G ) wih quadraic variaion proce M(f) = Y (cf 2 )d,. (4.5) Proof. The uniquene of (4.3) hold by Lemma 4.2. For he proof of exience, we fir conruc an approximaing equence. For each ineger n 1 le q n (, ) be defined a in Lemma 4.4. By Lemma 4.3 here i an unique coninuou oluion {Y (n) : } of (4.3) wih q(, ) replaced by q n (, ). Then, by Lemma 4.1, for k n, we have a.. Y (n) τ n = Y (k) τ n for each, where τ n = inf{ : Y (n) (1) n} = inf{ : Y (k) (1) n}. Since {Y (n) τ n : } and {Y (k) τ n : } have coninuou pah, hey are indiinguihable. Uing Theorem 3.2, condiion (4.1) and noicing ha q n (Y (n), ) = q(y (n), ) for [, τ n ] we ge {Y (n) τ n (1))} e b µ(1) + e b (µ(1) + K) + Ke b e b ( ) {m(q(y (n) τ n, ))}d {Y (n) τ n (1)}d. By Gronwall inequaliy here i a locally bounded funcion C( ) on [, ) independen of n 1 uch ha {Y (n) τ n (1)} C(). (4.6) By he definiion of τ n we have np { < τ n < } C(), and o P {τ n } = P (τ n = ) + P ( < τ n < ) 1 [n, ) (µ(1)) + n 1 C(), which goe o zero a n. Bu {τ n } i an increaing equence, o we conclude ha a.. τ n a n. Thu here i a coninuou proce {Y : } uch ha a.. Y (n) = Y for all [, τ n ]. Clearly, {Y : } aifie (4.3) wih probabiliy one. By (4.6) and Faou lemma, {Y (1)} C(). The maringale characerizaion (4.4) and (4.5) follow by Lemma 4.3. The rong Markov propery can be proved a [17, Theorem 4.4]. Suppoe ha c > i a conan, β( ) i a bounded Lipchiz funcion on [, ) and γ( ) i a non-negaive locally Lipchiz funcion on [, ) aifying he linear growh condiion. The ochaic differenial equaion dy() = cy()db() + β(y())y()d + γ(y())d,, (4.7) define diffuion proce {y() : }, which may be called a coninuou ae branching diffuion wih ineracive growh and immigraion. Seing b = inf z β(z) and q(z) = β(z)z + bz + γ(z), z, 16

17 we can rewrie (4.7) a dy() = cy()db() by()d + q(y())d,. (4.8) The la equaion may be regarded a he pecial cae of he maringale problem (4.4) and (4.5) wih = {a} being a ingleon. Thu equaion (4.3) give a decompoiion of he pah of {y() : } ino excurion of he diffuion proce {x() : } defined by dx() = cx()db() bx()d,. (4.9) Thi generalize a reul of [15], who conidered he cae where β( ) and γ( ) are conan and hence he righ hand ide of (4.3) i independen of {y() : }. See alo [9]. Reference [1] Dawon, D.A., Meaure-Valued Markov Procee, In: Lec. Noe. Mah. 1541, 1-26, Springer-Verlag, Berlin (1993). [2] Dawon, D.A. and Li, Z.H.: Conrucion of immigraion uperprocee wih dependen paial moion from one-dimenional excurion. Probab. Theory Relaed Field 127 (23), [3] hier, S.N. and Kurz, T.G., Markov Procee: Characerizaion and Convergence, Wiley, New York (1986). [4] van, S.N., Two repreenaion of a condiioned uperproce, Proc. Roy. Soc. dinburgh Sec. A 123 (1993), [5] van, S. N. and Perkin,., Meaure-valued Markov branching procee condiioned on nonexincion, Irael J. Mah. 71 (199), [6] Ikeda, N. and Waanabe, S., Sochaic Differenial quaion and Diffuion Procee, Norh-Holland/Kodanha, Amerdam/Tokyo (1989). [7] Jacod, J. and Shiryaev, A. N., Limi Theorem for Sochaic Procee, Springer-Verlag, Berlin (1987). [8] Konno, N. and Shiga, T., Sochaic parial differenial equaion for ome meaure-valued diffuion, Probab. Theory Relaed Field 79 (1988), [9] Le Gall, J.-F. and Yor, M., xcurion brownienne e carré de proceu de Beel, C. R. Acad. Sci. Pari Sér. I Mah. 33 (1986), [1] Li, Z.H., Meaure-valued immigraion diffuion and generalized Ornein-Uhlenbeck diffuion, Aca Mahemaicae Applicaae Sinica (nglih Serie) 15 (1999), [11] Li, Z.H., Skew convoluion emigroup and relaed immigraion procee, Theory Probab. Appl. 46 (22),

18 [12] Li, Z.H. and Shiga T., Meaure-valued branching diffuion: immigraion, excurion and limi heorem, J. Mah. Kyoo Univ. 35 (1995), [13] Méléard, S. and Roelly, S., Ineracing branching meaure procee, In: Sochaic Parial Differenial quaion and Applicaion, G. Da Prao and L. Tubaro ed., PRNMS 268, Longman Scienific and Technical, Harlow, (1992). [14] Méléard, S. and Roelly, S., Ineracing meaure branching procee; ome bound for he uppor, Soch. Soch. Repor 44 (1993), [15] Piman, J. and Yor, M., A decompoiion of Beel bridge, Z. Wahrch. verw. Geb. 59 (1982), [16] Roelly-Coppolea, S. and Rouaul, A., Proceu de Dawon-Waanabe condiionné par le fuur loinain, C. R. Acad. Sci. Pari Sér. I Mah. 39 (1989), [17] Shiga, T., A ochaic equaion baed on a Poion yem for a cla of meaure-valued diffuion procee, J. Mah. Kyoo Univ. 3 (199), [18] Walh, J.B., An Inroducion o Sochaic Parial Differenial quaion, In: Lec. Noe Mah. 118, , Springer-Verlag, Berlin (1986). Zongfei FU School of Informaion People Univeriy of China Beijing 1872, China fuzongfei@ina.com Zenghu LI Deparmen of Mahemaic Beijing Normal Univeriy Beijing 1875, China lizh@ .bnu.edu.cn 18