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Two Representations of a Conditioned Superprocess

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Two Representations of a Conditioned Superprocess Two Representations of a Conditioned Superprocess By Steven N. Evans* Technical Report No. 335 February 1992 *Research supported in part by a Presidential Young Investigator Award Department of Statistics University of California Berkeley, California 94720 Two representations of a conditioned superprocess Steven N. Evans* Department of Statistics University of California at Berkeley 367 Evans Hall Berkeley CA 94720 Abstract: We consider a class of measure-valued Markov processes that are constructed by taking a superprocess over some underlying Markov process and conditioning it to stay alive forever. We obtain two representations of such a process. The first representation is in terms of an "immortal particle" that moves around according to the underlying Markov process and throws of pieces of mass, which then proceed to evolve in the same way that mass evolves for the unconditioned superprocess. As a consequence of this representation, we show that the tail a-field of the conditioned superprocess is trivial if the tail a-field of the underlying process is trivial. The second representation is analogous to one obtained by LeGall in the unconditioned case. It represents the conditioned superprocess in terms of a certain process taking values in the path space of the underlying process. This repre- sentation is useful for studying the "transience" and "recurrence" properties of the closed support process. In particular, we find some evidence for the conjecture that the closed support of conditioned super Brownian motion is "transient" in more than 4 dimensions and "recurrent" otherwise. American Mathematical Society 1980 subject classifications: Primary 60G57, 60J80. Secondary 60F20, 60J25. Keywords and Phrases: superprocess, measure-valued, diffusion, branching process, tail event, Campbell measure, path-valued, transience, recurrence. * Research supported in part by a Presidential Young Investigator Award 1 1. Introduction We begin by recalling a special case of the superprocess construction of Fitzsimmons (1988) (see also Dynkin (1989)). We recommend Dawson (1991) as a general introduction to, and bibliographic survey of, the extensive literature on superprocesses and measure- valued Markov processes in general. In particular, the latter work provides a thorough discussion of the manner in which superprocesses arise as limits of branching particle systems. Suppose that E is a topological Lusin space and E is the Borel a-field of E. Denote by M(E) the class of finite Borel measures on E. Define M(E) to be the a-field of subsets of M(E) generated by the maps p '-+ (i, f)Jf u(dx)f(x) as f runs over bp£, the class of bounded, nonnegativle, E-measurable functions. Let C =(= , Ot,.F, at, PX) be a Borel right Markov process with state space (E, C) and semigroup (Pt). Assume Ptd = 1. For each f E bp£ the integral equation (1.1) vt(X) = Ptf(x) - ,/j P.(x, v2 ) ds has a unique solution, which we denote by (t, x) i-+ Vtf(x); and there exists a Markov semigroup (Qt) on (M(E), M(E)) with Laplace functionals (1.2) J Q(p,dv) exp(-(v,f)) = exp(-(p, Vtf)) for all P E M(E), t > 0 and f E bpE. Write Mo(E) for the set M(E) topologised by the weak topology. That is, give M(E) the weakest topology which makes all the maps p i-i (i, f) continuous, where f runs over the bounded, continuous functions on E. The Borel u-field of Mo(E) is M(E). There is a Markov process X (,7,t,It,Xt,Pi&) with state space (M(E), M(E)) and semigroup (Qt). This process is a right process on Mo(E). In the nomenclature of Fitzsimmons (1988), the process X is called the (C,-A2/2)-superprocess. Starting from any initial measure 1u the process X "dies out" PI-almost surely. That is, Xt is the null measure 0 for all t sufficiently large. We are interested in the process constructed by starting X at an initial measure p E M(E) = M(E)\{0} and conditioning 2 X to stay alive forever. This type of construction was carried out in Roelly-Coppoletta and Rouault (1989) and Evans and Perkins (1990). We will briefly recall the relevant details. If T > t and r is a bounded gt-measurable random variable, then P[rIX. # O, < s < T] = P.