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Donald By n iga ( Kingman and ) N ESR-AUDPROCESSES MEASURE-VALUED AND liosJ Math. J. Illinois n.Is.H ona´ rbb Statist. Poincar´e Probab. H. Inst. Ann. TCATCEUTOS FLOWS EQUATIONS, STOCHASTIC ls fsohsi oso rde eeintroduced were bridges of flows stochastic of class A tcatceuto,srn ouin tcatcflw coa flow, stochastic solution, strong equation, Stochastic 2003 2012 , 1999 hsrpitdffr rmteoiia npagination in original the from differs reprint This . osuytecaecn rcse ihmul- with processes coalescent the study to ) 50 rmr 00,6J8 eodr 02,92D25. 60J25, secondary 60J68; 60G09, Primary seas aio ( Sagitov also [see ) 1982 20)147–181]. (2006) in h naso Probability of Annals The 1 ( Gall Le and Bertoin fact, In )]. δ 1 0 n h otasnSnta co- Bolthausen–Sznitman the and n eguLi Zenghu and rnhn rcs,immigration, process, branching e 41 1999 , 20)307–333]. (2005) ][e otasnand Bolthausen [see 1] ] h a fsuch of law The )]. dflw of flows nd eo them of ne dz ihmul- with mea- dom nsuper- on oncin they connection, 2 swt multiple with ts to of ation , i and oin n[0 on ) osof ions flows t nand in of e om- eisof series , ] The 1]. 2003 les- ) 2 D. A. DAWSON AND Z. LI papers [see Bertoin and Le Gall (2003, 2005, 2006)]. We refer the reader to Le Jan and Raimond (2004), Ma and Xiang (2001) and Xiang (2009) for the study of stochastic flows of mappings and measures in abstract settings. Let Bs,t : function∈ v} B (v) ≥ 7→ −t,0 induces a random probability measure ρt(dv) on [0, 1]. The process ρt : t 0 was characterized in Bertoin and Le Gall (2003) as the unique solution{ of≥ a} martingale problem. In fact, this process is a measure-valued dual to the Λ-coalescent process. It was also pointed out in Bertoin and Le Gall (2003) that ρt : t 0 can be regarded as a generalized Fleming–Viot process [see also Donnelly{ ≥ } and Kurtz (1999a, 1999b)]. Let Λ(dz) be a finite measure on [0, 1] such that Λ( 0 ) = 0, and let M(ds, dz, du) be a Poisson random measure on (0, ) {(0}, 1]2 with inten- sity{ z−2 ds Λ(dz}) du. It was proved in Bertoin and Le∞ Gall× (2005) that there is weak solution flow X (v) : t 0, v [0, 1] to the stochastic equation { t ≥ ∈ } t 1 1 (1.1) X (v)= v + z[1 X (v)]M(ds, dz, du). t {u≤Xs−(v)} − s− Z0 Z0 Z0 Moreover, Bertoin and Le Gall (2005) showed that for any 0 r1 < < r 1 the p-point motion (B (r ),...,B (r )) : t 0 is≤ equivalent· · · p ≤ { −t,0 1 −t,0 p ≥ } to (Xt(r1),...,Xt(rp)) : t 0 . Therefore, the solutions of (1.1) give a re- alization{ of the flow of bridges≥ } associated with the Λ-coalescent process. A separate treatment for the Kingman coalescent flow was also given in Bertoin and Le Gall (2005). In that case they showed the p-point motion (B (r ),...,B (r )) : t 0 is a diffusion process in { −t,0 1 −t,0 p ≥ } D := x = (x ,...,x ) Rp : 0 x x 1 p { 1 p ∈ ≤ 1 ≤···≤ p ≤ } with generator A0 defined by p 1 ∂2f (1.2) A0f(x)= xi∧j(1 xi∨j) (x). 2 − ∂xi ∂xj i,jX=1 Given a Λ-coalescent flow Bs,t : v , v [0, 1), s,t { ∈ s,t } ∈ −1 −1 and Bs,t (1) = Bs,t (1 ). In the Kingman coalescent case, it was proved in − −1 −1 Bertoin and Le Gall (2005) that the p-point motion (B0,t (r1),...,B0,t (rp)) : t 0 is a diffusion process in D with generator A{ given by ≥ } p 1 p 1 ∂f (1.3) A1f(x)= A0f(x)+ xi (x), 2 − ∂xi Xi=1 STOCHASTIC EQUATIONS, FLOWS AND PROCESSES 3 where A0 is given by (1.2). The analogous characterization for the Λ-coa- lescent flow with Λ( 0 ) = 0 was also provided in Bertoin and Le Gall (2005). Those results give deep{ } insights into the structures of the stochastic flows associated with the Λ-coalescents. The asymptotic properties of Λ-coalescent flows were studied in Bertoin and Le Gall (2006). For each integer k 1 let Λ (dx) be a finite measure on ≥ k [0, 1] with Λk( 0 ) = 0 and let Xk(t, v) : t 0, v [0, 1] be defined by (1.1) { } { ≥ ∈ } 2 from a Poisson random measure Mk(ds, dz, du) on (0, ) (0, 1] with −2 { −2 } 2 ∞−1 × intensity z ds Λk(dz) du. Suppose that z (z z )Λk(k dz) converges weakly as k to a finite measure on (0, ) denoted∧ by z−2(z z2)Λ(dz). By a limit theorem→∞ of Bertoin and Le Gall∞ (2006) the rescaled p∧-point mo- tion (kXk(kt, r1/k),...,kXk(kt, rp/k)) : t 0 converges in distribution to that{ of the weak solution flow of the stochastic≥ } equation t ∞ ∞ ˜ (1.4) Yt(v)= v + x1{u≤Ys−(v)}N(ds, dx, du), Z0 Z0 Z0 where N˜ (ds, dx, du) is a compensated Poisson random measure on [0, ) (0, )2 with intensity z−2 ds Λ(dz) du. It was pointed out in Bertoin∞ and× Le∞ Gall (2006) that the solution of (1.4) is a special critical continuous-state branching process (CB-process). In this paper we study two classes of stochastic flows defined by stochas- tic equations that generalize (1.1) and (1.4). We shall first treat the gen- eralization of (1.4) since it involves simpler structures. Suppose that σ 0 and b are constants, v γ(v) is a nonnegative and nondecreasing continu-≥ ous function on [0, ) and7→ (z z2)m(dz) is a finite measures on (0, ). Let W (ds, du) be a white∞ noise∧ on (0, )2 based on the Lebesgue measure∞ { } ∞ ds du. Let N(ds, dz, du) be a Poisson random measure on (0, )3 with intensity ds{ m(dz) du. Let} N˜(ds, dz, du) be the compensated measure∞ of N(ds, dz, du) . We shall see{ that for any} v 0 there is a pathwise unique {nonnegative solution} of the stochastic equation≥
t Ys−(v) t Yt(v)= v + σ W (ds, du)+ [γ(v) bYs−(v)] ds 0 0 0 − (1.5) Z Z Z t ∞ Ys−(v) + zN˜ (ds, dz, du). Z0 Z0 Z0 It is not hard to show each solution Y (v)= Yt(v) : t 0 is a continuous- state branching process with immigration (CBI-process).{ ≥ Then} it is natural to call the two-parameter process Yt(v) : t 0, v 0 a flow of CBI-processes. We prove that the flow has a version{ with≥ the≥ following} properties: (i) for each v 0, t Y (v) is a c`adl`ag process on [0, ) and solves (1.5); ≥ 7→ t ∞ (ii) for each t 0, v Yt(v) is a nonnegative and nondecreasing c`adl`ag process on [0, ).≥ 7→ ∞ 4 D. A. DAWSON AND Z. LI
The proof of those properties is based on the observation that Y (v) : v 0 is a path-valued process with independent increments. For any t{ 0, the≥ ran-} dom function v Y (v) induces a random Radon measure Y (dv≥) on [0, ). 7→ t t ∞ We shall see that Yt : t 0 is actually an immigration superprocess in the sense of Li (2011){ with trivial≥ } underlying spatial motion. One could replace t the diffusion term in (1.5) by the stochastic integral σ 0 Ys−(v) dW (s) using a one-dimensional Brownian motion W (t) : t 0 as in Dawson and Li (2006). The resulted equation defines an{ equivalent≥ CBI-process} R p for any fixed v 0, but it does not give an equivalent flow. To describe≥ our generalization of (1.1), let us assume that σ 0 and b 0 are constants, v γ(v) is a nondecreasing continuous function≥ on [0, 1] such≥ 7→ that 0 γ(v) 1 for all 0 v 1 and z2ν(dz) is a finite measure on (0, 1]. Let B≤(ds, du≤) be a white≤ noise≤ on (0, ) (0, 1] based on ds du, and let {M(ds, dz, du} ) be a Poisson random∞ measure× on (0, ) (0, 1]2 with intensity{ ds ν(dz)}du. We show that for any v [0, 1] there∞ × is a pathwise unique solution X(v)= X (v) : t 0 to the equation∈ { t ≥ } t 1 X (v)= v + σ [1 X (v)]B(ds, du) t {u≤Xs−(v)} − s− Z0 Z0 t (1.6) + b [γ(v) X (v)] ds − s− Z0 t 1 1 + z[1 X (v)]M(ds, dz, du). {u≤Xs−(v)} − s− Z0 Z0 Z0 Clearly, the above equation unifies and generalizes the flows described by (1.1), (1.2) and (1.3). Here it is essential to use the white noise as the dif- fusion driving force. We show there is a version of the random field Xt(v) : t 0, 0 v 1 with the following properties: { ≥ ≤ ≤ } (i) for each v [0, 1], t X (v) is c`adl`ag on [0, ) and solves (1.6); ∈ 7→ t ∞ (ii) for each t 0, v Xt(v) is nondecreasing and c`adl`ag on [0, 1] with X (0) 0 and X (1)≥ 1.7→ t ≥ t ≤ We refer to Xt(v) : t 0, 0 v 1 as a generalized Fleming–Viot flow following Bertoin{ and≥ Le Gall≤ (2003≤ ,}2005, 2006). In particular, our result gives the strong existence of the flows associated with the coalescents with multiple collisions. The study of this flow is more involved than the one defined by (1.5) as the path-valued process X(v) : 0 v 1 does not have independent increments. However, we shall{ see it is still≤ ≤ an }inhomogeneous Markov process. From the random field X (v) : t 0, 0 v 1 we can { t ≥ ≤ ≤ } define a c`adl`ag sub-probability-valued process Xt : t 0 on [0, 1], which is a counterpart of the generalized Fleming–Viot process{ ≥ of Be} rtoin and Le Gall (2003). We prove two scaling limit theorems for the generalized Fleming– Viot processes, which lead to a special form of the immigration superprocess STOCHASTIC EQUATIONS, FLOWS AND PROCESSES 5 defined from (1.5). From the theorems we derive easily a generalization of the limit theorem for the finite point motions in Bertoin and Le Gall (2006). The techniques of this paper are mainly based on the strong solutions of (1.5) and (1.6), which are different from those of Bertoin and Le Gall (2005, 2006). In Section 2 we give some general results for the pathwise uniqueness, comparison property and existence of nonnegative strong solu- tions of stochastic equations driven by white noises and Poisson random measures. Those extend the results in Fu and Li (2010) and provide the basis for the investigation of the strong solution flows of (1.5) and (1.6). They should also be of interest on their own right. In Section 3 we study the flows of CBI-processes and their associated immigration superprocesses. The generalized Fleming–Viot flows are discussed in Section 4. Finally, we prove the scaling limit theorems in Section 5.
