Path-valued branching processes and nonlocal branching superprocesses\thanksrefT1
A family of continuous-state branching processes with immigration are constructed as the solution flow of a stochastic equation system driven by time–space noises. The family can be regarded as an inhomogeneous increasing path-valued branching process with immigration. Two nonlocal branching immigration superprocesses can be defined from the flow. We identify explicitly the branching and immigration mechanisms of those processes. The results provide new perspectives into the tree-valued Markov processes of Aldous and Pitman [Ann. Inst. Henri Poincaré Probab. Stat. 34 (1998) 637–686] and Abraham and Delmas [Ann. Probab. 40 (2012) 1167–1211].
10.1214/12-AOP759 \volume42 \issue1 2014 \firstpage41 \lastpage79
Path-valued processes and superprocesses
T1Supported by NSFC (No. 11131003), 973 Program (No. 2011CB808001) and 985 Program.
class=AMS] \kwd[Primary ]60J80 \kwd60J68 \kwd[; secondary ]60H20 \kwd92D25 Stochastic equation \kwdsolution flow \kwdcontinuous-state branching process \kwdpath-valued branching process \kwdimmigration \kwdnonlocal branching \kwdsuperprocess
Continuous-state branching processes (CB-processes) are positive Markov processes introduced by Jiřina (1958) to model the evolution of large populations of small particles. Continuous-state branching processes with immigration (CBI-processes) are generalizations of them describing the situation where immigrants may come from other sources of particles; see, for example, Kawazu and Watanabe (1971). The law of a CB-process is determined by its branching mechanism , which is a function with the representation
where and are constants, and is a finite measure on . In most cases, we only define the function on , but it can usually be extended to an analytic function on an interval strictly larger than . The branching mechanism is said to be critical, subcritical or supercritical according as , or .
A CB-process can be obtained as the small particle limit of a sequence of discrete Galton–Watson branching processes; see, for example, Lamperti (1967). A genealogical tree is naturally associated with a Galton–Watson process. The genealogical structures of CB-processes were investigated by introducing continuum random trees in the pioneer work of Aldous (1991, 1993), where the quadratic branching mechanism was considered. Continuum random trees corresponding to general branching mechanisms were constructed in Le Gall and Le Jan (1998a, 1998b) and were studied further in Duquesne and Le Gall (2002). By pruning a Galton–Watson tree, Aldous and Pitman (1998) constructed a decreasing tree-valued process. Then they used time-reversal to obtain an increasing tree-valued process starting with the trivial tree. They gave some characterizations of the increasing process up to the ascension time, the first time when the increasing tree becomes infinite.
Tree-valued processes associated with general CB-processes were studied in Abraham and Delmas (2012). By shifting a critical branching mechanism, they defined a family of branching mechanisms , where or for some . Abraham and Delmas (2012) constructed a decreasing tree-valued Markov process by pruning a continuum tree, where the tree has branching mechanism . The explosion time was defined as the smallest negative time when the tree (or the total mass of the corresponding CB-process) is finite. Abraham and Delmas (2012) gave some characterizations of the evolution of the tree after this time under an excursion law. For the quadratic branching mechanism, they obtained explicit expressions for some interesting distributions. Those extend the results of Aldous and Pitman (1998) on Galton–Watson trees in the time-reversed form. The main tool of Abraham and Delmas (2012) was the exploration process of Le Gall and Le Jan (1998a, 1998b) and Duquesne and Le Gall (2002). Some general ways of pruning random trees in discrete and continuous settings were introduced in Abraham, Delmas and He (2012), Abraham, Delmas and Voisin (2010).
In this paper, we study a class of increasing path-valued Markov processes using the techniques of stochastic equations and measure-valued processes developed in recent years. Those path-valued processes are counterparts of the tree-valued processes of Abraham and Delmas (2012). A special case of the model is described as follows. Let or or for some . Let be a continuous function on with the representation
where and is a finite kernel from to . Let be a branching mechanism given by (1). Under an integrability condition, the function
also has the representation (1) with the parameters depending on . Let be the measure on defined by
Let be a white noise on based on the Lebesgue measure, and let be a compensated Poisson random measure on with intensity . Let be a constant. For , we consider the stochastic equation
We shall see that there is a pathwise unique positive càdlàg solution to (1). Then we can talk about the solution flow of the equation system. We prove that each is a CB-process with branching mechanism , and is an inhomogeneous path-valued increasing Markov process with state space , the space of positive càdlàg paths on endowed with the Skorokhod topology.
