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. Finally, we mention that Dynkin (1991) obtains a necessary and sufficient condition for a given set A to be polar for {suppXt : t > 0} in the case where ( belongs to a certain class of diffusions on Rd, including the case where C is a Brownian motion. This is done by showing that the problem has close connections to certain (answered) questions in the theory of partial differential equations. It would be interesting to know whether there is some similar relationship for transience and recurrence.
2. The Immortal Particle Representation
For the moment, fix p E M(E) and t > 0. Suppose that on some probability space (S, X, Q) we construct a M(E)-valued random variable Y and a E-valued random variable C with a joint distribution specified by the requirements that
(2.1) Q(Y E A)-=P(Xt EA) and
(2.2) Q(( E BJY) = Y(B). Thus Q(Y E A, C E B) = P [1A(Xt)Xt(B)/Xt(E)] -= (E)-yP'O[1A(Xt)Xt(B)]. The joint distribution of Y and C is therefore the normalised Campbell measure associated with the random measure Xt under P.
Notation. Given v E M(E), set iv = (v, 1)-1v.
Note from Proposition 2.7 of Fitzsimmons (1988) that
Q(( E B) = y(E)-'P'[Xt(B)] = P'(C E B). 5 Fortunately, the description of the Campbell measure that we need has been worked out in §4 of Dawson and Perkins (1991), where there is also a remark (see after Theorem 4.1.7) pointing out some of the relevance of this description to the study of the process conditioned on non-extinction. In Dawson and Perkins (1991) the representation of the Campbell measure of the associated historical process was given under conditions on C that are somewhat stronger than those with which we are working. These more restrictive conditions are only used for the construction and regularity of the historical process, and it is easy to check that the same arguments carry through for the Campbell measure of the superprocess itself under our conditions. We will briefly recall the relevant details. Note that our notation is somewhat different to that in Dawson and Perkins (1991). There is a kernel IIt(x, A), x E E, A E M(E), such that
= PI[exp(-(Xt I f))] Qt(, dv)exp(-(v,f)) = exp (-l fV))Vt - exp(-Jl(dx) J 11(x, dv)[1 - exp(-(v, f))])
for f E bp£. In other words, Xt under PO has the same distribution as the random measure f L(dv)v, where L is a Poisson random measure on M(E) with intensity given by D .-. f p(dx)IIt(x,D). For the moment, fix a right continuous function z : [0, t] -- E. Let N* be a Poisson random measure on [0, t] x M(E) with intensity given by C x D |dsflt-.(z(s),D).
Write Q, for the distribution of the random measure Jjo,t] fJ(E) N(ds, dv)v. Thus Q- has the Laplace functional
J Qz(dv) exp(-(v, f)) = exp(- (J ds lIt-a(z(s), dv))[1 - exp(- (v, f)]) = exp(-| ds Vt-.f(z(s))) for f E bpE. 6 Given v E M(E) and x E E, construct on our generic probability space (, 7,Q) a M(E)-valued random variable X' with the same distribution as Xt under PI' and an independent M(E) x E-valued random variable (Y', C') with distribution
Q(Y' E A, (C E B) = P [Q^tt(A)1B(6t)],
where Ct ={(: s E [O,t]}. Let
Rt((v,x), A x B) = Q(X' + Y' E A, C' E B).
Thus
(2.3) J Rt ((v, x), (dp, dy)) [exp(-(p, f))g(y)] =exp(-(V, Vtf))Pz[exp(-| dsVt ff(C))g(Ct)] [O,t] for f, g E bpE. The key fact that we require from Dawson and Perkins (1991) is that
(2.4) Q(Y E A, C E B) = jj(dx)Rt((M,,x),A x B).
To complete our preparation, we recall the following two results giving sufficent con- ditions for a function of a Markov process to be also Markov. The first is a well-known criterion due to Dynkin and the second is Theorem 2 of Rogers and Pitman (1981). We remind the reader that concatenations of kernels should be read from left to right.
