测度值分枝过程与移民过程

本文介绍了测度值分枝过程和由斜卷积半群定义的伴随移民过程的基本理论和研究现状,主要内容包括：分枝粒子系统的收敛;超过程的基本正则性和极限定理;非线性微分方程;广义分枝模型;斜卷积半群和进入律;用Ｋｕｚｎｅｔｓｏｖ过程构造移民过程等。     

Stochastic partial differential equations for some measure-valued diffusions

Summary We consider two classes of measure-valued diffusion processes; measure-valued branching diffusions and Fleming-Viot diffusion models. When the basic space is R ¹, and the drift operator is a fractional Laplacian of order 1<α≦2, we derive stochastic partial differential equations based on a space-time white noise for these two processes. The former is the expected one by Dawson, but the latter is a new type of stochastic partial differential equation.     

State classification for a class of measure-valued branching diffusions in a Brownian medium

Summary. The spatial structure of a new class of measure-valued diffusions which arise as limits in distribution of a sequence of interacting branching particle systems is investigated. We obtain the following criterion of state classification for these superprocesses: their effective state space is contained in the set of purely atomic measures or the set of absolutely continuous measures according as ε=0 or ε≠0, when the coefficient of the motion generator is a smooth function. Received: 15 December 1995 / In revised form: 24 March 1997     

Conditional log-Laplace functional for a class of branching processes in random environments

A conditional log-Laplace functional(CLLF) for a class of branching processes in random environments is derived. The basic idea is the decomposition of a dependent branching dynamic into a no-interacting branching and an interacting dynamic generated by the random environments.CLLF will play an important role in the investigation of branching processes and superprocesses with interaction.     

Law of large numbers for supercritical superprocesses with non-local branching

In this paper we establish a weak and a strong law of large numbers for supercritical superprocesses with general non-local branching mechanisms. Our results complement earlier results obtained for superprocesses with only local branching. Several interesting examples are developed, including multitype continuous-state branching processes, multitype superdiffusions and superprocesses with discontinuous spatial motions and non-decomposable branching mechanisms.     

UNIQUENESS PROBLEM FOR SPDES FROM POPULATION MODELS

This is a survey on the strong uniqueness of the solutions to stochastic partial differential equations(SPDEs) related to two measure-valued processes: superprocess and Fleming-Viot process which are given as rescaling limits of population biology models. We summarize recent results for Konno-Shiga-Reimers' and Mytnik's SPDEs, and their related distribution-function-valued SPDEs.     

Gamma-Dirichlet algebra and applications

The Gamma-Dirichlet algebra corresponds to the decomposition of the gamma process into the independent product of a gamma random variable and a Dirichlet process. This structure allows us to study the properties of the Dirichlet process through the gamma process and vice versa. In this article, we begin with a brief survey of several existing results concerning this structure. New results are then obtained for the large deviations of the jump sizes of the gamma process and the quasi-invariance of the two-parameter Poisson-Dirichlet distribution. We finish the paper with the derivation of the transition function of the Fleming-Viot process with parent independent mutation from the transition function of the measure-valued branching diffusion with immigration by exploring the Gamma-Dirichlet algebra embedded in these processes. This last result is motivated by an open R. C. Gritfiths. problem proposed by S. N. Ethier and     

无限状态空间上的测度和鞅

建立无限状态空间上的Radon－Nikodym定理和Riesz表示定理，引进测度值鞅并应用于超过程．     

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     

A REMARK ON EXTINCTION OF A CLASS OF SUPERPROCESSES

1.IntroductionSuperprocesses,i.e.,measure-vauedMarkovprocesses,havebeenanattractivetopicintherecenty6ars,whichcanbedefinedasfollows.Let((,Px)..MbeaMarkovprocesswithstrongcontinuousMarkovsemigroupSt(denotebyAitsinfinitesimaloperator,andbyPt(x,y)theassociatedtransitiollfunction).DenotebyMPthesetofallRadonmeasurespsatisfyingandbypC(t)(resp.pCc(t))thesetofpositivelycontinuousfUnctions(resp.withcompactsupport)int-Thesthcalledsuperprocess(X,P#).6MPwithparameters(andopandtakingvalueinMrisdete…     

Modelling a Multitype Branching Brownian Motion: Filtering of a Measure-Valued Process

In the framework of marked trees, a multitype branching brownian motion, described by measure-valued processes, is studied. By applying the strong branching property, the Markov property and the expression of the generator are derived for the process whose components are the measure-valued processes associated to each type particles. The conditional law of the measure-valued process describing the whole population observing the cardinality of the subpopulation of a given type particles is characterized as the unique weak solution of the Kushner‐Stratonovich equation. An explicit representation of the filter is obtained by Feyman–Kac formula using the linearized filtering equation.     