Notation. For a measure µ and a function f on a measurable space (E, E ) write µ,f = E f dµ if the integral exists. For any a 0 let M[0, a] be the set ofh finitei measures on [0, a] endowed with the topology≥ of weak conver- R gence. Let M1[0, a] be the subspace of M[0, a] consisting of sub-probability measures. Let B[0, a] be the Banach space of bounded Borel functions on [0, a] endowed with the supremum norm , and let C[0, a] denote its sub- space of continuous functions. We use Bk[0 ·, k a]+ and C[0, a]+ to denote the subclasses of nonnegative elements. Throughout this paper, we make the conventions b ∞ = and = Za Z(a,b] Za Z(a,∞) for any b a 0. Given a function f defined on a subset of R, we write ≥ ≥ ∆ f(x)= f(x + z) f(x) and D f(x)=∆ f(x) f ′(x)z z − z z − for x, z R if the right-hand side is meaningful. Let λ denote the Lebesgue measure∈ on [0, ). ∞ 2. Strong solutions of stochastic equations. In this section, we prove some results on stochastic equations of one-dimensional processes driven by white noises and Poisson random measures. The results extend those of Fu and Li (2010). Since our aim is to apply the results to the generalized Fleming–Viot flows and the flows of CBI-processes, we only discuss equations of nonnegative processes. However, the arguments can be modified to deal with general one-dimensional equations. Let E, U0 and U1 be separable topological spaces whose topologies can be defined by complete metrics. Suppose that π(dz), µ0(du) and µ1(du) are σ-finite Borel measures on E, U0 and U1, respectively. We say the parame- ters (σ, b, g0, g1) are admissible if: x b(x) is a continuous function on R satisfying b(0) 0; • 7→ + ≥ 6 D. A. DAWSON AND Z. LI
(x, u) σ(x, u) is a Borel function on R+ E satisfying σ(0, u)=0 for • u E;7→ × ∈ (x, u) g0(x, u) is a Borel function on R+ U0 satisfying g0(0, u)=0 and • g (x, u7→)+ x 0 for x> 0 and u U ; × 0 ≥ ∈ 0 (x, u) g1(x, u) is a Borel function on R+ U1 satisfying g1(x, u)+ x 0 • for x 7→0 and u U . × ≥ ≥ ∈ 1 Let W (ds, du) be a white noise on (0, ) E with intensity ds π(dz). Let { } ∞ × N0(ds, du) and N1(ds, du) be Poisson random measures on (0, ) U0 {and (0, ) } U with{ intensities} ds µ (du) and ds µ (du), respectively.∞ × Sup- ∞ × 1 0 1 pose that W (ds, du) , N0(ds, du) and N1(ds, du) are defined on some complete probability{ space} { (Ω, F , P}) and are{ independent} of each other. Let N˜0(ds, du) denote the compensated measure of N0(ds, du) . A nonnega- tive{ c`adl`ag} process x(t) : t 0 is called a solution{ of } { ≥ } t x(t)= x(0) + σ(x(s ), u)W (ds, du) − Z0 ZE t t (2.1) + b(x(s )) ds + g (x(s ), u)N˜ (ds, du) − 0 − 0 Z0 Z0 ZU0 t + g (x(s ), u)N (ds, du), 1 − 1 Z0 ZU1 if it satisfies the stochastic equation almost surely for every t 0. We say x(t) : t 0 is a strong solution if, in addition, it is adapted≥ to the augmented{ natural≥ } filtration generated by W (ds, du) , N (ds, du) and { } { 0 } N1(ds, du) [see, e.g., Situ (2005), page 76]. Since x(s ) = x(s) for at most {countably many} s 0, we can also use x(s) instead of x−(s6 ) in the integrals with respect to W≥(ds, du) and ds on the right-hand side− of (2.1). For the convenience of the statements of the results, we write b(x)= b (x) b (x), 1 − 2 where x b1(x) is continuous, and x b2(x) is continuous and nondecreas- ing. Let7→ us formulate the following conditions:7→ (2.a) there is a constant K 0 so that ≥ b(x)+ g (x, u) µ (du) K(1 + x) | 1 | 1 ≤ ZU1 for every x 0; ≥ (2.b) there is a nondecreasing function x L(x) on R+ and a Borel R 7→ function (x, u) g¯0(x, u) on + U0 so that sup0≤y≤x g0(y, u) g¯0(x, u) and 7→ × | |≤
σ(x, u)2π(du)+ [¯g (x, u) g¯ (x, u)2]µ (du) L(x) 0 ∧ 0 0 ≤ ZE ZU0 for every x 0; ≥ STOCHASTIC EQUATIONS, FLOWS AND PROCESSES 7
(2.c) for each m 1 there is a nondecreasing concave function z rm(z) on R such that ≥r (z)−1 dz = and 7→ + 0+ m ∞ b (x) bR(y) + g (x, u) g (y, u) µ (du) r ( x y ) | 1 − 1 | | 1 − 1 | 1 ≤ m | − | ZU1 for every 0 x,y m; (2.d) for≤ each m≤ 1 there is a nonnegative nondecreasing function z ρ (z) on R so that≥ ρ (z)−2 dz = , 7→ m + 0+ m ∞ σR(x, u) σ(y, u) 2π(du) ρ ( x y )2 | − | ≤ m | − | ZE and 1 2 l0(x, y, u) (1 t)1{|l (x,y,u)|≤n} µ (du) − 0 dt c(m,n) 0 ρ ( (x y)+ tl (x, y, u) )2 ≤ ZU0 Z0 m | − 0 | for every n 1 and 0 x,y m, where l0(x, y, u)= g0(x, u) g0(y, u) and c(m,n) 0≥ is a constant.≤ ≤ − ≥
Theorem 2.1. Suppose that (σ, b, g0, g1) are admissible parameters sat- isfying conditions (2.a)–(2.d). Then the pathwise uniqueness of solutions holds for (2.1).
Proof. We first fix the integer m 1. Let a0 = 1 and choose ak 0 decreasingly so that ak−1 ρ (z)−2 dz = k≥for k 1. Let x ψ (x) bea non-→ ak m k R ≥ 7→ negative continuous functionR on which has support in (ak, ak−1) and satis- fies ak−1 ψ (x) dx = 1 and 0 ψ (x) 2k−1ρ (x)−2 for a < x < a . For ak k k m k k−1 each k 1 we define the nonnegative≤ ≤ and twice continuously differentiable functionR ≥ |z| y (2.2) φ (z)= dy ψ (x) dx, z R. k k ∈ Z0 Z0 ′ It is easy to see that φk(z) z nondecreasingly as k and 0 φk(z) 1 ′ → | | →∞ ≤ ≤ for z 0 and 1 φk(z) 0 for z 0. By condition (2.d) and the choice of x ψ≥(x), − ≤ ≤ ≤ 7→ k ′′ 2 φk(x y) σ(x, u) σ(y, u) π(du) − E | − | (2.3) Z 2 ψ ( x y )ρ ( x y )2 ≤ k | − | m | − | ≤ k for 0 x,y m. Then the left-hand side tends to zero uniformly in 0 x,y ≤m as k≤ . For h, ζ R, by Taylor’s expansion we have ≤ ≤ →∞ ∈ 1 1 D φ (ζ)= h2φ′′(ζ + th)(1 t) dt = h2ψ ( ζ + th )(1 t) dt. h k k − k | | − Z0 Z0 8 D. A. DAWSON AND Z. LI
It follows that 2 1 (2.4) D φ (ζ) h2ρ ( ζ + th )−2(1 t) dt. h k ≤ k m | | − Z0 Observe also that (2.5) D φ (ζ)=∆ φ (ζ) φ′ (ζ)h 2 h . h k h k − k ≤ | | For 0 x,y m and n 1 we can use (2.4) and (2.5) to get ≤ ≤ ≥ D φ (x y)µ (du) l0(x,y,u) k − 0 ZU0 1 2 2 l0(x, y, u) (1 t)1 − {|l0(x,y,u)|≤n} µ0(du) 2 dt ≤ k U0 0 ρm( (x y)+ tl0(x, y, u) ) (2.6) Z Z | − | + 2 l (x, y, u) 1 µ (du) | 0 | {|l0(x,y,u)|>n} 0 ZU0 2 c(m,n) + 4 g¯ (m, u)1 µ (du). ≤ k 0 {g¯0(m,u)>n/2} 0 ZU0 By conditions (2.b), (2.d) one sees the right-hand side tends to zero uni- formly in 0 x,y m as k . Then the pathwise uniqueness for (2.1) follows by a≤ trivial≤ modification→∞ of Theorem 3.1 in Fu and Li (2010).
The key difference between the above theorem and Theorems 3.2 and 3.3 of Fu and Li (2010) is that here we do not assume x g0(x, u) is nondecreas- ing. This is essential for the applications to stochastic7→ equations like (1.6).
′ ′ ′′ ′′ Theorem 2.2. Let (σ, b , g0, g1) and (σ, b , g0, g1 ) be two sets of admis- sible parameters satisfying conditions (2.a)–(2.d). In addition, assume that: ′ ′′ (i) for every u U1, x x + g1(x, u) or x x + g1 (x, u) is nondecreas- ing; ∈ 7→ 7→ (ii) b′(x) b′′(x) and g′ (x, u) g′′(x, u) for every x 0 and u U . ≤ 1 ≤ 1 ≥ ∈ 1 ′ ′ ′ Suppose that x (t) : t 0 is a solution of (2.1) with (b, g1) = (b , g1), and ′′ { ≥ } ′′ ′′ ′ x (t) : t 0 is a solution of the equation with (b, g1) = (b , g1 ). If x (0) x{′′(0), then≥ P} x′(t) x′′(t) for all t 0 = 1. ≤ { ≤ ≥ } ′ ′′ Proof. Let ζ(t)= x (t) x (t) for t 0. Let x ψk(x) be defined as in the proof of Theorem 2.1.− Instead of (2.2≥), for each7→k 1 we now define ≥ z y (2.7) φ (z)= dy ψ (x) dx, z R. k k ∈ Z0 Z0 Then φ (z) z+ := 0 z nondecreasingly as k . Let k → ∨ →∞ l (t, u)= g (x′(t), u) g (x′′(t), u), t 0, u U , 0 0 − 0 ≥ ∈ 0 STOCHASTIC EQUATIONS, FLOWS AND PROCESSES 9 and l (t, u)= g′ (x′(t), u) g′′(x′′(t), u), t 0, u U . 1 1 − 1 ≥ ∈ 1 ′ For ζ(s ) 0 we have φk(ζ(s )) = φk(ζ(s )) = 0. Since x x + f(x, u) − ≤ ′ −′′ − ′ ′′ 7→ is nondecreasing for f = g1 or g1 , for ζ(s )= x (s ) x (s ) 0 we also have − − − − ≤ ζ(s )+ l (s , u)= x′(s ) x′′(s )+ g′ (x′(s ), u) g′′(x′′(s ), u) − 1 − − − − 1 − − 1 − x′(s ) x′′(s )+ f(x′(s ), u) f(x′′(s ), u) 0. ≤ − − − − − − ≤ The latter implies ∆ φ (ζ(s )) = φ (ζ(s )+ l (s , u)) φ (ζ(s )) = 0. l1(s−,u) k − k − 1 − − k − By Itˆo’s formula we have 1 t φ (ζ(t)) = φ (ζ(0)) + ds φ′′(ζ(s ))[σ(x′(s ), u) k k 2 k − − Z0 ZE σ(x′′(s ), u)]2π(du) − − t ′ ′ ′ + φk(ζ(s ))[b (x (s )) 0 − − (2.8) Z b′′(x′′(s ))]1 ds − − {ζ(s−)>0} t + ds ∆ φ (ζ(s ))1 µ (du) l1(s−,u) k − {ζ(s−)>0} 1 Z0 ZU1 t + ds D φ (ζ(s ))µ (du)+ M (t), l0(s−,u) k − 0 m Z0 ZU0 where t M (t)= φ′ (ζ(s ))[σ(x′(s ), u) m k − − Z0 ZE σ(x′′(s ), u)]W (ds, du) − − t + ∆ φ (ζ(s ))N˜ (ds, du) l1(s−,u) k − 1 Z0 ZU1 t + ∆ φ (ζ(s ))N˜ (ds, du). l0(s−,u) k − 0 Z0 ZU0 ′ ′′ Let τm = inf t 0 : x (t) m or x (t) m for m 1. Under conditions (2.b), { ≥ ≥ ≥ } ≥ ′ (2.c) it is easy to show that Mm(t τm) is a martingale. Recall that b (x) b′′(x) and b′(x)= b′ (x) b′ ({x) for a∧ nondecreasing} function x b′ (x). Then≤ 1 − 2 7→ 2 10 D. A. DAWSON AND Z. LI under the restriction ζ(s ) > 0 we have − φ′ (ζ(s ))[b′(x′(s )) b′′(x′′(s ))] k − − − − φ′ (ζ(s ))[b′(x′(s )) b′(x′′(s ))] ≤ k − − − − φ′ (ζ(s ))[b′ (x′(s )) b′ (x′′(s ))] ≤ k − 1 − − 1 − b′ (x′(s )) b′ (x′′(s )) ≤ | 1 − − 1 − | and ∆ φ (ζ(s )) l1(s−,u) k − = φ (ζ(s )+ g′ (x′(s ), u) g′′(x′′(s ), u)) φ (ζ(s )) k − 1 − − 1 − − k − φ (ζ(s )+ g′ (x′(s ), u) g′ (x′′(s ), u)) φ (ζ(s )) ≤ k − 1 − − 1 − − k − g′ (x′(s ), u) g′ (x′′(s ), u) . ≤ | 1 − − 1 − | The estimates (2.3) and (2.6) are still valid. If x′(0) x′′(0), we can take the expectation in (2.8) and let k to get ≤ →∞ t∧τm E[ζ(t τ )+] E r ( ζ(s ) )1 ds ∧ m ≤ m | − | {ζ(s−)>0} Z0 t r (E[ζ(s τ )+]) ds, ≤ m ∧ m Z0 where the second inequality holds by the concaveness of z rm(z). Then + 7→ E[ζ(t τm) ] = 0 for all t 0. Since τm as m , we get the desired comparison∧ property. ≥ →∞ →∞
We say the comparison property of solutions holds for (2.1) if for any two solutions x1(t) : t 0 and x2(t) : t 0 satisfying x1(0) x2(0) we have P x (t) {x (t) for≥ all}t 0 {= 1. From≥ Theorem} 2.2 we get≤ the following: { 1 ≤ 2 ≥ }
Theorem 2.3. Let (σ, b, g0, g1) be admissible parameters satisfying con- ditions (2.a)–(2.d). In addition, assume that for every u U the function ∈ 1 x x + g1(x, u) is nondecreasing. Then the comparison property holds for the7→ solutions of (2.1).
