Criterion on the limits of superprocesses

Starting with super-diffusion processes, the asymptotic behavior of the superprocesses on finite and infinite measure spaces is systematically studied by general branching mechanisms. The complete features of their limits are described and a criterion on their behavior is presented. Project supported by the National Natural Science Foundation of China (Grant No. 19631010) and the Mathematical Centre of the State Education Commission.     

On regularity of superprocesses

Summary Three theorems on regularity of measure-valued processesX with branching property are established which improve earlier results of Fitzsimmons [F1] and the author [D5]. The main difference is that we treatX as a family of random measures associated with finely open setsQ in time-space. Heuristically,X describes an evolution of a cloud of infinitesimal particles. To everyQ there corresponds a random measureX which arises if each particle is observed at its first exit time fromQ. (The stateX _t at a fixed timet is a particular case.) We consider a monotone increasing familyQ _t of finely open sets and we establish regularity properties of as a function oft. The results are used in [D6], [D7] and [D10] for investigating the relations between superprocesses and non-linear partial differential equations. Basic definitions on Markov processes and superprocesses are introduced in Sect. 1. The next three sections are devoted to proving the regularity theorems. They are applied in Sect. 5 to study parts of superprocess. The relation to the previous work is discussed in more detail in the concluding section. It may be helpful to look briefly through this section before reading Sects. 2–5.Partially supported by the National Science Foundation Grant DMS-8802667 and by The US Army Research Office through the Mathematical Sciences Institute at Cornell University     

Markov snakes and superprocesses

Summary We suggest the name Markov snakes for a class of path-valued Markov processes introduced recently by J.-F. Le Gall in connection with the theory of branching measure-valued processes. Le Gall applied this class to investigate path properties of superdiffusions and to approach probabilistically partial differential equations involving a nonlinear operator v–v ². We establish an isomorphism theorem which allows to translate results on continuous superprocesses into the language of Markov snakes and vice versa. By using this theorem, we get limit theorems for discrete Markov snakes.Partially supported by National Science Foundation Grant DMS-9301315 and by The US Army Research Office through the Mathematical Sciences Institute at Cornell University     

Path processes and historical superprocesses

Summary A superprocessX over a Markov process can be obtained by a passage to the limit from a branching particle system for which describes the motion of individual particles.The historical process for is the process whose state at timet is the path of over time interval [0,t]. The superprocess over the historical superprocess over —reflects not only the particle distribution at any fixed time but also the structure of family trees. The principal property of a historical process is that is a function of for alls<t. Every process with this property is calleda path process. We develop a theory of superprocesses over path processes whose core is the integration with respect to measure-functionals. By applying this theory to historical superprocesses we construct the first hitting distributions and prove a special Markov property for superprocesses.Partially supported by National Science Foundation Grant DMS-8802667     

The prolific backbone for supercritical superprocesses

We develop an idea of Evans and O’Connell (1994) [13], Engländer and Pinsky (1999) [10] and Duquesne and Winkel (2007) [4] by giving a pathwise construction of the so-called ‘backbone’ decomposition for supercritical superprocesses. Our results also complement a related result for critical (1+β)-superprocesses given in Etheridge and Williams (2003) [11]. Our approach relies heavily on the use of Dynkin-Kuznetsov N-measures.     

A class of path-valued Markov processes and its applications to superprocesses

Summary Let ( _s ) be a continuous Markov process satisfying certain regularity assumptions. We introduce a path-valued strong Markov process associated with ( _s ), which is closely related to the so-called superprocess with spatial motion ( _s ). In particular, a subsetH of the state space of ( _s ) intersects the range of the superprocess if and only if the set of paths that hitH is not polar for the path-valued process. The latter property can be investigated using the tools of the potential theory of symmetric Markov processes: A set is not polar if and only if it supports a measure of finite energy. The same approach can be applied to study sets that are polar for the graph of the superprocess. In the special case when ( _s ) is a diffusion process, we recover certain results recently obtained by Dynkin.     

A note on pseudocompactifications

There exists a countable spaceV _ω such that:

1. V _ω has a single non-isolated point,
2. V _ω has no pseudocompactificationX witht(X)=ω.

     

A note on rank-additivity

A new criterion is given for rank additivity of a collection of m × n complex matrices.     

A note on P-convexity

In this paper we study and characterize the structure of the linear transformations which preserve P-convex sets.     

A note on merging

For a finite poset P and x, yP let pr(x>y) be the fraction of linear extensions which put x above y. N. Linial has shown that for posets of width 2 there is always a pair x, y with 1/3 pr(x>y)2/3. The disjoint union C ₁C ₂ of a 1-element chain with a 2-element chain shows that the bound 1/3 cannot be further increased. In this paper the extreme case is characterized: If P is a poset of width 2 then the bound 1/3 is exact iff P is an ordinal sum of C ₁C ₂'s and C ₁'s.     

A note on disconnectedness

The purpose of this note is to characterize various disconnectedness in terms of (star) refinement of covers.     

A note on p-curvatures

In this note, we give an arithmetic criterion for the Lie-irreducibility of linear differential equations based on p-curvatures.