The formulation of path-valued processes provides new perspectives into the evolution of the random trees of Aldous and Pitman (1998) and Abraham and Delmas (2012). From this formulation we can derive some structural properties of the model that have not been discovered before. For let us define the random measure on . We shall see that is an inhomogeneous increasing superprocess involving a nonlocal branching structure, and the total mass process
is an inhomogeneous CB-process. Then one can think of as a path-valued branching process. On the other hand, for each the random increasing function induces a random measure on such that for . We prove that is a homogeneous superprocess with both local and nonlocal branching structures. We also establish some properties of an excursion law for the superprocess . Given a branching mechanism of the form (1), for a suitable interval we can define a family of branching mechanisms by
where the two terms on the right-hand side are defined using (1). The family can be represented by (2) with . In this case, the path-valued process under the excursion law corresponds to the time-reversal of the tree-valued process of Abraham and Delmas (2012). In general, we may associate with a “forest-valued branching process.”
To make the exploration self-contained, we shall consider a slightly generalized form of the equation system (1) involving some additional immigration structures. In Section 2, we present some preliminary results on inhomogeneous immigration superprocesses and CBI-processes. In Section 3 a class of CBI-processes with predictable immigration rates are constructed as pathwise unique solutions of stochastic integral equations driven by time–space noises. In Section 4 we introduce the path-valued increasing Markov processes and identify them as path-valued branching processes with immigration. A construction of those processes is given in Section 5 using a system of stochastic equations generalizing (1). In Section 6 we derive a homogeneous nonlocal branching immigration superprocess from the flow. The properties of the process under an excursion law are studied in Section 7.
We sometimes write for . Let denote the set of positive right continuous increasing functions on an interval . For a measure and a function on a measurable space we write if the integral exists. Throughout this paper, we make the conventions
for any . Other notations are explained as they first appear.
2 Inhomogeneous immigration superprocesses
In this section, we present some preliminary results on inhomogeneous immigration superprocesses and CBI-processes. Suppose that is an interval, and is a Lusin topological space. Let . A function on is said to be locally bounded if for each compact interval the restriction of to is bounded. Let be the space of finite Borel measures on endowed with the topology of weak convergence. Let be the set of bounded positive Borel functions on . Let denote the set of all functionals on with the representation
where and is a finite measure on . Let denote the set of all functionals on of the form with and . By Theorems 1.35 and 1.37 in Li (2011) one can prove the following:
There is a one-to-one correspondence between functionals and infinitely divisible sub-probability measures on , which is determined by
If and if is an operator on such that for all , then .
Suppose that is an inhomogeneous Borel right transition semigroup on . Let be a right continuous inhomogeneous Markov process realizing . Let be a Borel function on , and let be a positive Borel function on . Let be a kernel from to , and let be a kernel from to . Suppose that the function
on is locally bounded, where denotes the restriction of to . For and define
Let for . By Theorem 6.10 in Li (2011) one can show there is an inhomogeneous Borel right transition semigroup on the state space defined by
where is the unique locally bounded positive solution to the integral equation
Let us consider a right continuous realization of the transition semigroup defined by (7). Suppose that is a locally bounded positive Borel function on . Let for . Following the proofs of Theorems 5.15 and 5.16 in Li (2011), one can see
where is the unique locally bounded positive solution to
Then there is an inhomogeneous Borel right sub-Markov transition semigroup on given by
A Markov process with transition semigroup given by (11) is called an inhomogeneous superprocess with branching mechanisms . The family of operators is called the cumulant semigroup of the superprocess. From (11) one can derive the following branching property:
for and , where “” denotes the convolution operation. Some special branching mechanisms are given in Dawson, Gorostiza and Li (2002), Dynkin (1993) and Li (1992, 2011). Clearly, the semigroup given by (7) corresponds to a conservative inhomogeneous superprocess. In general, the inhomogeneous superprocess is not necessarily conservative.
We can append an additional immigration structure to the inhomogeneous superprocess. Suppose that is a Radon measure on and is a family of functionals such that is a locally bounded Borel function on for each .