Lemma 2.5. Consider two measurable spaces F and G and a Markov process Z with state space F and transition semigroup (St). Given a measurable function 6: F -+ G, let A be the Markov kernel from F to G induced by 6. That is, f A(x, dy)h(y) = h(6(x)). Suppose that StA = ATt for some Markov semigroup (Tt) on G. Then S o Z is a Markov process with transition semigroup (TO).
7 Lemma 2.6. Consider two measurable spaces F and G and a Markov process Z with state space F and transition semigroup (St). Let r be the Markov kernel from F to G induced by a measurable function y : F -+ G, and let A be a Markov kernel from G to F. Suppose that:
i) the kernel Ar is the identity kernel on G; ii) for each t > 0, the Markov kernel Tt = AStI from G to G satisfies the identity ASt=TtA; iii) the process Z has initial distribution A(y, ) for some y E G. Then 7y o Z is a Markov process with initial state y and transition semigroup (Tt).
The promised representation is contained in the following two theorems.
Notation. If ,B is a point in some measurable space, write e for the unit point mass at
Theorem 2.7. The collection (Rt)t>o is a Borel measurable Markov semigroup. Let {(Yt, (t) : t > 0} be a Markov process with initial distribution7r and transition semigroup (Rt).
i) The process {(t : t > 0} is Markov with semigroup (Pt). ii) Suppose that 7r = e1A x /i for some i E M(E). Then {Yt : t > 0} is Markov with semigroup (Qi).
Proof. The relevant measurability properties of (Rt) are checked in Dawson and Perkins (1991). In order to verify that the Chapman-Kolmogorov equations hold for (Rt), it will suffice to show that RaRtHf,g = Rs+tHf,g for all s, t > 0 and f, g E bpE, where Hf,g(v, x) = exp(-(v, f))g(x). As VaVb = Va.b for all a, b > 0, we have from (2.3) and the 8 Markov property of C that R.RtiHf,g(v, x) = exp(- (v, V.+tf)) x P" [exp(- [|S du V_s-utf(Cu))P1- [exp(-|[lt du Vt-uf(Cu))g(Ct)]] exp(- (v, V.+tf)) x P'r[exp(-| du V,a+t-uf(Cu)) exp(-| du Vs+tl_uf(Cu))g(Cs+t)] Ra+tHf,g(v,x). Claim (i) is clear from the construction of (Rt) and Lemma 2.5. We will verify claim (ii) using Lemma 2.6. Let r1 denote the Markov kernel induced by the projection from M(E) x E onto M(E) and let A' denote the Markov kernel from M(E) to M(E) x E given by A'(v,.) = ev x v. Clearly, A'T' is the identity kernel on M(E). From (2.1) and (2.4) we have that Qt = A'RtrF. Moreover, from (2.2) and (2.4) we have that A'Rt = QtA'. Claim (ii) now follows from Lemma 2.6. 0
Theorem 2.8. Let {Xt: t > 0} be a Markov process on M(E) with transition semigroup (Qt), and let {(Yt', c:) t > 0} be an independent Markov process on M(E) x E with transition semigroup (Rt). Then {(Xt' + Yt', t) : t > 0} is a Markov process on M(E) x E with transition semigroup (Rt).
Proof. This follows immediately from (1.2), (2.3) and Lemma (2.5). 0
Using Theorems 2.7 and 2.8, we can show that (Rt) is the transition semigroup of a right process.
Theorem 2.9. There is a right Markov process (Y, () = (E, 'K, t, (Yt(t, It), Q(v'z)) on Mo(E) x E with transition semigroup (Rt).