The monotonicity assumption on the function x x + g1(x, u) in The- orem 2.3 is natural. To see this, suppose that x (t)7→ and x (t) are two { 1 } { 2 } solutions of (2.1) and (si, ui) : i 1 is the set of atoms of N1(ds, du) . The assumption guarantees{ that x≥ (s} ) x (s ) implies { } 1 i− ≤ 2 i− x (s )= x (s )+ g (x (s ), u ) 1 i 1 i− 1 1 i− i x (s )+ g (x (s ), u )= x (s ). ≤ 2 i− 1 2 i− i 2 i STOCHASTIC EQUATIONS, FLOWS AND PROCESSES 11
A similar explanation can be given to Theorem 2.2. In some applications the kernel x g0(x, u) may be nondecreasing. When this is true, we can replace (2.d)7→ by the following simpler condition: (2.e) For each u U the function x g (x, u) is nondecreasing, and for ∈ 0 7→ 0 each m 1 there is a nonnegative and nondecreasing function z ρm(z) on R so≥ that ρ (z)−2 dz = and 7→ + 0+ m ∞ σ(x, uR) σ(y, u) 2π(du)+ l (x, y, u) l (x, y, u) 2µ (du) | − | | 0 | ∧ | 0 | 0 ZE ZU0 ρ ( x y )2 ≤ m | − | for all 0 x,y m, where l (x, y, u)= g (x, u) g (y, u). ≤ ≤ 0 0 − 0
Proposition 2.4. Let (σ, b, g0, g1) be admissible parameters. If (2.e) holds, then (2.d) holds.
Proof. Since x g (x, u) is nondecreasing, it is not hard to see (x 7→ 0 | − y)+ tl0(x, y, u) x y . By condition (2.e) and the monotonicity of z ρ(z) we have | ≥ | − | 7→ 1 2 (1 t)l0(x, y, u) 1{|l (x,y,u)|≤n} dt − 0 µ (du) ρ ( (x y)+ tl (x, y, u) )2 0 Z0 ZU0 m | − 0 | 1 [ l (x, y, u) l (x, y, u)2] n dt | 0 |∧ 0 µ (du) n. ≤ ρ ( x y )2 0 ≤ Z0 ZU0 m | − | Then condition (2.d) is satisfied.
Theorem 2.5. Suppose that (σ, b, g0, g1) are admissible parameters sat- isfying conditions (2.a), (2.c), (2.e). Then there is a unique strong solution to (2.1).
Proof. We first note that (2.b) follows from (2.e). By Proposition 2.4, we also have (2.d) from (2.e). Let Vn be a nondecreasing sequence of ∞ { } Borel subsets of U0 so that n=1 Vn = U0 and µ0(Vn) < for every n 1. For m,n 1 one can use (2.e) to see ∞ ≥ ≥ S x β (x) := [g (x, u) g (x, u) m]µ (du) 7→ m 0 − 0 ∧ 0 ZU0 and x γ (x) := [g (x, u) m]µ (du) 7→ m,n 0 ∧ 0 ZVn are continuous nondecreasing functions. By the results for continuous-type stochastic equations as in Ikeda and Watanabe [(1989), page 169] one can 12 D. A. DAWSON AND Z. LI show there is a nonnegative weak solution to t x(t)= x(0) + σ(x(s) m, u)W (ds, du) ∧ Z0 ZE t t + b (x(s) m) ds γ (x(s) m) ds, m ∧ − m,n ∧ Z0 Z0 where b (x)= b(x) β (x). The pathwise uniqueness holds for the above m − m equation by Theorem 2.1. Then it has a unique strong solution. Let Wn be ∞ { } a nondecreasing sequence of Borel subsets of U1 so that n=1 Wn = U1 and µ1(Wn) < for every n 1. Following the proof of Proposition 2.2 of Fu and Li (2010) one∞ can show there≥ is a unique strong solutionSx (t) : t 0 to { m,n ≥ } t x(t)= x(0) + σ(x(s ) m, u)W (ds, du) − ∧ Z0 ZE t t + b (x(s ) m) ds γ (x(s) m) ds m − ∧ − m,n ∧ Z0 Z0 t + [g (x(s ) m, u) m]N (ds, du) 0 − ∧ ∧ 0 Z0 ZVn t + [g (x(s ) m, u) m]N (ds, du). 1 − ∧ ∧ 1 Z0 ZWn We can rewrite the above equation into t x(t)= x(0) + σ(x(s ) m, u)W (ds, du) − ∧ Z0 ZE t + b (x(s ) m) ds m − ∧ Z0 t + [g (x(s ) m, u) m]N˜ (ds, du) 0 − ∧ ∧ 0 Z0 ZVn t + [g (x(s ) m, u) m]N (ds, du). 1 − ∧ ∧ 1 Z0 ZWn As in the proof of Lemma 4.3 of Fu and Li (2010) one can see the sequence xm,n(t) : t 0 , n = 1, 2,..., is tight in D([0, ), R+). Following the proof of{ Theorem≥ 4.4} of Fu and Li (2010) it is easy∞ to show that any weak limit point x (t) : t 0 of the sequence is a nonnegative weak solution to { m ≥ } t x(t)= x(0) + σ(x(s ) m, u)W (ds, du) − ∧ Z0 ZE t + b (x(s ) m) ds m − ∧ Z0 STOCHASTIC EQUATIONS, FLOWS AND PROCESSES 13
(2.9) t + [g (x(s ) m, u) m]N˜ (ds, du) 0 − ∧ ∧ 0 Z0 ZU0 t + [g (x(s ) m, u) m]N (ds, du). 1 − ∧ ∧ 1 Z0 ZU1 By Theorem 2.1 the pathwise uniqueness holds for (2.9), so the equation has a unique strong solution [see, e.g., Situ (2005), page 104]. Then the result follows by a simple modification of the proof of Proposition 2.4 of Fu and Li (2010). See El Karoui and M´el´eard (1990) and Kurtz (2007, 2010) for the general theory of stochastic equations driven by white noises and Poisson random measures.
3. Stochastic flows of CBI-processes. In this section, we give the con- structions and characterizations of the flow of CBI-processes and the asso- ciated immigration superprocess. Suppose that σ 0 and b are constants, and (u u2)m(du) is a finite measure on (0, ). Let≥ φ be a function given by ∧ ∞ 1 ∞ (3.1) φ(z)= bz + σ2z2 + (e−zu 1+ zu)m(du), z 0. 2 − ≥ Z0 A Markov process with state space R := [0, ) is called a CB-process with + ∞ branching mechanism φ if it has transition semigroup (pt)t≥0 given by
−λy −xvt(λ) (3.2) e pt(x,dy)= e , λ 0, R ≥ Z + where (t, λ) v (λ) is the unique locally bounded nonnegative solution of 7→ t d v (λ)= φ(v (λ)), v (λ)= λ, t 0. dt t − t 0 ≥ Given any β 0 we can also define a transition semigroup (q ) on R by ≥ t t≥0 + t −λy (3.3) e qt(x,dy)=exp xvt(λ) βvs(λ) ds . R − − Z + Z0 A nonnegative real-valued Markov process with transition semigroup (qt)t≥0 is called a CBI-process with branching mechanism φ and immigration rate β. It is easy to see that both (pt)t≥0 and (qt)t≥0 are Feller semigroups. See, for example, Kawazu and Watanabe (1971) and Li (2011), Chapter 3. Let W (ds, du) be a white noise on (0, )2 based on the Lebesgue mea- sure ds{ du, and let} N(ds, dz, du) be Poisson∞ random measure on (0, )3 with intensity ds m({dz) du. Let N˜}(ds, dz, du) be the compensated measure∞ of N(ds, dz, du) . { } { } 14 D. A. DAWSON AND Z. LI
Theorem 3.1. There is a unique nonnegative strong solution of the stochastic equation
t Ys− t Y = Y + σ W (ds, du)+ (β bY ) ds t 0 − s− Z0 Z0 Z0 t ∞ Ys− + zN˜(ds, dz, du). Z0 Z0 Z0 Moreover, the solution Yt : t 0 is a CBI-process with branching mecha- nism φ and immigration{ rate ≥β. }
Proof. The existence and uniqueness of the strong solution follows by an application of Theorem 2.5 [see also Dawson and Li (2006)]. Using Itˆo’s formula one can see that Yt(v) : t 0 solves the martingale problem asso- ciated with the generator{L defined≥ by} 1 ∞ (3.4) Lf(x)= σ2xf ′′(x) + (β bx)f ′(x)+ x D f(x)m(dz). 2 − z Z0 Then it is a CBI-process with branching mechanism φ and immigration rate β [see Kawazu and Watanabe (1971) and Li (2011), Section 9.5].
Let v γ(v) be a nonnegative and nondecreasing continuous function on [0, ). We7→ denote by γ(dv) the Radon measure on [0, ) so that γ([0, v]) = γ(v∞) for v 0. By Theorem 3.1 for each v 0 there∞ is a pathwise unique nonnegative≥ solution Y (v)= Y (v) : t 0 to≥ the stochastic equation { t ≥ } t Ys−(v) t Yt(v)= v + σ W (ds, du)+ [γ(v) bYs−(v)] ds 0 0 0 − (3.5) Z Z Z t ∞ Ys−(v) + zN˜ (ds, dz, du). Z0 Z0 Z0
Theorem 3.2. For any v v 0 we have P Y (v ) Y (v ) for all 2 ≥ 1 ≥ { t 2 ≥ t 1 t 0 = 1 and Yt(v2) Yt(v1) : t 0 is a CBI-process with branching mech- anism≥ } φ and immigration{ − rate β≥:= γ}(v ) γ(v ) 0. 2 − 1 ≥ Proof. The comparison property follows by applying Theorem 2.2 and Proposition 2.4 to (3.5). Let Z = Y (v ) Y (v ) for t 0. From (3.5) we have t t 2 − t 1 ≥ t Ys−(v2) t Z = v v + σ W (ds, du)+ (β bZ ) ds t 2 − 1 − s− Z0 ZYs−(v1) Z0 t ∞ Ys−(v2) + zN˜ (ds, dz, du) Z0 Z0 ZYs−(v1) STOCHASTIC EQUATIONS, FLOWS AND PROCESSES 15
(3.6) t Zs− t = v v + σ W (ds, du)+ (β bZ ) ds 2 − 1 1 − s− Z0 Z0 Z0 t ∞ Zs− + zN˜1(ds, dz, du), Z0 Z0 Z0 where
W1(ds, du)= W (ds,Ys−(v1)+ du) is a white noise with intensity ds du, and
N1(ds, dz, du)= N(ds,dz,Ys−(v1)+ du) is a Poisson random measure with intensity ds m(dz) du. That shows Zt : t 0 is a weak solution of (3.5). Then it a CBI-process with branching{ mech-≥ anism} φ and immigration rate β.
