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Immigration Structures Associated with Dawson-Watanabe Superprocesses

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Immigration Structures Associated with Dawson-Watanabe Superprocesses View metadata, citation and similar papers at core.ac.uk brought to you COREby provided by Elsevier - Publisher Connector stochastic processes and their applications ELSEVIER StochasticProcesses and their Applications 62 (1996) 73 86 Immigration structures associated with Dawson-Watanabe superprocesses Zeng-Hu Li Department of Mathematics, Beijing Normal University, Be!ring 100875, Peoples Republic of China Received March 1995: revised November 1995 Abstract The immigration structure associated with a measure-valued branching process may be described by a skew convolution semigroup. For the special type of measure-valued branching process, the Dawson Watanabe superprocess, we show that a skew convolution semigroup corresponds uniquely to an infinitely divisible probability measure on the space of entrance laws for the underlying process. An immigration process associated with a Borel right superpro- cess does not always have a right continuous realization, but it can always be obtained by transformation from a Borel right one in an enlarged state space. Keywords: Dawson-Watanabe superprocess; Skew convolution semigroup; Entrance law; Immigration process; Borel right process AMS classifications: Primary 60J80; Secondary 60G57, 60J50, 60J40 I. Introduction Let E be a Lusin topological space, i.e., a homeomorphism of a Borel subset of a compact metric space, with the Borel a-algebra .N(E). Let B(E) denote the set of all bounded ~(E)-measurable functions on E and B(E) + the subspace of B(E) compris- ing non-negative elements. Denote by M(E) the space of finite measures on (E, ~(E)) equipped with the topology of weak convergence. Forf e B(E) and # e M(E), write ~l(f) for ~Efd/~. Suppose that ~ = (~?,,N, St, ~,,Px) is a Borel right process in E with semigroup (Pt)t ~ o and 4> is a branching mechanism given by 4)(x,z)=b(x)z +c(x)z 2+ (e .... 1 + zu)m(x, du), xeE, z >~O, (1.1) Research supported by the National Education Committee of China. 0304-4149/96/$15.00 ~" 1996 Elsevier Science B.V. All rights reserved SSDI 0304-4149(95)00087-9 74 Z-H. Li / Stochastic Processes and their Applications 62 (1996) 73-86 where b • B(E), c • B(E) + and [u A u 2 ] m(x, du) is a bounded kernel from E to (0, oo). From a general construction in Fitzsimmons (1988, 1992) we have that for each f• B(E) + the evolution equation v,f(x)+ fldsf 4)(y,VJ(y))P, ~(x, dy)=Ptf(x), t>~O, xeE, (1.2) has a unique solution Vtf• B(E) +, and there is a Markov semigroup (Qt)t~>o on M(E) such that f~t e-"lS)Qt(/t, dv ) = exp{ -/t(V,f)}, t >~ O, ll M(E). (1.3) (E) Furthermore, (Q,),~o is the transition semigroup of a Borel right process X = (W, ~, ~, Xt, Q,). The process X is called a Dawson-Watanabe superprocess with parameters (~, qS), or simply a (~, c~)-superprocess. The (~, ~b)-superprocess is a special form of the measure-valued branching process (MB-process), which is a kind of measure-valued Markov process with transition semi- group satisfying the branching property (1.3) with Vt fdefined by Vtf(x)= -log[ e-~J)@(bx, dv), t>~O,x•E, (1.4) JM ~E) where 6x denotes the unit mass at x • E. See e.g. Dawson (1993) and Watanabe (1968). An MB-process X is the mathematical model for the evolution of a population in some region whose growth and decay is subject to the law of chance. If we consider a situation where there are some additional sources of population from which immigration into the region occurs during the evolution, we need to introduce a measure-valued immigration process. Several authors have constructed measure- valued immigration processes under different hypotheses; see e.g. Dynkin (1991), Gorostiza et al. (1990), Li (1992) and Shiga (1990). As observed in Li (1995), a special type of immigration associated with the MB-process may be described by a flow of probability measures that solves an equation with a kind of skew product: Let (Nt)t >1o be probability measures on M(E). We call (Nt)t >~o a skew convolution semigroup associated with X or (Qt)t >1o if Nr+,=(NcQ,),N,, r,t >~O, (1.5) where "," denotes the convolution operation. The relation (1.5) holds if and only if QN(#,'):=Q,(I~,')*N,, t >~O,t~eM(E), (1.6) defines a Markov semigroup (Q~)~/> o on M(E). In view of (1.6), if Y is a Markov process in M(E) having