There is an inhomogeneous transition semigroup on given by
where is the unique locally bounded positive solution to (10).
It is easy to check that
defines an inhomogeneous sub-Markov transition semigroup on . Clearly, the Laplace functional of this transition semigroup is given by (13).
If a Markov process with state space has transition semigroup given by (13), we call it an inhomogeneous immigration superprocess with immigration mechanisms and immigration measure . The intuitive meaning of the model is clear in view of (14). That is, the population at any time is made up of two parts, the native part generated by the mass at time has distribution and the immigration in the time interval gives the distribution . When shrinks to a singleton, we can identify with the positive half line . In this case, the transition semigroups given by (11) and (13) determine one-dimensional CB- and CBI-processes, respectively.
Now let us consider a branching mechanism of the form (1). We can define the transition semigroup of a homogeneous CB-process by
where is the unique locally bounded positive solution of
which is essentially a special form of (8). We can write the above integral equation into its differential form
The Chapman–Kolmogorov equation of implies for all . The set of functions is the cumulant semigroup. Observe that is continuously differentiable with
It is easy to see that is a Feller semigroup. Let us consider a càdlàg realization of the corresponding CB-process with an arbitrary initial time . Let be a Radon measure on . By Theorem 5.15 in Li (2011), for and , we have
where is the unique bounded positive solution to
In particular, for and with compact support, we have
where is the unique compactly supported bounded positive function on solving
It is not hard to see that for . For any let
Suppose that as . Then for any we have
where is the right inverse of defined by
A proof of (22) was already given in Abraham and Delmas (2012). We here give a simple derivation of the result since the argument is also useful to prove the next theorem. By (18) and (19), for any and we have
where is the unique bounded positive solution to
Then one can see . It follows that
Since implies , if , we get
That gives (22) first for and then for all .
Let be a locally bounded positive Borel function on . Suppose that is a constant and is a finite measure on . Let be an immigration mechanism given by
By Theorem 2.3 we can define an inhomogeneous transition semigroup on by
A positive Markov process with transition semigroup is called an inhomogeneous CBI-process with immigration rate . It is easy to see that the homogeneous time–space semigroup associated with is a Feller transition semigroup. Then has a càdlàg realization . A modification of the proof of Theorem 5.15 in Li (2011) shows that, for and ,
where is the unique bounded positive solution to (19). In particular, for and with compact support, we have
where is the unique compactly supported bounded positive solution to (21). For any let
By a modification of the proof of Theorem 2.4, we get the following:
Suppose that as . Then for any we have
where is defined by (23).
3 The predictable immigration rate
The main purpose of this section is to give a construction of the CBI-process with transition semigroup defined by (25) as the pathwise unique solution of a stochastic integral equation driven by time–space noises. For the convenience of applications, we shall generalize the model slightly by considering a random immigration rate. This is essential for our study of the path-valued Markov processes. The reader is referred to Bertoin and Le Gall (2006), Dawson and Li (2006, 2012), Fu and Li (2010) and Li and Mytnik (2011) for some related results.
Suppose that is a filtered probability space satisfying the usual hypotheses. Let be an -white noise on based on the Lebesgue measure and let and be -Poisson point processes on with characteristic measures and , respectively. We assume that the white noise and the Poisson processes are independent of each other. Let denote the stochastic integral on with respect to the white noise. Let and denote the Poisson random measures on associated with and , respectively. Let denote the compensated random measure associated with . Suppose that is a positive -predictable process such that is locally bounded. We are interested in positive càdlàg solutions of the stochastic equation
For any positive càdlàg solution of (3) satisfying , one can use a standard stopping time argument to show that is locally bounded and
By Itô’s formula, it is easy to see that solves the following martingale problem: for every ,
Suppose that is a positive càdlàg solution of (3) and is a positive càdlàg solution of the equation with replaced by . Then we have
For each integer define . Then decreasingly as and
Let be a positive continuous function supported by , so that
and for every . Let
It is easy to see that and
Moreover, we have increasingly as . Let for . From (3) we have
By this and Itô’s formula,
It is easy to see that for any . If , we have
Taking the expectation in both sides of (3) gives
Then we get the desired estimate by letting .
Suppose that is a positive càdlàg solution of (3), and is a positive càdlàg solution of the equation with replaced by . Then we have