Theorem 3.3. Let v v u u 0. Then Y (u ) Y (u ) : t 0 2 ≥ 1 ≥ 2 ≥ 1 ≥ { t 2 − t 1 ≥ } and Yt(v2) Yt(v1) : t 0 are independent CBI-processes with immigration rates{α := γ−(u ) γ(u≥) and} β := γ(v ) γ(v ), respectively. 2 − 1 2 − 1
Proof. Let Lα and Lβ denote the generators of the CBI-processes with immigration rates α and β, respectively. Let Xt = Yt(u2) Yt(u1) and Zt = Y (v ) Y (v ). For any G C2(R2 ) one can use Itˆo’s formula− to show t 2 − t 1 ∈ + t G(Xt,Zt)= G(X0,Z0)+ LαG(Xs,Zs) ds 0 (3.7) Z t + LβG(Xs,Zs) ds + local mart., Z0 where Lα and Lβ act on the first and second coordinates of G, respectively. Then Xt : t 0 and Zt : t 0 are independent CBI-processes with immi- gration{ rates≥α }and β,{ respectively.≥ }
Proposition 3.4. There is a locally bounded nonnegative function t C(t) on [0, ) so that 7→ ∞ E sup [Ys(v2) Ys(v1)] C(t) (v2 v1) + [γ(v2) γ(v1)] 0≤s≤t − ≤ { − − (3.8) n o + √v v + γ(v ) γ(v ) 2 − 1 2 − 1 } for t 0 and v v 0. ≥ 2 ≥ 1 ≥ p Proof. Let Zt = Yt(v2) Yt(v1) for t 0. Taking the expectation in (3.6) we have − ≥ t E(Z ) = (v v )+ t[γ(v ) γ(v )] b E(Z ) ds. t 2 − 1 2 − 1 − s Z0 16 D. A. DAWSON AND Z. LI
Solving the above integral equation gives (3.9) E(Z ) = (v v )e−bt + [γ(v ) γ(v )]b−1(1 e−bt) t 2 − 1 2 − 1 − with b−1(1 e−bt)= t for b = 0 by convention. By (3.6) and Doob’s martin- gale inequality,−
t Ys−(v2) 2 1/2 E sup Zs (v2 v1) + 2σE W (ds, du) 0≤s≤t ≤ − 0 Ys−(v ) n o Z Z 1 t + [γ(v ) γ(v )] + b E(Z ) ds { 2 − 1 | | s } Z0 t 1 Ys−(v2) 2 + 2E1/2 zN˜(ds, dz, du) Z0 Z0 ZYs−(v1) t ∞ Ys−(v2) + E zN(ds, dz, du) Z0 Z1 ZYs−(v1) t 1/2 (v v )+ t[γ(v ) γ(v )] + 2σ E(Z ) ds ≤ 2 − 1 2 − 1 s Z0 1 1/2 t 1/2 2 + 2 z ν(dz) E(Zs) ds Z0 Z0 ∞ t + b + zν(dz) E(Z ) ds. | | s Z1 Z0 Then (3.8) follows by (3.9).
Suppose that (E, ρ) is a complete metric space. Let F be a subset of [0, ) such that 0 F and let t x(t) be a path from F to E. For any ε> 0∞ the number of ε∈-oscillations of7→ this path on F is defined as µ(ε) := sup n 0:there are 0= t
Lemma 3.5. Suppose that (Ω, G , Gt, P) is a filtered probability space and X : t 0 is a (G )-Markov process with state space (E, E ) and transition { t ≥ } t semigroup (Ps,t)t≥s. Suppose that ρ is a complete metric on E so that: (i) for ε> 0 and 0 s,t u we have ω Ω : ρ(X (ω), X (ω)) < ε G ; ≤ ≤ { ∈ s t }∈ u STOCHASTIC EQUATIONS, FLOWS AND PROCESSES 17
(ii) for ε> 0 and x E we have U (x) := y E : ρ(x,y) < ε E and ∈ ε { ∈ }∈ c (3.10) αε(h):= sup sup Ps,t(x, Uε(x) ) 0 (h 0). 0≤t−s≤h x∈E → → Then X : t 0 has a ρ-c`adl`ag modification. { t ≥ }
Proof. Let F = 0, r1, r2,... be a countable dense subset of [0, ) and let F = 0, r , . . .{ , r . For ε>} 0 and a> 0 let νa(ε) and νa(ε) denote,∞ n { 1 n} n respectively, the numbers of ε-oscillations of t Xt on F [0, a] and Fn a a 7→ ε ∩ ∩ [0, a]. Then νn(ε) ν (ε) increasingly as n . Let τn(0) = 0 and for k 0 define → →∞ ≥ ε ε τ (k +1)=min t F (τ (k), ) : ρ(X ε , X ) ε , n { ∈ n ∩ n ∞ τn(k) t ≥ } ε ε ε if τn(k) < and τn(k +1)= if τn(k)= . Since Fn is discrete, for any a 0 we have∞ ∞ ∞ ≥ τ ε(k + 1) a = ( τ ε(k)= s ρ(X , X ) ε ). { n ≤ } { n } ∩ { s t ≥ } s
Theorem 3.6. The path-valued process Y (v) : v 0 has a ρ-c`adl`ag mo- { ≥ } dification. Consequently, there is a version of the solution flow Yt(v) : t 0, v 0 of (3.5) with the following properties: { ≥ ≥ } (i) for each v 0, t Y (v) is a c`adl`ag process on [0, ) and solves (3.5); ≥ 7→ t ∞ (ii) for each t 0, v Yt(v) is a nonnegative and nondecreasing c`adl`ag process on [0, ).≥ 7→ ∞ Proof. Step 1. For any T 0 let D[0, T ] be the space of nonnegative c`adl`ag functions on [0, T ], and≥ let B(D[0, T ]) be its σ-algebra generated T by the Skorokhod topology. For v 0 let Y (v)= Yt(v) : 0 t T . The- orem 3.3 implies that Y T (v) : v ≥0 is a process{ in (D[0, T≤], B≤(D}[0, T ])) with independent increments.{ ≥ } Step 2. Let F = T, r , r ,... be a countable dense subset of [0, T ]. We T { 1 2 } consider the metric ρT on D[0, T ] defined by
ρT (ξ,ζ)= sup ξ(s) ζ(s) = sup ξ(s) ζ(s) . 0≤s≤T | − | r∈FT | − | For any ε> 0 and ξ D[0, T ] we have ∈ U¯ (ξ):= ζ D[0, T ] : ρ (ξ,ζ) ε ε { ∈ T ≤ } = ζ D[0, T ] : ξ ζ ε . { ∈ | r − r|≤ } r∈\FT Then the above set belongs to B(D[0, T ]) [see, e.g., Ethier and Kurtz (1986), page 127]. It follows that ∞ U (ξ) := ζ D[0, T ] : ρ (ξ,ζ) < ε = U¯ (ξ) ε { ∈ T } ε−1/n n=1 [ also belongs to B(D[0, T ]). F T T Step 3. Let ( v )v≥0 be the natural filtration of Y (v) : v 0 . For any ε> 0 and 0 s,t v we have { ≥ } ≤ ≤ T T ρT (Y (s),Y (t))= sup Yr(s) Yr(t) . r∈FT | − | Then one can show ω Ω : ρ (Y T (ω,s),Y T (ω,t)) < ε F T . { ∈ T }∈ v STOCHASTIC EQUATIONS, FLOWS AND PROCESSES 19
T T Step 4. Let (Pu,v)v≥u denote the transition semigroup of Y (v) : v 0 . By Proposition 3.4 for ε> 0 and ξ D[0, ) we have { ≥ } ∈ ∞ c Pu,v(ξ, Uε(ξ) )= P sup [Ys(v) Ys(u)] ε 0≤s≤T − ≥ n o −1 ε E sup [Ys(v) Ys(u)] ≤ 0≤s≤T − n o ε−1C(t) (v u) + [γ(v) γ(u)] ≤ { − − + √v u + γ(v) γ(u) . − − } Since v γ(v) is uniformly continuous on each boundedp interval, Lemma 3.5 7→ T implies that Y (v) : v 0 has a ρT -c`adl`ag modification. That implies the existence of a{ ρ-c`adl`ag≥ modification} of Y (v) : v 0 . { ≥ }
In the situation of Theorem 3.6 we call the solution Yt(v) : t 0, v 0 of (3.5) a flow of CBI-processes. Let F [0, ) be the set{ of nonnegative≥ ≥ and} nondecreasing c`adl`ag functions on [0, ). Given∞ a finite stopping time τ and a function µ F [0, ) let Y µ (v) : t ∞0 be the solution of ∈ ∞ { τ,t ≥ } τ+t Y µ (v) µ τ,s− Yτ,t(v)= µ(v)+ σ W (ds, du) Zτ Z0 τ+t µ (3.12) + [γ(v) bYτ,s−(v)] ds τ − Z µ τ+t ∞ Yτ,s−(v) + zN˜ (ds, dz, du) Zτ Z0 Z0 µ µ and write simply Yt (v) : t 0 instead of Y0,t(v) : t 0 . The pathwise uniqueness for the{ above equation≥ } follows from{ that of≥ (3.5}) since W (τ + ds, du) is a white noise based on ds dz, and N(τ + ds, dz, du) is a{ Poisson } { } random measure with intensity ds m(dz) du. Let Gτ,t be the random operator on F [0, ) that maps µ to Y µ . ∞ τ,t P µ µ Theorem 3.7. For any finite stopping time τ we have Yτ+t = Gτ,tYτ for all t 0 = 1. { ≥ }
Proof. By the sample path regularity of (t, v) Yt(v) we only need P µ µ 7→ to show Yτ+t(v)= Gτ,tYτ (v) = 1 for every t 0 and v 0. By (3.5) we have { } ≥ ≥ µ τ+t Ys−(v) µ µ Yτ+t(v)= Yτ (v)+ σ W (ds, du) Zτ Z0 τ+t + [γ(v) bY µ (v)] ds − s− Zτ 20 D. A. DAWSON AND Z. LI
µ τ+t ∞ Ys−(v) + zN˜ (ds, dz, du). Zτ Z0 Z0 By the pathwise uniqueness for (3.12) we get the desired result.
For any Radon measure µ(dv) on [0, ) with distribution function v µ(v), µ ∞ 7→µ the random function v Yt (v) induces a random Radon measure Yt (dv) µ 7→ µ on [0, ) so that Yt ([0, v]) = Yt (v) for v 0. We shall give some character- ∞ ≥µ izations of the measure-valued process Yt : t 0 . For simplicity, we fix a constant a 0{ and consider≥ } the restrictions of µ(dv), µ ≥ γ(dv) and Yt : t 0 to [0, a] without changing the notation. Let us con- sider the step{ function≥ } n (3.13) f(x)= c 1 (x)+ c 1 (x), x [0, a], 0 {0} i (ai−1,ai] ∈ Xi=1 where c0, c1,...,cn R and 0= a0 < a1 < < an = a is a partition of [0, a]. For{ this function}⊂ we have{ · · · } n (3.14) Y µ,f = c Y µ(0) + c [Y µ(a ) Y µ(a )]. h t i 0 t i t i − t i−1 Xi=1 From (3.12) and (3.14) it is simple to see t ∞ Y µ,f = µ,f + σ gµ (u)W (ds, du) h t i h i s− Z0 Z0 t (3.15) + [ γ,f b Y µ ,f ] ds h i− h s− i Z0 t ∞ ∞ µ ˜ + zgs−(u)N(ds, dz, du), Z0 Z0 Z0 where n µ µ µ µ (3.16) gs (u)= c01{u≤Ys (0)} + ci1{Ys (ai−1) Proposition 3.8. For any t 0 and f B[0, a] we have ≥ ∈ (3.17) E[ Y µ,f ]= µ,f e−bt + γ,f b−1(1 e−bt) h t i h i h i − with b−1(1 e−bt)= t for b = 0 by convention. − Proof. We first consider the step function (3.13). By taking the expec- tation in (3.15) we obtain t E[ Y µ,f ]= µ,f + t γ,f b E[ Y µ,f ] ds. h t i h i h i− h s i Z0 STOCHASTIC EQUATIONS, FLOWS AND PROCESSES 21 The above integral equation has the unique solution given by (3.17). For a general function f B[0, a] we get (3.17) by a monotone class argument. ∈ µ Theorem 3.9. The measure-valued process Yt : t 0 is a c`adl`ag strong µ { ≥ } Markov process in M[0, a] with Y0 = µ. µ Proof. In view of (3.14), the process t Yt ,f is c`adl`ag for the step function (3.13). Since any function in C[07→, a] h can bei approximated by a sequence of step functions in the supremum norm, it is easy to conclude µ t Yt ,f is c`adl`ag for all f C[0, a]. By Theorem 3.7, for any finite stop- 7→ h i µ ∈ µ ping time τ we have Yτ+t = Gτ,tYτ almost surely. That clearly implies the strong Markov property of Y µ : t 0 . { t ≥ } µ Theorem 3.10. For any f B[0, a] the process Yt ,f : t 0 has a c`adl`ag modification. Moreover, there∈ is a locally bounde{hd functioni ≥t }C(t) so that 7→ E µ 2 1/2 2 1/2 (3.18) sup Ys ,f C(t)[ µ,f + γ,f + µ,f + γ,f ] 0≤s≤th i ≤ h i h i h i h i h i for every t 0 and f B[0, a]+. ≥ ∈ Proof. We first consider a nonnegative step function given by (3.13) with constants c0, c1,...,cn R+. By (3.15) and Doob’s martingale in- equality, { }⊂ E µ sup Ys ,f 0≤s≤th i h i t ∞ 2 µ,f + 2σE1/2 gµ (u)W (ds, du) ≤ h i s− Z0 Z0 t + t γ,f + b E[ Y µ,f ] ds h i | | h s i Z0 t 1 ∞ 2 E1/2 µ ˜ + 2 zgs−(u)N(ds, dz, du) Z0 Z0 Z0 t ∞ ∞ E µ + zgs−(u)N(ds, dz, du) Z0 Z1 Z0 t ∞ = µ,f + 2σE1/2 ds gµ(u)2 du h i s Z0 Z0 t + t γ,f + b E[ Y µ,f ] ds h i | | h s i Z0 22 D. A. DAWSON AND Z. LI t 1 ∞ E1/2 2 µ 2 + 2 ds z m(dz) gs (u) du Z0 Z0 Z0 t ∞ ∞ E µ + ds zm(dz) gs (u) du Z0 Z1 Z0 t 1/2 1 1/2 µ,f + 2 E[ Y µ,f 2 ] ds σ + z2m(dz) ≤ h i h s i Z0 Z0 t ∞ + t γ,f + E[ Y µ,f ] ds b + zm(dz) . h i h s i | | Z0 Z1 In view of (3.17) we get (3.18) for the step function. Now let η(dv)= µ(dv)+ γ(dv) and choose a bounded sequence of step functions fn so that fn f in L2(η) as n . By applying (3.18) to the nonnegative{ } step function→ f f we get→∞ | n − m| E µ 2 1/2 sup Ys , fn fm C(t)[ η, fn fm + 2 η, fn fm ]. 