transition semigroup (Q,)tN >~o, we call it an immigration process associated with X. It was proved in Li (1995) that the family of probability measures (Nt),/> o is a skew convolution semigroup associated with (Q,), >~o if and only if there is an infinitely divisible probability entrance law (Kt)t > o for (Qt), ~> o such that logf~,,e)e-"~Z'Nt(dv) = f] [logfM~)e-"~z)K~(dv)]ds (1.7) Z-H. Li / Stochastic Processes and their Applications 62 (/996) 73 86 75 for all t>~ 0 and feB(E) +. Therefore, the immigration structures associated with an M B-process may be characterized by its infinitely divisible probability entrance laws. The purpose of this paper is to describe the set of infinitely divisible probability entrance laws for the (~, ~b)-superprocess and to discuss the regularities of the corres- ponding immigration processes. By characterizing all those entrance laws we find some immigration processes which have not been studied in the literature. An example given at the end of the paper shows that an immigration process associated with the Borel right (~, 4))-superprocess may have no right continuous realization. The general theory of Markov processes developed in Sharpe (1988) provides important tools for the study. In Section 2 we prove a 1-1 correspondence between minimal probability entrance laws for the (~, ~b)-superprocess and entrance laws for the underlying process ~, using an argument of lifting and projecting adapted from Dynkin (1989). In Section 3 we show that each infinitely divisible probability entrance law for the superprocess is determined uniquely by an infinitely divisible probability measure on the space of entrance laws for the underlying process. In Section 4 we study the basic regularities of the immigration processes. If the infinitely divisible probability entrance law for the superprocess can be closed by a probability measure on M(E), it yields a Borel right immigration process. In Section 5 it is shown that a "good" version of the general immigration process may be obtained by transformation from a Borel right one in an enlarged state space. The problems considered in this paper are similar to those in Li et al. (1993) and Li and Shiga (1995), although the basic hypotheses and formulations are different. In Li et al. (1993) we discussed Feller processes and in Li and Shiga (1995) we were only interested in diffusions. 2. Minimal probability entrance laws Let us consider a change of form of (1.2). Define the semigroup of bounded kernels (P~),~ob on E by P~.f(x) = Pxf(~t) exp{- fl b(~.~)ds}. (2.1) Then (1.2) is equivalent to Vt.f(x) + fl ds fw q~o(Y, Vs f(y))P~ ,~(x,dy)= P~.f(x), (2.2) where ~bo(X,Z) = c(x)z 2 + (e -z" - 1 + zu)m(x, du). (2.3) 76 Z-H. Li / Stochastic Processes and their Applications 62 (1996) 73-86 Note that the first moments of the (~, q~)-superprocess are given by MIE) V(f) Q,(p, dv) =/~(P~ f). (2.4) See e.g. Fitzsimmons (1988). Given a semigroup of bounded kernels (Tt)t ~> o on E, we denote by .;¢'(T) the set of entrance laws x = (x,),>o for (Tt)t ~ o that satisfy flKs(E)ds ,< oc for all t > 0. (2.5) Let Y{ ~(Q) denote the set of probability entrance laws K = (K,)t > o for the semigroup (Qt)t ~ o such that itds(v(E)K~(dv) < Go for all t > 0, (2.6) 3o dM (E) and let Yd~(Q) denote the subset of .~ ~(Q) comprising minimal elements. See e.g. Sharpe (1988) for the definition of an entrance law. Theorem 2.1. There is a one-to-one correspondence between K~ff~(Q) and ~ )ff(pb), which is given by 7,(f) = ( v(f)Kt(dv), (2.7) dM (E) and f~t e-"(f)Kt(dv) = exp{ - Stb(7, f)}, (2.8) (E) where SP(7, f) = 7t(f) -- fo;ds ~bo(y, Vs f(y))Tt-s(dy). (2.9) We omit the proof of the above theorem, which follows from (1.3) and (2.4) by the same argument as Dynkin (1989). To describe the class yg~(Q) we need to clarify a connection between scg(P) and yg(pb). Lemma 2.1. There is a one-to-one correspondence between Ic ~ ~U (P) and 7 ~ •(pb), which is given by 7t = lim tcrP~_, and ~:t = lim 7,P,-,. (2.10) r~O r~O Moreover, if the two entrance laws t¢ and 7 are related by (2.10), then we have e-IJbHtKt(f) <~ 7,(f) ~< ellblltKt(f), t > O, f6 B(E) +. (2.11) Z-H. Li / Stochastic" Processes and their Applications 62 (l 996) 73-86 77 Proof. The assertions follow from the inequalities e-IlhlltPtf ~ P~f <~ ellblltPtf, t >10, feB(E) +. We omit the details. [] For K e .~'(P) we note St(K, f) = Kt(f) -- ds (p(y, V; f(y))Kt s(dy), t > 0, .fie B(E) +.
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