LONG-TIME ASYMPTOTIC OF STABLE DAWSON-WATANABE PROCESSES IN SUPERCRITICAL REGIMES?

Khoa L?E

Department of Mathematical and Statistical Sciences,University of Alberta,632 Central Academic,Edmonton,AB T6G 2R3,Canada

Department of Mathematics,South Kensington Campus,Imperial College London,London,SW7 2AZ,United Kingdom

E-mail:n.le@imperial.ac.uk

Abstract Let W=(Wt)t≥0be a supercritical α-stable Dawson-Watanabe process(with α ∈ (0,2])and f be a test function in the domain of?(??)α2satisfying some integrability condition.Assuming the initial measure W0has a fi nite positive moment,we determine the long-time asymptotic of arbitrary order of Wt(f).In particular,it is shown that the local behavior of Wtin long-time is completely determined by the asymptotic of the total mass Wt(1),a global characteristic.

Key words Dawson-Watanabe process;α-stable process

1 Introduction

Let W=(Wt,t≥0)be a Dawson-Watanabe process starting from a fi nite measure m with motion generator ?(??)α2on Rd(α ∈ (0,2])and linear growth β >0.More precisely,W is a measure-valued Markov process such that the process

is a martingale with quadratic variationThe law of W is denoted by Pm.Throughout the paper,we assume that the initial measure m has a fi nite positive moment,that is

The case when β >0 is known as supercritical branching regime.The cases β <0 and β=0 are known respectively as subcritical and critical branching regimes which,however,are not considered in the current article.For a fi xed test function f with sufficient regularity and integrability,we investigate the long-time asymptotic of Wt(f)in supercritical branching regimes.To state the main result precisely,we prepare some notation.For each multi-index k=(k1,···,kd)∈ Ndand x=(x1,x2,···,xd)∈ Rd,we denote

Herein,D(?(??)α2)denotes the domain of the weak generator of the α-stable process(see the following section for a precise de fi nition).Theorem 1.1 extends results of Kouritzin and Ren[5]in which the fi rst order asymptotic(N=0)was identi fi ed.For a heuristic explanation of long-time limits of supercritical superprocesses and their connection with strong laws of large numbers,we refer to[4,Section 2]and[3,Subsection 2.2].

The higher order asymptotic expansions(1.5)and(1.6)are obtained by combining the method initiated by Asmussen and Hering[2]and an asymptotic expansion of the α-stable semigroup(see Proposition 4.1 below).When ?(??)α2is replaced by the generator of an Ornstein-Uhlenbeck process,similar results have been obtained by Adamczak and Mi lo′s[1].In such case,because of the exponential rates in the expansion of the Ornstein-Uhlenbeck semigroup,convergences in distribution are expected in the asymptotic of high orders(which are called central limit theorems).On the other hand,the rates in the asymptotic expansion of the α-stable semigroup are those of polynomials(see(4.1)below)and are negligible under the exponential growing expected total mass Wt(1),which leads to almost sure limits in the asymptotic of all high orders.In view of Theorem 1.1,it is interesting to observe that the local behavior of Wtin long time is completely determined by the asymptotic of the total mass Wt(1),which is a global characteristic.

We conclude the introduction with an outline of the article.Section 2 reviews the martingale formulations of Dawson-Watanabe processes.In Section 3,we investigate the long-time asymptotic of Wtagainst some special test functions.The proof of Theorem 1.1 is presented in Section 4.

2 Martingale Formulations

We use ν(f)and hf,νi to denoteRRdfdν for a measure ν and an integrable function f.Let Ttbe the semigroup corresponding to a symmetric α-stable process acting on bE(Rd),the space of bounded Borel measurable functions on Rd.In particular,for every f∈bE(Rd),

which is called the Green function representation.From(2.6),one derives the following two important identities

and

which are valid for all 0≤s≤t and f∈bE(Rd).The following estimate is intrinsic to supercritical regimes and plays a central role in our approach.

3 Characteristic Martingales

For every x,θ ∈ Rd,we denote eθ(x)=eiθ·x,cosθ(x)=cos(θ·x)and sinθ(x)=sin(θ·x)and recall assumption(1.2)on m and the de fi nition ofin(1.3).We investigate the long-time asymptotic of

Note that for every x∈Rd

Finally,combining(3.5),(3.6),(3.7)and(3.8)yields

Equality(3.3)follows from the above relation and(3.2),after observing that Xρ(t)(?kpt?ρ(t))is a real number. ?

4 Proof of the Main Result

We begin with an asymptotic expansion of Ttas t → ∞.If k=(k1,···,kd) ∈ Ndis a multi-index and f is a sufficiently smooth test function,we de fi nefollowing semigroup expansion is proved in[3,Proposition 3.2].

Proposition 4.1(Semigroup expansion) Let f be a measurable function on Rdand N be a non-negative integer such that(1.4)holds.Then,we have

Remark 4.2For semigroups with discrete spectra such as the Ornstein-Uhlenbeck semigroup,similar asymptotic expansions can be obtained via spectral decompositions.Although the α-stable semigroup does not belong to this class,such expansion can be obtained using Taylor’s expansion.We refer to[3]for a proof of the above result.

Set tn=nδfor some δ∈ (0,1)sufficiently small so that

We fi rst show that the sequence{tn}ndetermines the long-time asymptotic of.

which converges to 0 because of the range of δ in(4.2).In addition,

Fixing ε>0 and applying martingale maximal inequality as well as(2.5)and(2.8),we have

As in the proof of Lemma 2.1,an application of Jensen’s inequality gives

Applying Lemma 4.3,we fi nd that the above limit implies(1.5). ?

Acknowledgements The author thanks PIMS for its support through the Postdoctoral Training Centre in Stochastics during the completion of the paper.