0≤s≤th | − |i ≤ h | − |i h | − | i h i The right-hand side tends to zero as m,n . Then there is a c`adl`ag process Y µ(f) : t 0 so that →∞ { t ≥ } E µ µ (3.19) sup Ys ,fn Ys (f) 0, n . 0≤s≤t |h i− | → →∞ h i On the other hand, from (3.17) we have E[ Y µ, f f ]= µ, f f e−bt + b−1(1 e−bt) γ, f f , h t | n − |i h | n − |i − h | n − |i µ which tends to zero as n . Then Yt (f) : t 0 is a modification of µ →∞ { ≥ }+ Yt ,f : t 0 . Finally, we get (3.18) for f B[0, a] by using (3.19) and the{h resulti for≥ step} functions. ∈ µ Theorem 3.11. The process Yt : t 0 is the unique solution of the following martingale problem: for every{ G≥ C} 2(R) and f B[0, a], ∈ ∈ G( Y µ,f ) h t i 1 t = G( µ,f )+ σ2 G′′( Y µ,f ) Y µ,f 2 ds h i 2 h s i h s i Z0 t ′ µ µ + G ( Ys ,f )[ γ,f b Ys ,f ] ds 0 h i h i− h i (3.20) Z t ∞ + ds Y µ(dx) [G( Y µ,f + zf(x)) s h s i Z0 Z[0,a] Z0 G( Y µ,f ) zf(x)G′( Y µ,f )]m(dz) − h s i − h s i + local mart. STOCHASTIC EQUATIONS, FLOWS AND PROCESSES 23 Proof. Again we start with the step function (3.13). Using (3.15) and Itˆo’s formula, G( Y µ,f ) h t i 1 t ∞ = G( µ,f )+ σ2 ds G′′( Y µ ,f )gµ (u)2 du h i 2 h s− i s− Z0 Z0 t + G′( Y µ ,f )[ γ,f b Y µ ,f ] ds h s− i h i− h s− i Z0 t ∞ ∞ + ds m(dz) [G( Y µ,f + zgµ(u)) h s i s Z0 Z0 Z0 G( Y µ,f ) G′( Y µ,f )zgµ(u)] du − h s i − h s i s + local mart. 1 t = G( µ,f )+ σ2 G′′( Y µ,f ) Y µ,f 2 ds h i 2 h s i h s i Z0 t + G′( Y µ,f )[ γ,f b Y µ,f ] ds h s i h i− h s i Z0 t ∞ ∞ + ds Y µ(dx) [G( Y µ,f + zf(x)) s h s i Z0 Z0 Z0 G( Y µ,f ) G′( Y µ,f )zf(x)]m(dz) − h s i − h s i + local mart. That proves (3.20) for step functions. For f B[0, a] we get the martingale problem using (3.19). The uniqueness of the∈ solution follows from a result in Li (2011), Section 9.3. The solution of the martingale problem (3.20) is the special case of the immigration superprocess studied in Li (2011) with trivial spatial motion. More precisely, the infinitesimal particles propagate in [0, a] without migra- tion. Then for any disjoint bounded Borel subsets B1 and B2 of [0, a], the µ µ nonnegative real-valued processes Yt (B1) : t 0 and Yt (B2) : t 0 are { ≥ } µ { ≥ } independent. That explains why the restriction of Yt : t 0 to the interval [0, a] is still a Markov process. To consider the process{ ≥ of measur} es on the half line [0, ) we need to introduce a weight function as follows. Let h be∞ a strictly positive continuous function on [0, ) vanishing at infinity. Let M [0, ) be the space of Radon measures µ on∞ [0, ) so that h ∞ ∞ µ, h < . Let Bh[0, ) be the set of Borel functions on [0, ) bounded by consth i h,∞ and let C [0,∞ ) denote its subset of continuous functions.∞ A topo- · h ∞ logy on Mh[0, ) can be defined by the convention µn µ in Mh[0, ) if and only if ∞µ ,f µ,f for every f C [0, ). Suppose→ that µ∞ h n i → h i ∈ h ∞ ∈ 24 D. A. DAWSON AND Z. LI M [0, ) and γ M [0, ). It is easy to show that Y µ : t 0 is a c`adl`ag h ∞ ∈ h ∞ { t ≥ } strong Markov process in Mh[0, ), and the results of Theorem 3.10 and Theorem 3.11 are also true for B∞[0, ). h ∞ 4. Generalized Fleming–Viot flows. In this section we give a construc- tion of the generalized Fleming–Viot flow as the strong solution of a stochas- tic integral equation. Let σ 0, b 0 and 0 β 1 be constants, and let z2ν(dz) be a finite measure≥ on (0,≥1]. Suppose≤ that≤ B(ds, du) is a white noise on (0, )2 with intensity ds du, and M(ds, dz, du{ ) is a Poisson} ran- dom measure∞ on (0, ) (0, 1] (0, ) with{ intensity ds} ν(dz) du. Let ∞ × × ∞ q(x, u) = 1 (1 x), x 0, u (0, 1]. {u≤1∧x} − ∧ ≥ ∈ We first consider the stochastic integral equation t 1 t Xt = X0 + σq(Xs−, u)B(ds, du)+ b(β Xs−) ds 0 0 0 − (4.1) Z Z Z t 1 1 + zq(Xs−, u)M˜ (ds, dz, du), Z0 Z0 Z0 where M˜ (ds, dz, du) denotes the compensated measure of M(ds, dz, du). In fact, the compensation in (4.1) can be disregarded as 1 1 q(X , u) du = [1 (X 1)] du = 0. s− {u≤Xs−∧1} − s− ∧ Z0 Z0 Theorem 4.1. There is a unique nonnegative strong solution to (4.1). Proof. We first show the pathwise uniqueness for (4.1). Set l(x, y, u)= q(x, u) q(y, u). For x,y 0 and 0 z,t 1 we have − ≥ ≤ ≤ (x y)+ ztl(x, y, u) − = [(x 1 x) (y 1 y)] + (1 zt)(1 x 1 y) − ∧ − − ∧ − ∧ − ∧ + zt(1 1 ). {u≤x∧1} − {u≤y∧1} It is then easy to see (x y)+ ztl(x, y, u) (1 zt) 1 x 1 y . | − |≥ − | ∧ − ∧ | Moreover, we have 1 l(x, y, u)2 du = (1 x 1 y) (1 x 1 y)2 ∧ − ∧ − ∧ − ∧ Z0 1 x 1 y . ≤ | ∧ − ∧ | STOCHASTIC EQUATIONS, FLOWS AND PROCESSES 25 Using the above two inequalities, 1 1 1 z2l(x, y, u)2 (1 t) dt ν(dz) du − (x y)+ ztl(x, y, u) Z0 Z0 Z0 | − | 1 1 1 t 1 l(x, y, u)2 z2ν(dz) − dt du ≤ 1 zt 1 x 1 y Z0 Z0 − Z0 | ∧ − ∧ | 1 1 1 t 1 z2ν(dz) − dt z2ν(dz). ≤ 1 zt ≤ Z0 Z0 − Z0 Then condition (2.d) is satisfied with ρ(z)= √z. Other conditions of The- orem 2.1 can be checked easily. Then we have the pathwise uniqueness for (4.1). To show the existence of the solution, we may assume X0 = v 0 is a deterministic constant. By Theorem 2.5 there a unique nonnegative strong≥ solution of (4.1) if the Poisson integral term is removed. Then for each k 1 there is a unique nonnegative strong solution to ≥ t 1 Zt = Z0 + σq(Zs−, u)B(ds, du) Z0 Z0 t (4.2) + b(β Z ) ds − s− Z0 t 1 1 + zq(Zs−, u)M(ds, dz, du), Z0 Z1/k Z0 because the last term on the right-hand side gives at most a finite number of jumps on each bounded time interval. Let Z (t) : t 0 be the solution { k ≥ } of (4.2) with Zk(0) = v. Let T1 = inf t 0 : Zk(t) 1 . On the time interval [0, T ], the stochastic integral terms{ in≥ (4.2) vanish.≤ } Then t Z (t) is non- 1 7→ k increasing on [0, T1]. By modifying the proof of Proposition 2.1 in Fu and Li (2010) one can see Z (t) 1 for t T . Thus Z (t) (Z (0) 1) = (v 1) for k ≤ ≥ 1 k ≤ k ∨ ∨ all t 0. Let τk be a bounded sequence of stopping times. Note that the last term≥ on the{ } right-hand side of (4.2) can be considered as a stochastic integral with respect to the compensated Poisson random measure. Then for any t 0 we have ≥ E [Z (τ + t) Z (τ )]2 { k k − k k } t 1 3σ2E ds q(Z (τ + s), u)2 du + 3b2t2(v 1)2 ≤ k k ∨ Z0 Z0 t 1 1 2 2 + 3E ds z ν(dz) q(Zk(τk + s), u) du Z0 Z0 Z0 1 3t σ2 + tb2(v 1)2 + z2ν(dz) . ≤ ∨ Z0 26 D. A. DAWSON AND Z. LI The right-hand side tends to zero as t 0. By a criterion of Aldous (1978), → the sequence Zk(t) : t 0 is tight in D([0, ), R+) [see also Ethier and Kurtz (1986),{ pages 137≥ and} 138]. By a modification∞ of the proof of Theo- rem 4.4 in Fu and Li (2010) one sees that any limit point of this sequence is a weak solution of (4.1). Now let v γ(v) be a nondecreasing continuous function on [0, 1] so that 0 γ(v) 7→1 for all 0 v 1. We denote by γ(dv) the sub-probability measure≤ on [0≤, 1] so that γ≤([0,≤ v]) = γ(v) for 0 v 1. By Theorem 4.1 for ≤ ≤ each v 0 there is a pathwise unique nonnegative solution Xt(v) : t 0 to the equation≥ { ≥ } t 1 X (v)= v + σ[1 X (v)]B(ds, du) t {u≤Xs−(v)} − s− Z0 Z0 t (4.3) + b[γ(v) X (v)] ds − s− Z0 t 1 1 + z[1 X (v)]M˜ (ds, dz, du). {u≤Xs−(v)} − s− Z0 Z0 Z0 It is not hard to see that 0 v 1 implies P 0 Xt(v) 1 for all t 0 = 1. The compensation for the≤ Poisson≤ random{ measure≤ can≤ be disregarded,≥ } so this equation just coincides with (1.6). By Theorem 2.2 for any 0 v1 v 1 we have ≤ ≤ 2 ≤ P X (v ) X (v ) for all t 0 = 1. { t 1 ≤ t 2 ≥ } Therefore X(v) : 0 v 1 is a nondecreasing path-valued process in D[0, ). { ≤ ≤ } ∞ Proposition 4.2. There is a locally bounded nonnegative function t C(t) on [0, ) so that 7→ ∞ E sup [Xs(v2) Xs(v1)] 0≤s≤t − n o (4.4) C(t) (v v ) + [γ(v ) γ(v )] ≤ { 2 − 1 2 − 1 + √v v + γ(v ) γ(v ) 2 − 1 2 − 1 } for t 0 and 0 v v 1. ≥ ≤ 1 ≤ 2 ≤ p Proof. Let Z = X (v ) X (v ) for t 0. From (4.3) we have t t 2 − t 1 ≥ t 1 Z = (v v )+ σ[Y (u) Z ]B(ds, du) t 2 − 1 s− − s− Z0 Z0 t (4.5) + b [γ(v ) γ(v )] Z ds { 2 − 1 − s−} Z0 t 1 1 + z[Y (u) Z ]M˜ (ds, dz, du), s− − s− Z0 Z0 Z0 STOCHASTIC EQUATIONS, FLOWS AND PROCESSES 27 where Ys(u) = 1{Xs(v1) Recall that D[0, ) is the space of nonnegative c`adl`ag functions on [0, ) endowed with the∞ Borel σ-algebra generated by the Skorokhod topology.