The complete characterization of a general class of superprocesses

In an important paper, [7], Dynkin, Kuznetsov and Skorohod showed that, under mild conditions, the log-Laplace functional of every branching measure-valued process is the solution of an evolution equation determined by three parameters, ξQ and ℓ. This paper essentially deals with the converse of this result. We consider a general class ℋ of BMV (subject to some mild conditions). First, we derive from [7] that every process X∈ℋ is a (ξΦ, K)-superprocesses, where the triples (ξ, Φ, K) are subject to some conditions ANS1-ANS3. Conversely, we show that, for each of these triples satisfying ANS1-ANS3, there exists a version X of the (ξ, Φ, K)-superprocess which belongs to ℋ. Consequently, ANS1-ANS3 fully characterizes ℋ, and subject to mild conditions, BMV and superprocesses are equivalent concepts. This requires to prove a general existence theorem for superprocesses, the existence of a regular version of these processes and that for processes in ℋ, branching characteristics Q and ℓ are continuous. Received: 11 April 1996 / Revised version: 25 May 1999     

Limit theorems for flows of branching processes

We construct two kinds of stochastic flows of discrete Galton-Watson branching processes. Some scaling limit theorems for the flows are proved, which lead to local and nonlocal branching superprocesses over the positive half line.     

Discontinuous Measure-Valued Branching Processes and Generalized Stochastic Equations

We study a class of integrable and discontinuous measure-valued branching processes. They are constructed as limits of renormalized spatial branching processes, the underlying branching distribution belonging to the domain of attraction of a stable law. These processes, computed on a test function f, are semimartingales whose martingale terms are identified with integrals of f with respect to a martingale measure. According to a representation theorem of continuous (respectively purely discontinuous) martingale measures as stochastic integrals with respect to a white noise (resp. to a POISSON process), we prove that the measure-valued processes that we consider are solutions of stochastic differential equations in the space of L² (Ω)-valued vector measures.     

Construction and regularity of measure-valued markov branching processes

A construction is given for a general class of measure-valued Markov branching processes. The underlying spatial motion process is an arbitrary Borel right Markov process, and state-dependent offspring laws are allowed. It is shown that such processes are Hunt processes in the Ray weak* topology, and have continuous paths if and only if the total mass process is continuous. The entrance spaces of such processes are described explicitly. Research supported in part by NSF Grant DMS 87-21237.     

Properties of Super-Poisson Processes and Super-Random Walks with Spatially Dependent Branching Rates

The global supports of super-Poisson processes and super-random walks with a branching mechanism ψ（z）=z^2 and constant branching rate are known to be noncompact. It turns out that, for any spatially dependent branching rate, this property remains true. However, the asymptotic extinction property for these two kinds of superprocesses depends on the decay rate of the branching-rate function at infinity.     

GIRSANOV'S THEOREM ON ABSTRACT WIENER SPACES

ＧＩＲＳＡＮＯＶ’ＳＴＨＥＯＲＥＭＯＮＡＢＳＴＲＡＣＴＷＩＥＮＥＲＳＰＡＣＥＳＺＨＡＮＧＹＩＮＮＡＮＡｂｓｔｒａｃｔＬｅｔ（Ｅ，Ｈ，μ）ｂｅａｎａｂｓｔｒａｃｔＷｉｅｎｅｒｓｐａｃｅｉｎｔｈｅｓｅｎｓｅｏｆＬ．Ｇｒｏｓ．Ｉｔｉｓｐｒｏｖｅｄｔｈ...     

Nonlocal Branching Superprocesses and Some Related Models

A new formulation of nonlocal branching superprocesses is given from which we derive as special cases the rebirth, the multitype, the mass-structured, the multilevel and the age-reproduction-structured superprocesses and the superprocess-controlled immigration process. This unified treatment simplifies considerably the proof of existence of the old classes of superprocesses and also gives rise to some new ones.     

Comparing Fleming-Viot and Dawson-Watanabe processes

Fleming-Viot processes and Dawson-Watanabe processes are two classes of “superprocesses” that have received a great deal of attention in recent years. These processes have many properties in common. In this paper, we prove a result that helps to explain why this is so. It allows one to prove certain theorems for one class when they are true for the other. More specifically, we show that product moments of a Fleming-Viot process can be bounded above by the corresponding moments of the Dawson-Watanabe process with the same “underlying particle motion”, and vice versa except for a multiplicative constant. As an application, we establish existence and continuity properties of local time for certain Fleming-Viot processes.