Proof. Let us begin by showing that for any (v, x) E M(E) x E there is a right continuous Markov process with initial state (v, x) and transition semigroup (Rt). From Theorem 2.8 and the right continuity of the paths of X, it suffices to consider the case when v = 0. 9 From Theorem 2.7 and the right continuity of the paths of X and (, we see that there is a right continuous Markov process with initial state (e.,x) and transition semigroup (Rt). Again applying Theorem 2.8 and the fact that the extinction time of X under P1ED has an absolutely continuous distribution supported on the whole of [0, oo4, it is follows that there is a Markov process with initial state (0, x) and transition semigroup (Rt) such that the M(E)-valued component is continuous on ]O, oo[ and the E-valued component is continuous on [0, oo[. To finish the proof of right continuity, it suffices to note that the total mass of the M(E)-valued component of the process with initial state (0, x) and transition semigroup (Rt) has the same finite dimensional distributions as the square of a 4-dimensional Bessel process with initial state 0 (cf. Evans (1992)). As Mo(E) xE is Lusin and (Rt) is Borel measurable, the proof can now be completed in the same manner as the proof of Theorem 2.17 in Fitzsimmons (1988). To avoid introducing the extra machinery of Ray-Knight compactifications, we will only consider the case when E is compact. The general case then follows in exactly the same way as in Fitzsimmons (1988). Following Fitzsimmons (1988), we need to verify that if {(Yt, (t) : t > 0} is a right continuous Markov process with transition semigroup (Rt) and arbitrary initial state (v',x') and H is a continuous function on M0(E) x E that vanishes at infinity, then s '-4 Rt-..(Y., C.) is almost surely right continuous on [0, t[ for each t > 0. It suffices to consider the case H = Hf,g, where Hf,g(v,x) = exp(-(v,f))g(x) for nonnegative, continuous functions f and g on E. We know from Fitzsimmons (1988) that s i-4 exp(-(Xq,Vt_,f)) is Pr-almost surely right continuous on [0, t[ for any v E M(E), and an argument similar to the one above using Theorems 2.7 and 2.8 gives that s i-4 exp(-(Y8,Vt.j.f)) is almost surely right continuous on [0, t[. So, from (2.3), it suffices to check that s ~-. h((C, s) is almost surely right continuous on [0, t[, where h(x, s) PX[exp(-J_0,t-8] du Vt-s-uf(tu))g(Ct_s)] for x E E and s E [0,t[. Let (C', T') denote the process with state space E x [0, t[ that is obtained by taking the Cartesian product (see, for example, §15 of Sharpe (1988)) of C with uniform motion to the right on [O,t[ killed on exiting [O,t[. What we wish to show is equivalent to proving that s + h(C', 'r) is almost surely right continuous when the initial state of (C', r') is (x', 0). 10 As (C', r') is a right process (see, for example, Theorem 15.2 of Sharpe (1988)), it suffices to show that lIhIl - h is an excessive function for ((',r'), where Ihlh sup,,8 h(x,s). Let (P') be the transition semigroup of (c', '). For r < t - s we have P,'h(x, s) = P[exp(- du Vt_8_uf(Cu))g(Ct-8)]j Ir,t-45] while for r > t -s we have P,'h(x,s) = 0, and it is clear that llhll - h is, indeed, excessive. 0
Proof of Theorem 1.4. Let (Y, () be the right Markov process described in Theorem 2.9. By Theorem 2.7, it suffices to show that if C E nt>o a{(Y., cJ) : S> t}, then QES XIS(C) is either 0 or 1. Observe that QPXJL(C) = Jim QE (XI(CI1u) = lim Q(YU,CU)(q!E C) QeXJuxalmost surely. Now OuC E nt>o a{(Y.,C.) s > t} for all u > 0, and hence, from Theorem 2.8 and the fact that X eventually dies out, we have that Q(V,z)(OuC) - Q(OX)(quC) for all (v, x) E M(E) x E. Thus
Q COxP(C) = lim Q(O°Cu)(OuC) E fl {(C.) $ > t}, t>o and the result follows from Theorem 2.7. 0
3. The Path-Valued Process Representation