∞ Let ρ be the metric on D[0, ) defined by (3.11). ∞ Theorem 4.3. The path-valued process X(v) : 0 v 1 is a Markov process in D[0, ). { ≤ ≤ } ∞ Proof. Let 0 t∧τn + bX (v)−1[γ(p) γ(v)X (v)−1X (p)] ds s− − s− s− Z0 t∧τn 1 σ2X (p) + ds s− [1 X (v)]2 du X (v)3 {u≤Xs−(v)} − s− Z0 Z0 s− t∧τn 1 σ2 ds [1 X (p)] − X (v)2 {u≤Xs−(p)} − s− Z0 Z0 s− [1 X (v)] du × {u≤Xs−(v)} − s− t∧τn 1 1 Xs−(p)(1 z)+ z1{u≤X (p)} X (p) + − s− s− X (v)(1 z)+ z1 − X (v) Z0 Z0 Z0 s− − {u≤Xs−(v)} s− M(ds, dz, du) × p t∧τn Xs−(v) = + σX (v)−1[1 X (v)−1X (p)] v s− {u≤Xs−(p)} − s− s− Z0 Z0 B(ds, du) × t∧τn + bX (v)−1[γ(p) γ(v)X (v)−1X (p)] ds s− − s− s− Z0 t∧τn 1 Xs−(v) Xs−(p)(1 z)+ z1{u≤X (p)} X (p) + − s− s− X (v)(1 z)+ z − X (v) Z0 Z0 Z0 s− − s− M(ds, dz, du), × where the two terms involving σ2 counteract each other. Observe also that the last integral does not change if we replace M(ds, dz, du) by the compen- sated measure M˜ (ds, dz, du). Then we get the equation p t∧τn Xs−(v) Z (t)= + σX (v)−1[1 Z (s )] n v s− {u≤Xs−(v)Zn(s−)} − n − Z0 Z0 B(ds, du) × t∧τn 1 Xs−(v) 1{u≤X (v)Z (s−)} Z (s ) (4.7) + z s− n n − z + (1 z)X (v) − z + (1 z)X (v) Z0 Z0 Z0 − s− − s− M˜ (ds, dz, du) × t∧τn + bX (v)−1[γ(p) γ(v)Z (s )] ds. s− − n − Z0 Since X (v) 1/n for 0 (4.8) X (p)= Z X (v)1 + X (v)1 , t 0. t t t {t<τ∞} t {t≥τ∞} ≥ STOCHASTIC EQUATIONS, FLOWS AND PROCESSES 29 Now from (4.7) and (4.8) we infer that X (p) : t 0 is measurable with { t ≥ } respect to the σ-algebra Fv generated by the process Xt(v) : t 0 and the restricted martingale measures { ≥ } ˜ 1{u≤Xs−(v)}B(ds, du), 1{u≤Xs−(v)}M(ds, dz, du). By similar arguments, for any q (v, 1] one can see 1 X (q) : t 0 is mea- ∈ { − t ≥ } surable with respect to the σ-algebra Gv generated by the process 1 Xt(v) : t 0 and the restricted martingale measures { − ≥ } ˜ 1{Xs−(v) We call the solution flow Xt(v) : t 0, v [0, 1] of (4.3) specified in Theorem 4.4 a generalized Fleming–Viot{ ≥ flow following∈ } Bertoin and Le Gall (2003, 2005, 2006). The law of the flow is determined by the parameters (σ, b, γ, ν). Let F [0, 1] be the set of nondecreasing c`adl`ag functions f on [0, 1] such that 0 f(0) f(1) 1. Given a finite stopping time τ and a function µ F [0≤, 1], let ≤Xµ (v≤) : t 0 be the solution of ∈ { τ,t ≥ } τ+t 1 µ µ X (v)= µ(v)+ σ[1 µ X (v)]B(ds, du) τ,t {u≤Xτ,s−(v)} − τ,s− Zτ Z0 τ+t (4.9) + b[γ(v) Xµ (v)] ds − τ,s− Zτ τ+t 1 1 µ + z[1 µ X (v)]M˜ (ds, dz, du) {u≤Xτ,s−(v)} − τ,s− Zτ Z0 Z0 and write simply Xµ(v) : t 0 instead of Xµ (v) : t 0 . The pathwise { t ≥ } { 0,t ≥ } uniqueness for the above equation follows from that of (4.3). Let Fτ,t be 30 D. A. DAWSON AND Z. LI µ the random operator on F [0, 1] that maps µ to Xτ,t. As for the flow of CBI-processes we have P µ µ Theorem 4.5. For any finite stopping time τ we have Xτ+t = Fτ,tXt for all t 0 = 1. { ≥ } For any sub-probability measure µ(dv) on [0, 1] with distribution func- µ tion v µ(v), we write Xt (dv) for the random sub-probability measure on 7→ µ µ [0, 1] determined by the random function v Xt (v). We call Xt : t 0 the 7→ {µ ≥ } generalized Fleming–Viot process associated with the flow Xt (v) : t 0, v [0, 1] . The reader may refer to Dawson (1993) and Ethier{ and Kurtz≥ (1993∈) for the} theory of classical Fleming–Viot processes. To give some character- izations of the generalized Fleming–Viot process, let us consider the step function n (4.10) f(u)= c 1 (u)+ c 1 (u), u [0, 1], 0 {0} i (ai−1,ai] ∈ Xi=1 where c0, c1,...,cn R and 0= a0 < a1 < < an = 1 is a partition of [0, 1]. For{ this function}⊂ we have{ · · · } n (4.11) Xµ,f = c Xµ(0) + c [Xµ(a ) Xµ(a )]. h t i 0 t i t i − t i−1 Xi=1 By (4.9) and (4.11) we have t 1 Xµ,f = µ,f + σ[gµ (u) Xµ ,f ]B(ds, du) h t i h i s− − h s− i Z0 Z0 t (4.12) + b[ γ,f Xµ ,f ] ds h i − h s− i Z0 t 1 1 + z[gµ (u) Xµ ,f ]M˜ (ds, dz, du), s− − h s− i Z0 Z0 Z0 where n µ µ µ µ (4.13) gs (u)= c01{u≤Xs (0)} + ci1{Xs (ai−1) µ Theorem 4.6. The generalized Fleming–Viot process Xt : t 0 de- { ≥µ } fined above is an almost surely c`adl`ag strong Markov process with X0 = µ. Proposition 4.7. For any t 0 and f B[0, 1] we have ≥ ∈ (4.14) E[ Xµ,f ]= µ,f e−bt + γ,f (1 e−bt). h t i h i h i − STOCHASTIC EQUATIONS, FLOWS AND PROCESSES 31 µ Theorem 4.8. For any f B[0, 1] the process Xt ,f : t 0 has a c`adl`ag modification. Moreover, there∈ is a locally bounded{h functioni ≥t }C(t) so that 7→ E µ 2 1/2 2 1/2 (4.15) sup Xs ,f C(t)[ µ,f + γ,f + µ,f + γ,f ] 0≤s≤th i ≤ h i h i h i h i h i for any t 0 and f B[0, 1]+. ≥ ∈ The generalized Fleming–Viot process can be characterized in terms of a martingale problem. Given any finite family f ,...,f B[0, 1], write { 1 p}⊂ p (4.16) G (η)= η,f , η M [0, 1]. p,{fi} h ii ∈ 1 Yi=1 Let D1(L) be the linear span of the functions on M1[0, 1] of the form (4.16), and let L be the linear operator on D1(L) defined by p 2 LGp,{fi}(η)= σ η,fifj η,fk η,fk "h i h i− h i# Xi (4.17) + βp,|I| η, fi η,fj η,fk " h i− h i# I⊂{1,...,pX},|I|≥2 Yi∈I Yj∈ /I kY=1 p p + b γ,fi η,fk η,fk , "h i h i− h i# Xi=1 Yk6=i kY=1 where I denotes the cardinality of I 1,...,p and | | ⊂ { } 1 β = z|I|(1 z)p−|I|ν(dz). p,|I| − Z0 µ Theorem 4.9. The generalized Fleming–Viot process Xt : t 0 is the unique solution of the following martingale problem: for{ any p ≥1 }and f ,...,f B[0, 1], ≥ { 1 p}⊂ t µ µ (4.18) Gp,{fi}(Xt )= Gp,{fi}(µ)+ LGp,{fi}(Xs ) ds + mart. Z0 Proof. We first consider a collection of step functions f1,...,fp . Let µ { } gi (s, u) be defined by (4.13) with f = fi. Since the compensation of the Poisson random measure in (4.12) can be disregarded, by Itˆo’s formula we get µ Gp,{fi}(Xt ) t 1 2 µ µ µ = Gp,{fi}(µ)+ σ ds hi (s, u)hj (s, u) Xs ,fk du 0 0 h i Z Z Xi t 1 1 p p µ µ µ + ds ν(dz) [ Xs ,fk + zhk (s, u)] Xs ,fk du 0 0 0 ( h i − h i) Z Z Z kY=1 kY=1 t p µ µ + b [ γ,fi Xs ,fi ] Xs ,fk ds + mart. 0 h i − h i h i Z Xi=1 Yk6=i t 1 2 µ µ µ µ = Gp,{fi}(µ)+ σ ds li (u)lj (u) Xs ,fk Xs (du) 0 0 h i Z Z Xi t p p µ µ + b γ,fi Xs ,fk Xs ,fk ds + mart. 0 "h i h i− h i# Z Xi=1 Yk6=i kY=1 t p 2 µ µ µ = Gp,{fi}(µ)+ σ Xs ,fifj Xs ,fk Xs ,fk ds 0 "h i h i− h i# Z Xi µ In particular, if µ(1) = γ(1) = 1, we have Xt (1) = 1 for all t 0, and the µ ≥ corresponding generalized Fleming–Viot process Xt : t 0 is a probability- valued Markov process with generator L defined{ by ≥ } p 2 LGp,{fi}(η)= σ η,fifj η,fk η,fk "h i h i− h i# Xi (4.19) + βp,|I| η, fi η,fj η,fk " h i− h i# I⊂{1,...,pX},|I|≥2 Yi∈I Yj∈ /I kY=1 p + η,Af η,f , h ii h ki Xi=1 Yk6=i where Af(x)= b [f(y) f(x)]γ(dy), x [0, 1]. − ∈ Z[0,1] This is a generalization of a classical Fleming–Viot process [see, e.g., Ethier and Kurtz (1993), page 351]. On the other hand, for b = 0 the solution flow µ Xt (v) : t 0, 0 v 1 of (4.3) corresponds to the Λ-coalescent process { ≥ 2 ≤ ≤2 } with Λ(dz)= σ δ0 + z ν(dz), which is clear from (4.18) and the martingale 34 D. A. DAWSON AND Z. LI problem given by Theorem 1 in Bertoin and Le Gall (2005). For b> 0 it seems the flow determines a coalescent process with a spatial structure. A serious exploration in the subject would be of interest to the understanding of the related dynamic systems. 