[r{1-exp(-4(Xt, 1)(T-t1) }' -H ~~~~ exp-4[(Xt I 1)]T as T - oo. Let M(E) denote the trace of M(E) on M(E). If we set (1.3) QtF(1) = (y, 1)-1Pi[F(Xt)(Xt, 1)] for each bounded M(E)-measurable functions F, then (Qt) is a Markov semigroup that is the transition semigroup of a Markov process X = (_, I, t, It,X,t,pi) with state space (M(E),M(E)). The process X is a right process on M0(E), where Mo(E) is M(E) equipped with relative topology inherited from Mo(E). Besides Roelly-Coppoletta and Rouault (1989) and Evans and Perkins (1990), the process X has also been investigated in Evans (1991, 1992) and Aldous (1991). A noteworthy feature of the work on conditioned superprocesses is that, in some sense, the process X inherits some of the properties of the underlying process 6. For instance, it is shown by analytic arguments in Evans and Perkins (1990) that, if Pt converges in a suitable sense to an invariant probability measure v as t -4 oo, then (XAt, 1)- Xt converges in probability to v as t - oo. As part of his work on continuum random trees, Aldous (1991) offers a somewhat heuristic explanation of such results. He sketches an argument that X under F should be distributed as the sum of two independent measure-valued process. The first component is a copy of X under PIt. Roughly speaking, the second component is produced by taking an "immortal particle" that moves around as a copy of 6 under PA, where /i = (t, 1)-,s, and throws off pieces of mass that continue to evolve according to the dynamics under which mass evolves for X. Our first aim here is to give a rigorous treatment of this representation. We do this in §2. As a consequence, it would be possible to obtain more probabilistic proofs of some existing results along the lines set out in Aldous (1991). Rather than do this, we prove the following result as an example of the usefulness of this representation. 3 Theorem 1.4. Suppose that p E M(E). Ifthe tail a-field nt,o au{, : s > t} is PA-trivial, then the tail a-field nt>oa{X S > t} is Pt-trial. The most obvious question to ask about the tail of X is whether a given set A E E is intersected by the closed support of Xt for arbitrarily large values of t. Because of the rather pathological behaviour of the support when the sample paths of C contain jumps (see, for example, Perkins (1990) or Evans and Perkins (1991)), this question is only of interest in the case of those g with sample paths that are sufficiently continuous so that the closed support supp Xt is a compact set for all t > 0. As yet, a necessary and sufficient condition on C for suppXt, t > 0, to be compact is not known. LeGall (1991) gives a sufficient condition. Brownian motion in Rd for any d > 1 is an example of a C with this compact support property. We can think of the study of the probability of the event nt>0{13 > t: supp XJ f A # 0} as being about "recurrence" or "transience" of the set A for the closed support process {suppXt : t > 0}. LeGall (1991) considers the analogously defined question of whether or not a set A is "polar" for {suppXt : t > 0}; that is, whether or not the probability of the event {3t > 0 : suppXt n A # 0} is zero. As a first step in that investigation, a construction of X is given in terms of a Markov process Z taking values in the space of continuous E-valued stopped paths. (An E-valued stopped path is a pair w = (f, P), where ,B 2 0 and f : [0, oo[-- E is a continuous function such that f(s) = f(3) for every s > P.) Letting (ft,8t) = Zt, it is then shown that the probability of the event in question is also the probability of the event {3t > O: ft(/t) E A}. The latter probability can then be studied with the tools of probabilistic potential theory. We remark that the path-valued process construction of X is of independent interest and has applications other than the one we have just outlined. We carry out the analogue of this programme in §3. Firstly, we obtain a similar construction of X in terms of two identically distributed, dependent path-valued Markov processes Z+ and Z- and an independent copy of the process Z. Letting (f+, P+) - Z+, we then show that the probability of the event nt>ol{3s > t suppX,. n A 0 0} is zero if and only if the probability of the event ntfl{3s > t f:(fl:) E A} is zero. 4 Unfortunately, we are as yet unable to apply this criterion in the simplest case - when ( is Brownian motion in Rd and A is a bounded open set. However, this approach suggests the conjecture that in this case the probability starting from any initial state of the event nflo>{3s > t : suppX,. n A :# 0} is either 1 of 0 depending on whether d < 4 or d > 4.
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