Let us begin by recalling the path-valued process representation of the (,-A2/2)- superprocess given in LeGall (1991). Suppose now that E is a Polish space with metric d. Moreover, suppose that ( has continuous paths and satisfies the condition that there is an integer k > 3 and constants C, e > 0 such that for every x E E and t > 0 we have
P s[Sup d(x, .)k] . Ct2+. O<8
6((f,/),(f',/3')) = sup d(f(s),f'(s)) + 1/3- P,1, a8> then (W, 6) is a Polish space. For wo = (fo,/3o) E W, a E [0,Po] and b E [a,oo[, let Q(a,b,fo;-) denote the proba- bility measure on W defined by the following requirements: i) / = b, Q(a, b, fo; )-a.s.; ii) f(s) = fo(s), Vs E [O, a], Q(a, b, fo; )-a.s.; iii) the distribution of {f(s) : s > a} under Q(a, b, fo; ) coincides with the distribution of {'(.-a)A(b-a): s > a} under pfo(a). Define a kernel Qit(w, dw') on W as follows. Fix w = (f, /) E W and t > 0. Let {B.} be a reflecting one-dimensional Brownian motion started at /3 that is independent of 6, and write m = info<,
Theorem 3.1. For any wo = (fo,/3o) E W there is a continuous, timne-homogeneous strong Markov process {Zt : t > 0} defined on some probability space (E, ', Q) such that Zo = wo and Q[F(Zt)1{Z. : 0 < u < s}] = JQt-.(Za. dw)F(w) for all 0 < s < t and every nonnegative measurable function F on W. Set (ft, /3t) = Zt. The process {/3t : t > 0} is a one-dimensional reflecting Brownian motion. The conditional distzibution of {ft : t > 01 given {/3,, : u > 0} is that of a 12 time-inhomogeneous Markov process with
Q[F(ft)I{fu:: < u
for all 0 < s < t and every nonnegative measurable function G, where mr,t = inf Theorem 3.2. Fix x E E and p > 0. Suppose, in the notation of Theorem 3.1, that Zo = wo is the trivial path with fo = 0 and fo(O) = x. Let t e tf denote the local time of :$t t > 0} at a > 0. Set u=- inf{t > oe0 > 4p}. The measure valued process {X. : s > 0} obtained by setting (X.) h) 4 ][O,] det h(ft(s)) is continuous and has the same distzibution as {X. : s > 0} under PPEz. We will now use use these results to obtain a similar representation of X. Suppose from now on that we axe in the set-up described in Theorem 3.2 for some x E E and p > 0. Fix T > 0. Observe that the distribution of {X. : s > 0} conditional on XT # 0 is the same as the distribution of {X, : s > 0} conditional on supo<*<,3P. > T. Recall the excursion theory construction of the reflecting, one-dimensional Brownian motion {fJ. : 0 < s < a} in terms of a Poisson random measure, N, on [0, 4p] x U, where U is the space of continuous excursions from 0 into [0, oo[ (see §VI.8 of Rogers and Williams (1987) or Chap. XII of Revuz and Yor (1991) for comprehensive accounts of the excursion theory we will need). The random measure N has a-finite intensity A x n, where A is Lebesgue measure on [0, 4p] and n is the Ito excursion law. Let UT denote the subset of U consisting of excursions that reach a level greater than T. Define two measures n' and n4 on U by setting nT = nT(fnUT) and n4 = n-ny. Let NT be a Poisson random measure on [0, 4p] x U with intensity A x n4 that is conditioned to be non-zero, and let N4T be an independent Poisson random measure on [0, 4p] x U with intensity A x n4. If we construct a process {f,T,. : 0 < s < a} from NT + NT in the same 13 manner that { 0 < s < a} is constructed from N, then {/3T : 0 < s < a} has the same distribution as {/: < s < a} conditional on supo<.