5. Scaling limit theorems. In this section, we prove some limit theorems for the generalized Fleming–Viot flows. We shall present the results in the setting of measure-valued processes and through the use of Markov process arguments. These are different from the approach of Bertoin and Le Gall (2006), who used the analysis of characteristics of semimartingales. For each k 1 let σ 0 and b 0 be two constants, let z2ν (dz) be a finite mea- ≥ k ≥ k ≥ k sure on (0, 1] and let v γk(v) be a nondecreasing continuous function on [0, 1] so that 0 γ (v7→) 1 for all 0 v 1. We denote by γ (dv) the ≤ k ≤ ≤ ≤ k sub-probability measure on [0, 1] so that γk([0, v]) = γk(v) for 0 v 1. Let k ≤ ≤ Xt (v) : t 0, v [0, 1] be a generalized Fleming–Viot flow with parameters { ≥ ∈ } k k −1 (σk, bk, γk,νk) and with X0 (v)= v for v [0, 1]. Let Yk(t, v)= kXkt(k v) for t 0 and v [0, k]. Let η (z)= kγ (k∈−1z) and m (dz)= ν (k−1dz) for ≥ ∈ k k k k z (0, k]. In view of (4.3), we can also define Yk(t, v) : t 0, v [0, k] di- rectly∈ by { ≥ ∈ } t k Y (t, v)= v + kσ [1 k−1Y (s , v)]W (ds, du) k k {u≤Yk(s−,v)} − k − k Z0 Z0 t (5.1) + kb [η (v) Y (s , v)] ds k k − k − Z0 t k k + z[1 k−1Y (s , v)]N˜ (ds, dz, du), {u≤Yk(s−,v)} − k − k Z0 Z0 Z0 where Wk(ds, du) is a white noise on (0, ) (0, k] with intensity ds du, { } ∞ × 2 and Nk(ds, dz, du) is a Poisson random measure on (0, ) (0, k] with intensity{ ds m (dz)}du. In the sequel, we assume k a for∞ fixed× a constant k ≥ a 0. Then the rescaled flow Yk(t, v) : t 0, v [0, k] induces an M[0, a]- ≥ a { ≥ ∈ } valued process Yk (t) : t 0 . We are interested in the asymptotic behavior a { ≥ } of Yk (t) : t 0 as k . Recall that λ denotes the Lebesgue measure on [0,{ ). ≥ } →∞ ∞ Lemma 5.1. For any G C2(R) and f C[0, a] we have ∈ ∈ G( Y a(t),f ) h k i t = G( λ, f )+ kb G′( Y a(s),f ) η ,f ds h i k h k i h k i Z0 t kb G′( Y a(s),f ) Y a(s),f ds − k h k i h k i Z0 STOCHASTIC EQUATIONS, FLOWS AND PROCESSES 35 1 t + k2σ2 G′′( Y a(s),f ) Y a(s),f 2 ds 2 k h k i h k i Z0 1 t kσ2 G′′( Y a(s),f ) Y a(s),f 2 ds − 2 k h k i h k i Z0 t k + ds m (dz) G( Y a(s),f + zf(x)) G( Y a(s),f ) k { h k i − h k i Z0 Z0 Z[0,a] G′( Y a(s),f )zf(x) Y a(s,dx) − h k i } k t k + ds [εk(s, z)+ ξk(s, z)]mk(dz)+ local mart., Z0 Z0 where k ε (s, z)= G( Y a(s),f + z[f(x) k−1 Y a(s),f ]) k { h k i − h k i Z0 G( Y a(s),f + zf(x)) − h k i k−1G′( Y a(s),f )z Y a(s),f Y a(s,dx) − h k i h k i} k and ξ (s, z) = [k Y (s, a)] k − k [G( Y a(s),f k−1z Y a(s),f ) × h k i− h k i G( Y a(s),f )+ k−1G′( Y a(s),f )z Y a(s),f ]. − h k i h k i h k i Proof. For the step function defined by (3.13) we get from (5.1) that t k Y a(t),f = λ, f + kσ h (s , u)W (ds, du) h k i h i k k − k Z0 Z0 t (5.2) + kb [ η ,f Y a(s ),f ] ds k h k i − h k − i Z0 t k k + zh (s , u)N˜ (ds, dz, du), k − k Z0 Z0 Z0 where h (s, u)= g (s, u) k−1 Y a(s),f and k k − h k i n (5.3) gk(s, u)= c01{u≤Yk(s,0)} + ci1{Yk(s,ai−1) 1 t k + k2σ2 G′′( Y a(s),f ) ds h (s, u)2 du 2 k h k i k Z0 Z0 t k k + ds m (dz) G( Y a(s),f + zh (s, u)) k { h k i k Z0 Z0 Z0 G( Y a(s),f ) − h k i G′( Y a(s),f )zh (s, u) du − h k i k } + local mart. t = G( λ, f )+ kb G′( Y a(s),f )[ η ,f Y a(s),f ] ds h i k h k i h k i − h k i Z0 1 t + k2σ2 G′′( Y a(s),f )[ Y a(s),f 2 k−1 Y a(s),f ] ds 2 k h k i h k i− h k i Z0 t k + ds m (dz) G( Y a(s),f + zl (s,x)) k { h k i k Z0 Z0 Z[0,a] G( Y a(s),f ) − h k i G′( Y a(s),f )zl (s,x) Y a(s,dx) − h k i k } k t k + [k Y (s, a)] ds G( Y a(s),f k−1z Y a(s),f ) − k { h k i− h k i Z0 Z0 G( Y a(s),f ) − h k i + k−1G′( Y a(s),f )z Y a(s),f m (dz) h k i h k i} k + local mart. That gives the desired result for the step function. For f C[0, a] it follows by approximating the function by a sequence of step function∈ s. Lemma 5.2. For t 0 and f C[0, a]+ we have ≥ ∈ E a sup Yk (s),f 0≤s≤th i h i k λ, f + kb η ,f t + 4t[ λ, f + η ,f ] zm (dz) ≤ h i kh k i h i h k i k Z1 1 1/2 + 2√t[ λ, f 2 + η ,f 2 ]1/2 σ + z2m (dz) . h i h k i k Z0 Proof. We first consider a nonnegative step function given by (3.13) with c , c ,...,c R . Let g (s, u) and h (s, u) be defined as in the proof { 0 1 n}⊂ + k k STOCHASTIC EQUATIONS, FLOWS AND PROCESSES 37 of Lemma 5.1. By (5.2) and Doob’s martingale inequality we get E a sup Yk (s),f 0≤s≤th i h i t k 2 λ, f + 2kσ E1/2 h (s , u)W (ds, du) ≤ h i k k − Z0 Z0 t k k + kb η ,f t + E ds zm (dz) h (s , u) du kh k i k | k − | Z0 Z1 Z0 t k k + E z h (s , u) N (ds, dz, du) | k − | k Z0 Z1 Z0 t 1 k 2 + 2E1/2 zh (s , u)N˜ (ds, dz, du) . k − k Z0 Z0 Z0 It then follows that E a sup Yk (s),f 0≤s≤th i h i t k λ, f + 2kσ E1/2 ds h (s, u)2 du ≤ h i k k Z0 Z0 t k k + kb η ,f t + 2E ds zm (dz) h (s, u) du kh k i k | k | Z0 Z1 Z0 t 1 k 1/2 2 2 + 2E ds z mk(dz) hk(s, u) du Z0 Z0 Z0 t k λ, f + kb η ,f t + 4E Y a(s),f ds zm (dz) ≤ h i kh k i h k i k Z0 Z1 t 1 1/2 + 2E1/2 Y a(s),f 2 ds kσ + z2m (dz) . h k i k k Z0 Z0 By Proposition 4.7 one can see E[ Y a(t),f ]= λ, f e−kbkt + η ,f (1 e−kbkt) λ, f + η ,f . h k i h i h k i − ≤ h i h k i Then we have the desired inequality for the step function. The inequality for f C[0, a]+ follows by approximating this function with a bounded sequence of∈ positive step functions. a Lemma 5.3. Let τk be a bounded stopping time for Yk (t) : t 0 . Then for any t 0 and f C[0, a] we have { ≥ } ≥ ∈ E Y a(τ + t),f Y a(τ ),f {|h k k i − h k k i|} t 1 1/2 E1/2 Y a(τ + s),f 2 ds kσ + z2m (dz) ≤ h k k i k k Z0 Z0 38 D. A. DAWSON AND Z. LI (5.4) t + kb E ( η , f + Y a(τ + s), f ) ds k h k | |i h k k | |i Z0 t k + 4E Y a(τ + s), f ds zm (dz) . h k k | |i k Z0 Z1 Proof. We first consider the step function given by (3.13). Let gk(s, u) and hk(s, u) be defined as in the proof of Lemma 5.1. From (5.2) we have E Y a(τ + t),f Y a(τ ),f {|h k k i − h k k i|} t k 2 kσ E1/2 h (τ + s , u)W (τ + ds, du) ≤ k k k − k Z0 Z0 t + kb E η ,f Y a(τ + s ),f ds k |h k i − h k k − i| Z0 t 1 k 2 + E1/2 zh (τ + s , u)N˜ (τ + ds, dz, du) k k − k k Z0 Z0 Z0 t k k + E z h (τ + s , u) N (τ + ds, dz, du) | k k − | k k Z0 Z1 Z0 t k k + E ds zm (dz) h (τ + s , u) du . k | k k − | Z0 Z1 Z0 By the property of independent increments of the white noise and the Pois- son random measure, E Y a(τ + t),f Y a(τ ),f {|h k k i − h k k i|} t k kσ E1/2 ds h (τ + s, u)2 du ≤ k k k Z0 Z0 t + kb E ( η , f + Y a(τ + s), f ) ds k h k | |i h k k | |i Z0 t 1 k 1/2 2 2 + E ds z mk(dz) hk(τk + s, u) du Z0 Z0 Z0 t k k + 2E ds zm (dz) h (τ + s, u) du k | k k | Z0 Z1 Z0 t 1 1/2 E1/2 Y a(τ + s),f 2 ds kσ + z2m (dz) ≤ h k k i k k Z0 Z0 t + kb E ( η , f + Y a(τ + s), f ) ds k h k | |i h k k | |i Z0 t k + 4E Y a(τ + s), f ds zm (dz) . h k k | |i k Z0 Z1 STOCHASTIC EQUATIONS, FLOWS AND PROCESSES 39 Then (5.4) holds for the step function. For f C[0, a] the inequality follows by an approximation argument. ∈ 2 2 Lemma 5.4. Suppose that kbk b, ηk η weakly on [0, a] and k σk 2 → → 2 × δ0(dz) + (z z )mk(dz) converges weakly on [0, ) to a finite measure σ ∧ 2 ∞ × δ0(dz)+(z z )m(dz) as k . Let 0 a1 < < an be an ordered set of ∧ a1 →∞an { ≤ · · · } constants. Then (Yk (t),...,Yk (t)) : t 0 , k = 1, 2,... is a tight sequence in D([0, ), M[0{, a ] M[0, a ]). ≥ } ∞ 1 ×···× n Proof. Let τ be a bounded stopping time for Y a(t) : t 0 and as- k { k ≥ } sume the sequence τk : k = 1, 2,... is uniformly bounded. Let fi C[0, ai] for i = 1,...,n. By ({5.4) we see } ∈ n E ai ai Yk (τk + t),fi Yk (τk),fi ( |h i − h i|) Xi=1 n t 1 1/2 E1/2 Y ai (τ + s),f 2 ds kσ + z2m (dz) ≤ h k k i i k k i=1 Z0 Z0 (5.5) X n t E ai + kbk ( ηk, fi + Yk (τk + s), fi ) ds 0 h | |i h | |i Xi=1 Z n t k E ai + 4 Yk (τk + s), fi ds zmk(dz) . 0 h | |i 1 Xi=1 Z Z Then the inequality in Lemma 5.2 implies n E ai ai lim sup Yk (τk + t),fi Yk (τk),fi = 0. t→0 k≥1 ( |h i − h i|) Xi=1 a1 an By a criterion of Aldous (1978), the sequence ( Yk (t),f1 ,..., Yk (t),fn ) : t 0 is tight in D([0, ), Rn) [see also Ethier{ h and Kurtzi (h1986), pagesi 137≥ and} 138]. Then a simple∞ extension of the tightness criterion of Roelly- a1 an Coppoletta (1986) implies (Yk (t),...,Yk (t)) : t 0 is tight in D([0, ), M[0, a ] M[0, a ]). { ≥ } ∞ 1 ×···× n Suppose that σ 0 and b 0 are two constants, v η(v) is a nonnega- tive and nondecreasing≥ continuous≥ function on [0, )7→ and (z z2)m(dz) is a finite measure on (0, ). Let η(dv) be the Radon∞ measure∧ on [0, ) so that η([0, v]) = η(v) for ∞v 0. Suppose that W (ds, du) is a white noise∞ on (0, )2 with intensity ds≥ dz and N(ds, dz, du{ ) is a Poisson} random mea- sure∞ on (0, )3 with intensity ds{ m(dz) du. Let} X (v) : t 0, v 0 be the ∞ { t ≥ ≥ } 40 D. A. DAWSON AND Z. LI solution flow of the stochastic equation t Xs−(v) t Xt(v)= v + σ W (ds, du)+ b [η(v) Xs−(v)] ds 0 0 0 − (5.6) Z Z Z t ∞ Xs−(v) + zN˜ (ds, dz, du). Z0 Z0 Z0 By Theorem 3.11, for each a 0 the flow Xt(v) : t 0, v 0 induces an ≥ { a ≥ ≥ } M[0, a]-valued immigration superprocess Xt : t 0 which is the unique solution of the following martingale problem:{ for≥ every} G C2(R) and f C[0, a], ∈ ∈ G( X ,f ) h t i t = G( λ, f )+ b G′( X ,f )[ η,f X ,f ] ds h i h s i h i − h s i Z0 t 1 2 ′′ 2 + σ G ( Xs,f ) Xs,f ds 2 0 h i h i (5.7) Z t ∞ + ds m(dz) [G( X ,f + zf(x)) h s i Z0 Z0 Z[0,a] G( X ,f ) G′( X ,f )zf(x)]X (dx) − h s i − h s i s + local mart. Theorem 5.5. Suppose that kbk b, ηk η weakly on [0, a] and 2 2 2 → → k σkδ0(dz) + (z z )mk(dz) converges weakly on [0, ) to a finite measure 2 ∧2 a ∞ σ δ0(dz) + (z z )m(dz) as k . Then Yk (t) : t 0 converges to the immigration superprocess∧ Xa :→∞t 0 in distribution{ on≥ D} ([0, ), M[0, a]). { t ≥ } ∞ For the proof of the above theorem, let us make some preparations. Since the solution of the martingale problem (5.7) is unique, it suffices to a a prove any weak limit point Zt : t 0 of the sequence Yk (t) : t 0 is the solution of the martingale{ problem.