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A Martingale Transformation for Superprocesses∗

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A Martingale Transformation for Superprocesses∗ A martingale transformation for superprocesses∗ March 31, 2012 Hui He, Nana Luan Abstract In this note, we prove a new martingale transformation for superprocesses with general branching mechanisms which could be regarded as a Girsanov's type trans- formation in measure-valued setting. As an application, a criterion for global ex- tinction of superprocesses with spatially dependent branching mechanisms is given. AMS 2010 Subject Classifications: 60J68 Keywords: superprocess, martingale transformation, Girsanov trans- formation, extinction probability. 1 Introduction and main results A Girsanov's type theorem for superprocesses was first proved by Dawson (1978). A bi- variate version was given by Evans and Perkins (1994); see also Theorem IV.1.6 in Perkins (2002). Motivated by some recent works on Girsanov's type theorems for continuous state branching processes and L´evytrees; see Abraham and Delmas (2010), in this work, we will prove a Girsanov's type theorem for superprocesses with general branching mechanisms. The processes could be discontinuous and have non-local branchings. Suppose that E is a Lusin topological space and d is a metric for its topology so that the d-completion of E is compact. Let ξ = (Ω; G; Gt; ξt; Px) be a conservative Borel right process in E with transition semigroup (Pt)t≥0. We denote by B(E) the Banach space of bounded Borel functions on E endowed with the supremum/uniform norm \k · k". Let C(E) be the subspace of B(E) consisting of continuous functions. We use the superscript \+" to denote the subset of positive elements of the function spaces, e.g., B+(E) and C+(E). For a measure µ and a Borel function f on E write µ(f) or hµ, fi for R fdµ if the integral exists. Let M(E) denote the space of finite measures on E endowed with the topology of weak convergence. ∗Supported by NSFC(11071021, 11126037) and Ministry of Education (985 Project). 1 To describe our superprocesses, we first define an operator f 7! (·; f) from B(E)+ to B(E) as follows. Let b 2 B(E) and c 2 B(E)+. Let η(x; dy) be a bounded kernel on E and H(x; dν) a σ-finite kernel from E to M(E)o := M(E)nf0g, where 0 is the null measure. Suppose that Z 2 sup ν(1) ^ ν(1) + νx(1) H(x; dν) < 1; (1.1) x2E M(E)o + where νx(dy) denotes the restriction of ν(dy) to Enfxg. For x 2 E and f 2 B(E) write Z (x; f) = b(x)f(x) + c(x)f 2(x) − f(y)η(x; dy) Z E + e−ν(f) − 1 + ν(fxg)f(x) H(x; dν): (1.2) M(E)o Let (A;D(A)) denote the weak generator of (Pt)t≥0, i.e., for f 2 D(A) and every x 2 E, Z t f(ξt) − f(x) − Af(ξs)ds; t ≥ 0 (1.3) 0 is a Gt-martingale under Px. We further assume that PtC(E) ⊂ C(E) and D(A) ⊂ C(E). Definition 1.1 We call an M(E)-valued Markov process the (ξ; )-superprocess if its transition semigroup (Qt)t≥0 on M(E) is given by Z −ν(f) + e Qt(µ, dν) = exp{−µ(Vtf)g; f 2 B(E) ; (1.4) M(E) where (t; x) 7! Vt(x; f) is the unique locally bounded positive solution to Z t Z Vt(x) = Ptf(x) − ds (y; Vs)Pt−s(x; dy); t ≥ 0; x 2 E: (1.5) 0 E The ξ and will be refereed to as the underlying motion and the branching mechanism of the (ξ; )-superprocesses. More details could be found in Section 2.3 of Li (2011). Let X be the set of all M(E)-valued right continuous functions on [0; 1). Let F be the σ-field generated by all finite dimensional cylinder sets in X . We denote by X = fXt : t ≥ 0g the coordinate process on X and let fFt : t ≥ 0g be its natural filtration. By Theorem 5.12 in Li (2011), the (ξ; )-superprocess has a right realization in M(E). Thus, for µ 2 M(E), we have a probability measure Pµ on (X ; F) such that under Pµ , X = fXt : t ≥ 0g is a (ξ; )-superprocess and Pµ [X0 = µ] = 1. For q 2 B(E), assume Z −ν(q) sup [ν(1) + νx(1)] e H(x; dν) < 1: (1.6) x2E fν:ν(1)>1g Then q(x; f) := (x; q + f) − (x; q) 2 Z = b(x) + 2c(x)q(x) + ν(fxg)(1 − e−ν(q))H(x; dν) f(x) M(E)o Z +c(x)f 2(x) − f(y)η(x; dy) Z E + e−ν(f) − 1 + ν(fxg)f(x) e−ν(q)H(x; dν) M(E)o is well-defined and thus it is also a branching mechanism. For q 2 D(A) and t ≥ 0, define Z t q Ht = exp µ(q) − Xt(q) + Xs(Aq − (·; q))ds : (1.7) 0 According to Proposition 2.27 and Lemma 2.17 in Li (2011) we have that, under Pµ , for any 0 ≤ t ≤ T , Z t Xt(1) < 1 and Xs(1)ds < 1: 0 + + q Thus if q 2 D(A) and (·; q) − Aq 2 B (E), then Ht > 0 for 0 ≤ t ≤ T . Theorem 1.1 For q 2 D(A), assume that (1.6) holds. If lim inf (x; l) = +1; (1.8) l!1 x2E then q i) under Pµ , fHt : t ≥ 0g is an Ft-martingale; ii) for each t ≥ 0, dP q j µ Ft = Hq: (1.9) t dPµ jFt R Remark 1.1 (1.8) holds if infx2E c(x) > 0 or infx2E M(E)o ν(1)H(x; dν) = +1: Remark 1.2 When satisfies Conditions 7.1 and 7.2 in Li (2011) and q belongs to the q domain of the strong generator of ξ, Theorem 7.13 in Li (2011) proved that fHt : t ≥ 0g is a local martingale where it does not require (1.6) and (1.8). In the following, we make the convention that 1 · 0 = 0. We give a criterion for global extinction of (ξ; )-superprocesses. + + q Corollary 1.1 Assume q 2 D(A) and (·; q) − Aq 2 B (E). Then, under Pµ , H q converges a.s. to a limit H1 such that Z 1 q H11fτ0<+1g = exp µ(q) + dsXs(Aq − (·; q)) 1fτ0<+1g; 0 where τ0 = infft ≥ 0 : Xt(1) = 0g. Furthermore, we have q dPµ q = H11fτ0<+1g (1.10) dPµ and q q −1 Pµ [τ0 < 1] = Pµ (H1) 1fτ0<+1g : (1.11) 3 Remark 1.3 When (x; f) = b(x)f(x) + c(x)f(x)2, for some very specialized q; b and c, (1.10) has been proved in Delmas and H´enard (2011). (1.11) means that if one wants to know the global extinction probability of (ξ; )- superprocesses, we only need to calculate the r.h.s of (1.11). However, it is not so easy to know its exact value. In the next section, we will give an example to show how to make it easier in calculating the r.h.s. of (1.11) in some cases. The proofs of Theorem 1.1 and Corollary 1.1 will be given in Section 3. 2 An example Recall that A denotes the weak generator of ξ. In this section, we will refer the (ξ; )- superprocess as (A; )-superprocess for convenience. Let us consider a special case of 2 + (x; f) = b(x)f(x) + c(x)f(x) with b 2 C(E); c 2 C (E). We further assume that infx2E c(x) > 0 and 1=c 2 D(A). Define 1 Xc(dx) = X (dx); t ≥ 0: t c(x) t c c c Then by Corollary 3.6 in Delmas and H´enard(2011), fXt : t ≥ 0g is a (A ; )- superprocess under Pµ , where 1 c(x; f) = b(x) − c(x)A f(x) + f(x)2 =: ~b(x)f(x) + f(x)2 c c u and A u = c(A(u=c) − A(1=c)u) for u 2 Dc(L) := fu 2 C(E): c 2 D(A)g: Assume ~b 2 D(Ac). Define q b − ~b ~ ~ 2 c~ 0 b0 = sup max b(x); (b(x) − 2A b(x)) _ 0 and q = : x2E 2 Then q 2 D(Ac)+ and (·; q) − Acq 2 B+(E). Furthermore, we have c 2 q(x; f) = b0f(x) + f(x) : Recall τ0 = infft ≥ 0 : Xt(1) = 0g. ~ c Proposition 2.1 Assume that infx2E c(x) > 0 and 1=c 2 D(A) and b 2 D(A ). Then c Z 1 q c c 2 Pµ [τ0 < 1] = Pµc exp −µ (q) + dsXs(A q − b0q − q ) ; (2.1) 0 c 1 where µ (dx) = c(x) µ(dx): c Proof. Noting that q is a homogeneous branching mechanism and b0 > 0, we have c q Pµ [τ0 < 1] = 1 c c for any µ 2 M(E). Meanwhile, we also note that τ0 := infft ≥ 0 : Xt (1) = 0g = τ0 and c c c fXt : t ≥ 0g is a (A ; )-superprocess under Pµ . Thus c Pµ [τ0 < 1] = Pµc [τ0 < 1]: Then the desired result follows from (1.11). 4 c Remark 2.1 Since q is a spatially homogeneous branching mechanism, it will be much easier to calculate the r.h.s of (2.1) whose value is just the global extinction probability of (A; )-superprocess.
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