<0,/3 > T. From a variant of Williams' path decomposition of the Brownian excursion law (see, for example, Theorem XII.4.5 in Revuz and Yor (1991) and the remarks preceeding it) we know that n(UT) = T1, and hence x = 4pT-1 exp(-4pT-1) Q(NT'([0°4p] U) 1) 1 -exp(-4pT-1) Note that the right hand side converges to 1 as T --+ oo. Moreover, the probability measure n'(U)'4n' has the following description. Let C+ and C- be two independent 3-dimensional Bessel processes starting from 0, and let M be an independent real random variable with distribution Q(M > y) = y/T, y > T. Set r+ = inf{t: Ct+ = M} and r= inf{t: Ct = M}. Then nT(U)-1n' is the distribution of process {Dt: t > O} defined by [C+, if O Theorem 3.3. On some probabi1ity space (E, 7, Q) there is a pair of continuous, W- valued stochastic processes {Z+ = (ft+,/t+) t > 0} and {Zj = (f-,/fl) t > 0} with the following properties. i) The processes {3t+ : t > 0} and {/ t > 0} are independent 3-dimensional Bessel processes started at 0. ii) Letting wo = (fo, P3o) denote the trivial path with ,Bo = 0 and fo(O) = x, Z+ =Z0- = wo. iii) For each S > 0, let y+(S) = sup{t : 6t+ = S} and y(S) = sup{t : / = S}. Then Z4+(s) = Z-(S) almost surely. Moreover, if we define {ZtS = (fts,/ts) : t > 0} by (Zr+ if0 Q[F(Z+)lI{Z+: 0 < U < s}] = JQt-.(Z,+ dw)F(w), where the kernels (Qt) are defined in the same manner as (Qt), with the exception that reflecting one-dimensional Brownian motion is replaced by the 3-dimensional Bessel pro- cess. Theorem 3.4. Let Z be as im Theorem 3.2 and let (Z+,Z-) be as in Theorem 3.3. Suppose that Z and (Z+,Z-) are independent. Let t et+a (respectively, t t a) denote the local time of {P+: t > 0} (respectively, { :>t O) at a > 0.The measure valued process {X S > 0} obtained by setting (Xs,h) = de h(ft(s))+ dt+t h(fti(s)) + det' h(fr (s))] 4 J[O,a] J[0o,[ J[o'o[ is continuous and has the same distribution as {X. : s > 0} under P r. Remarks. i) Using the technique in the proof of Theorem 1.1 in LeGall (1991), it is possible to show that Z+ and Z- are actually strong Markov. ii) If (Y, () is the process described in Theorem 2.9, then the process {.X. : s > 0} obtained by setting (X, h) =[J 8d4' h(f+(s)) + ide-' h(ft (s))] has the same distribution as {Y. : s > 0} under Q(O°x). The decomposition XJ = X"S + XS is a special case of the decomposition of Theorem 2.8. 15 iii) Noting the construction in §2.5 of Aldous (1991) of the self-similar continuum random tree in terms of two independent 3-dimensional Bessel processes, the conclusion of Theorem 3.4 is closely related to the heuristic description in §5.1 in Aldous (1991) of (Y, () under Q(O1x) as a Markov process indexed by the self-similar continuum random tree. The following necessary and sufficient condition for a set to be "transient" for the closed support of X now follows immediately from arguments similar to those in the proof of Proposition 2.2 of LeGall (1991) and the fact that the 3-dimensional Bessel process is transient. Corollary 3.5. For all A E E, Pz(f{3r> s: suppXrlnA#0 5>0 =Q(fl [{3t > u: ft+(+) E A} U {3t > u: ft(Pt) e A}]) In particular, X (fl{3r >8 :sUppXr nfA #00}) 8>0 is zero if and only if fn3t > u:f+(#+)EA}) ts>0 is zero. A straightforward calculation shows that if C is Brownian motion in Rd and A is a bounded, open set, then Q[ O I dtlA(ft+(ft))] is infinite or finite depending on whether d < 4 or d > 4. It is tempting to conjecture that, for any p E M(Rd), P-(fl>0{3r > s: suppXr n B # 0}) is either 1 or 0 depending on whether d < 4 or d > 4. Of course, this probability is 1 when d < 2, as a simple argument using the representation in §2 shows; but, unfortunately, we are unable to provide a proof of the remainder of this conjecture. 16 References D. Aldous (1991). The continuum random tree II: an overview. In Stochastic Analysis (M.T. Barlow and N.H. Bingham, eds.). Cambridge University Press. D.A. Dawson (1991). Measure- Valued Markov Processes. To appear in Lecture Notes in Mathematics. Springer-Verlag. D.A. Dawson and E.A. Perkins (1991). Historical Processes. Mem. Amer. Math. Soc. 93, No. 454. E.B. Dynkin (1989). Regular transition functions and regular superprocesses. Trans. Amer. Math. 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