≥ } To simplify the nota{ tion we≥ pass} to a a a subsequence and simply assume Yk (t) : t 0 converges to Zt : t 0 in distribution. Using Skorokhod’s representation{ ≥ } theorem, we can{ also≥ as-} a a sume Yk (t) : t 0 and Zt : t 0 are defined on the same probability { a ≥ } { ≥ } a space and Yk (t) : t 0 converges a.s. to Zt : t 0 in the topology of D([0, ), M{[0, a]). For≥ n} 1 let { ≥ } ∞ ≥ t τ = inf t 0:sup [1 + Y a(s)+ Za, 1 2] ds n . n ≥ h k s i ≥ k≥1 Z0 It is easy to see that τ as n . n →∞ →∞ STOCHASTIC EQUATIONS, FLOWS AND PROCESSES 41 2 2 Lemma 5.6. Suppose that kbk b, ηk η weakly on [0, a] and k σk 2 → → 2 × δ0(dz) + (z z )mk(dz) converges weakly on [0, ) to a finite measure σ ∧ 2 ∞ × δ0(dz) + (z z )m(dz) as k . Let εk(s, z) be defined as in Lemma 5.1. Then for each∧ n 1 we have→∞ ≥ t∧τn k E ds ε (s, z) m (dz) 0, k . | k | k → →∞ Z0 Z0 Proof. By the mean-value theorem, we have 1 ε (s, z)= z Y a(s),f k k h k i k [G′( Y a(s),f + zθ (s,x)) G′( Y a(s),f )]Y a(s,dx), × h k i k − h k i k Z0 −1 a where θk(s,x) takes values between f(x) and f(x) k Yk (s),f . Conse- quently, − h i 2 2 ε (s, z) G′ z Y a(s), f Y a(s), 1 G′ f z Y a(s), 1 2. | k |≤ k k k h k | |ih k i≤ k k kk k h k i Moreover, since Y a(s), 1 k, we get h k i≤ 1 k ε (s, z) G′′ z2 Y a(s), f θ (s,x) Y a(s,dx) | k |≤ k k k h k | |i | k | k Z0 1 k G′′ z2 Y a(s), f [ f(x) + k−1 Y a(s), f ]Y a(s,dx) ≤ k k k h k | |i | | h k | |i k Z0 2 f 2 G′′ z2 Y a(s), 1 2. ≤ k k k k k h k i It follows that t∧τn k E ds ε (s, z) m (dz) | k | k Z0 Z0 C k t∧τn (z z2)m (dz)E Y a(s), 1 2 ds ≤ k ∧ k h k i Z0 Z0 nC k (z z2)m (dz), ≤ k ∧ k Z0 where C = 2 f ( G′ + G′′ f ). The right-hand side goes to zero as k . k k k k k kk k → ∞ Lemma 5.7. Suppose that kb b, η η weakly on [0, a] and k2σ2 k → k → k × δ (dz) + (z z2)m (dz) converges weakly on [0, ) to a finite measure σ2 0 ∧ k ∞ × 42 D. A. DAWSON AND Z. LI 2 δ0(dz) + (z z )m(dz) as k . Let ξk(s, z) be defined as in Lemma 5.1. Then for each∧ n 1 we have→∞ ≥ t∧τn k E ds ξ (s, z) m (dz) 0, k . | k | k → →∞ Z0 Z0 Proof. It is elementary to see that ξ (s, z) k G( Y a(s),f k−1z Y a(s),f ) | k |≤ | h k i− h k i G( Y a(s),f )+ k−1G′( Y a(s),f )z Y a(s),f − h k i h k i h k i| 1 min 2 G′ z Y a(s), f , G′′ z2 Y a(s), f 2 ≤ k k h k | |i 2k k k h k | |i C[1 + Y a(s), 1 2](z k−1z2), ≤ h k i ∧ where C = f (2 G′ + f G′′ /2). Then we have k k k k k kk k t∧τn k E ds ξ (s, z) m (dz) | k | k Z0 Z0 k t∧τn C (z k−1z2)m (dz)E [1 + Y a(s), 1 2] ds ≤ ∧ k h k i Z0 Z0 k nC (z k−1z2)m (dz). ≤ ∧ k Z0 The right-hand side tends to zero as k . →∞ a Proof of Theorem 5.5. Let f C[0, a]. Then Yk (t),f : t 0 con- a ∈ {h R i ≥ } verges a.s. to Zt ,f : t 0 in the topology of D([0, ), ). Consequently, {h a i ≥ }a ∞ we have a.s. Yk (t),f Zt ,f for a.e. t 0 [see, e.g., Ethier and Kurtz (1986), page 118].h Byi Lemma → h 5.1i, ≥ t G( Y a(t),f )= G( λ, f )+ kb G′( Y a(s),f ) η ,f ds h k i h i k h k i h k i Z0 t kb G′( Y a(s),f ) Y a(s),f ds − k h k i h k i Z0 1 t + k2σ2 G′′( Y a(s),f ) Y a(s),f 2 ds 2 k h k i h k i Z0 1 t (5.8) kσ2 G′′( Y a(s),f ) Y a(s),f 2 ds − 2 k h k i h k i Z0 t k + ds m (dz) H(x,z, Za,f )Y a(s,dx) k h s i k Z0 Z0 Z[0,a] STOCHASTIC EQUATIONS, FLOWS AND PROCESSES 43 t k + ds [εk(s, z)+ ξk(s, z)+ ζk(s, z)]mk(dz) Z0 Z0 + local mart., where H(x, z, u)= G(u + zf(x)) G(u) G′(u)zf(x) − − and ζ (s, z)= [H(x,z, Y a(s),f ) H(x,z, Za,f )]Y a(s,dx). k h k i − h s i k Z[0,a] By the mean-value theorem, ζ (s, z) H′ (x,z,θ (s)) Y a(s) Za,f Y a(s,dx), | k |≤ | u k h k − s i| k Z[0,k] a a 3 R where θk(s) takes values between Yk (s),f and Zs ,f . For G C ( ) we have h i h i ∈ H′ (x,z,θ (s)) = G′(θ (s)+ zf(x)) G′(θ (s)) G′′(θ (s))zf(x) | u k | | k − k − k | f (2 G′′ + 1 f G′′′ )(z z2). ≤ k k k k 2 k kk k ∧ It follows that ′′ 1 ′′′ 2 ζk(s) f (2 G + 2 f G )(z z ) (5.9) | | ≤ k k k k k kk k ∧ Y a(s), 1 Y a(s) Za,f . × h k i|h k − s i| By (5.9) and Schwarz’s inequality, t∧τn k E ds ζ (s) m (dz) | k | k Z0 Z0 t∧τn 1/2 C (t) E Y a(s) Za,f 2 ds ≤ k h k − s i Z0 t∧τn 1/2 E Y a(s), 1 2 ds × h k i Z0 t∧τn 1/2 √nC (t) E Y a(s) Za,f 2 ds , ≤ k h k − s i Z0 where k C (t)= f (2 G′′ + 1 G′′′ f ) (z z2)m (dz). k k k k k 2 k kk k ∧ k Z0 44 D. A. DAWSON AND Z. LI Note that sup C (t) < . It then follows that k≥1 k ∞ t∧τn k E ds ζ (s) m (dz) 0, k . | k | k → →∞ Z0 Z0 Now letting k in (5.8) and using Lemmas 5.6 and 5.7 we obtain (5.7) for G C3(R). A→∞ simple approximation shows the martingale problem actually holds∈ for any G C2(R). ∈ Theorem 5.8. Suppose that kbk b, ηk η weakly on [0, a] and 2 2 2 → → k σkδ0(dz) + (z z )mk(dz) converges weakly on [0, ) to a finite mea- 2 ∧ 2 ∞ sure σ δ0(dz) + (z z )m(dz) as k . Let 0 a1 < < an = a be ∧ →∞a1 {an≤ · · · } an ordered set of constants. Then (Yk (t),...,Yk (t)) : t 0 converges to (Xa1 ,...,Xan ) : t 0 in distribution{ on D([0, ), M[0, a ≥] } M[0, a ]). { t t ≥ } ∞ 1 ×···× n a1 an Proof. By Lemma 5.4 the sequence (Yk (t),...,Yk (t)) : t 0 is tight { a1 an ≥ } in D([0, ), M[0, a1] M[0, an]). Let (Zt ,...,Zt ) : t 0 be a weak ∞ a1 ×···× an { ≥ } limit point of (Yk (t),...,Yk (t)) : t 0 . To get the result, we only need to show (Za1{,...,Zan ) : t 0 and ≥(X}a1 ,...,Xan ) : t 0 have identical { t t ≥ } { t t ≥ } distributions on D([0, ), M[0, a1] M[0, an]). By passing to a subse- ∞ ×···× a1 quence and using Skorokhod’s representation, we can assume (Yk (t),..., an a1 an { Yk (t)) : t 0 converges to (Zt ,...,Zt ) : t 0 almost surely in the ≥ } { ≥ } an topology of D([0, ), M[0, a1] M[0, an]). Theorem 5.5 implies Zt : t 0 is an immigration∞ superprocess×···× solving the martingale problem{ (5.7) ≥ } ¯ai an an ¯an with a = an. Let Zt denote the restriction of Zt to [0, ai]. Then Zt = Zt a1 an ¯a1 ¯an in particular. We will show (Zt ,...,Zt ) : t 0 and (Zt ,..., Zt ) : t 0 { ≥ } { a1 ≥ } are indistinguishable. That will imply the desired result since (Xt ,..., Xan ) : t 0 and (Z¯a1 ,..., Z¯an ) : t 0 clearly have identical distributions{ t ≥ } { t t ≥ } on D([0, ), M[0, a1] M[0, an]). By the general theory of c`adl`ag pro- cesses, the∞ complement×···× in [0, ) of ∞ D(Z) := t 0 : P(Za1 = Za1 ,...,Zan = Zan ) = 1 { ≥ t t− t t− } is at most countable [see Ethier and Kurtz (1986), page 131]. For any t ai ai ∈ D(Z) we have almost surely limk→∞ Yk (t)= Zt for each i = 1,...,n [see Ethier and Kurtz (1986), page 118]. By an elementary property of weak convergence, for any t D(Z) we almost surely have ∈ ai ai an Zt ([0, ai]) = lim Y (t, [0, ai]) = lim Y (t, [0, ai]) k→∞ k k→∞ k Zan ([0, a ]) = Z¯an ([0, a ]) = Z¯ai ([0, a ]). ≤ t i t i t i Since Theorem 5.5 implies Zai : t 0 is equivalent to Z¯ai : t 0 , we have { t ≥ } { t ≥ } E ai E ¯ai [Zt ([0, ai])] = [Zt ([0, ai])]. STOCHASTIC EQUATIONS, FLOWS AND PROCESSES 45 It then follows that almost surely ai ¯ai (5.10) lim Y (t, [0, ai]) = Zt ([0, ai]). k→∞ k an ¯an On the other hand, since Yk (t) Zt , for any closed set C [0, ai] we have → ⊂ ai an ¯an ¯ai (5.11) lim sup Yk (t,C) = lim Yk (t,C) Zt (C)= Zt (C). k→∞ k→∞ ≤ ai ai ¯ai ai By (5.10) and (5.11) we have Zt = limk→∞ Yk (t)= Zt . Then Zt : t 0 and Z¯ai : t 0 are indistinguishable since both processes are c`adl`ag.{ ≥} { t ≥ } Let M be the space of Radon measures on [0, ) furnished with a metric compatible with the vague convergence. The result∞ of Theorem 5.8 clearly implies the convergence of Yk(t) : t 0 in distribution on D([0, ), M ) with the Skorokhod topology.{ From≥ Theorem} 5.8 we can also derive∞ the following generalization of a result of Bertoin and Le Gall (2006) [see also Bertoin and Le Gall (2000) for an earlier result]. Corollary 5.9. Suppose that kbk b, ηk η weakly on [0, a] and 2 2 2 → → k σkδ0(dz) + (z z )mk(dz) converges weakly on [0, ) to a finite mea- sure σ2δ (dz) +∧ (z z2)m(dz) as k . Let 0 a∞< < a be an 0 ∧ →∞ { ≤ 1 · · · n} ordered set of constants. Then (Yk(t, a1),...,Yk(t, an)) : t 0 converges to (X (a ),...,X (a )) : t 0 in{ distribution on D([0, ), R≥n )}. { t 1 t n ≥ } ∞ + Acknowledgment. We are very grateful to the referee for a careful read- ing of the paper and helpful comments. 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China Canada K1S 5B6 E-mail: [email protected] E-mail: [email protected] URL: http://math.bnu.edu.cn/˜lizh/ URL: http://lrsp.carleton.ca/directors/dawson/