Dual Processes and Compapison Theorem of Superprocesses Random Environents

In this paper, we give the dual processes for superprocesses in random environments constructed by L Mytnik and the comparison theorem of these superprocesses with Dawson-Watanabe superprocesses.

ENTRANCE LAWS FOR DAWSON-WATABE SUPERPROCESSES WITH NONLOCAL BRANCHING

This paper proves a 1-1 correspondence between minimal probability entrance laws for the superprocess and entrance laws for its underlying process. From this the author deduces that an infinitely divisible probability entrance law for the superprocess is uniquely determined by an infinitely divisible probability measure on the space of the underlying entrance laws. Under an additional condition, a characterization is given for all entrance laws for the superprocess, generalizing the results of Dynkin (1989). An application to immigration processes is also discussed.

SOME PROPERTIES OF (α, d, β)-SUPERPROCESSES

In this paper we show that tile diameter of tile support of (2, d, β)-superprocesses tends to zero a.s. at the time of extinction, and give tile probability distribution of hitting single point. For (α, d, β)-superprocesses, we obtain a limit theorem and some properties of the local time of it.

State Descomposition of Superprocesses of Stochastic Flows

In this paper, we reconstruct the superprocesses of stochastic flows by martingale method, and prove that if and only if the infinitesimal particles never hit each other, then atomic part and diffuse part of this kind of superprocesses will be also superprocesses of stochastic flows. This result completely answers the open problem in .

Central limit type theorem for Ornstein-Uhlenbeck superprocess

Ornstein-Uhlenbeck superprocess (O-U superprocess for short) first given in ref. [1], is a Gaussian Markov process with values in Schwartz distributions. The background for this process is as the fluctuation limit of some rescaled particle system, and describes the fluctuation of particle system around its macroscopic flow. Since O-U superprocess can be identified with the solution of some generalized Langevin equation, it is a kind of

Superprocesses arising from interactive stochastic flows

In this paper, we give a unified construction for superprocesses with dependent spatial motion constructed by Dawson, Li, Wang and superprocesses of stochastic flows constructed by Ma and Xiang. Furthermore, we also give some examples and rescaled limits of the new class of su perprocesses.

A REMARK ON EXTINCTION OF A CLASS OF SUPERPROCESSES

The extinction of a class of superprocesses associated with general branching characterstics and underlying Markov processes is investigated. The extinction is closely associated with the branching characteristics and the recurrence and transience of underlying processes.

Supercritical superprocesses: Proper normalization and non-degenerate strong limit

Suppose that X ={Xt, t≥0;Pμ} is a supercritical superprocess in a locally compact separable metric space E. Let φ0 be a positive eigenfunction corresponding to the first eigenvalue λ0 of the generator of the mean semigroup of X. Then Mt := e-λ0t〈φ0,Xt〉 is a positive martingale. Let M∞ be the limit of Mt. It is known(see Liu et al.(2009)) that M∞ is non-degenerate if and only if the L log L condition is satisfied. In this paper we are mainly interested in the case when the L log L condition is not satisfied. We prove that, under some conditions, there exist a positive function γt on [0,∞) and a non-degenerate random variable W such that for any finite nonzero Borel measure μ on E,lim/t→∞γt〈φ0,Xt〉=W, a.s.-Pμ.We also give the almost sure limit of γt〈f, Xt〉for a class of general test functions f.

Strong law of large numbers for supercritical superprocesses under second moment condition

Consider a supercritical superprocess X = {Xt, t 〉~ O} on a locally compact separable metric space （E, m）. Suppose that the spatial motion of X is a Hunt process satisfying certain conditions and that the branching mechanism is of the form

Harmonic functions of superprocesses and conditioned superprocesses'

The harmonic functions for superprocesses are defined by applying the martingale property. Under a general condition, a classification theorem of harmonic functions for homogeneous superprocesses is obtained in terms of the solutions to a measure functional equation. The conditioned superdiffusions in a regular domain by Doob’s harmonic transform are defined and investigated.

Limit Theorems for the Superprocess With Immigration

In this note we study the limiting behavior of a certain class of superprocess with immigration. Such a process Xt is non-extinction almost surely due to external particles’ immigrating. Under suitable conditions, which include the convergence of the semigroup for the underlying process to some limiting probability measure v, we show that the random measure t-1Xt converges in distribution to Zcv as t→∞, where Zc is a random variable possessing Γ-distribution with parameter c. Moreover, for the weighted

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.

Some properties of superprocesses conditioned on non-extinction

We simply call a superprocess conditioned on non-extinction a conditioned superprocess. In this study, we investigate some properties of the conditioned superprocesses (subcritical or critical). Firstly, we give an equivalent description of the probability of the event that the total occupation time measure on a compact set is finite and some applications of this equivalent description. Our results are extensions of those of Krone (1995) from particular branching mechanisms to general branching mechanisms. We also prove a claim of Krone for the cases of d = 3, 4. Secondly, we study the local extinction property of the conditioned binary super-Brownian motion {X t , P μ ∞ }. When d = 1, as t goes to infinity, X t / $ \sqrt t $ converges to ηλ in weak sense under P μ ∞ , where η is a nonnegative random variable and λ is the Lebesgue measure on ?. When d ? 2, the conditioned binary super-Brownian motion is locally extinct under P μ ∞ .

Superprocesses on Two Space Forms

Superprocess is one class of measure-valued branching Markov processes. Many results of the process on abstract spaces and Euclidean spaces are obtained in the literature. In this paper, we discuss super-Brownian motions on two space forms and reveal partially the relationship between the properties of superprocesses and the geometric structure of the underlying state spaces. Finally, we also presents some open problems.

On Tanaka formulae for (α,d,β)-superprocesses

In this paper, Tanaka formulae for (α, d,β)-superprocesses in the dimensions where the local time exists are established under the optimal initial condition.

CONDITIONED SUPERPROCESSES

A class of superprocesses which dies out is investigated.Under the condition of norextinction, a new superprocess is constructed,its life time is infinite, and its distribution is determined by the moment function. Several limit theorems about this superprocess and its occupation time process are obtained.

Dirichlet problem related to a class of superprocesses

ASSUME that ξ = （ξt, Ⅱx） is a right Markov process in Rd. Let φ（x, z） = a（x）z+b（x）z2+integral from n=0 to ∞ (e-uz-1+uz)nx（du）, x∈ Rd, z∈R+. （1）Consider the following Dirichlet problem:

A LIMIT THEOREM FOR INTERACTING MEASURE-VALUED BRANCHING PROCESSES

It is well known that Fleming-Viot superprocesses can be obtained from the Dawson-Watanabe superprocesses by conditioning the latter to have constant total mass. The same question is investigated for measure-valued branching processes with interacting intensity independent of the geographical position. It is showed that a sequence of conditioned probability laws of this kind of interacting measure-valued branching processes also approximates to the probability law of Fleming-Viot superprocesses.

Fleming－Viot测度值过程

Ｆｖ－超过程是测度值过程研究的重要内容之一，近两年有较大的发展，本文系统地介绍了有关的知识，并对其研究的方法及存在的问题进行了简单的讨论。

一般超过程灭绝时的分布及其矩性质

本文研究了分枝特征为ψ（ｚ） ＝ ｂｚ ＋ ｃｚ２ ＋ ∫∞０ （ｅｕｚ － １＋ ｕｚ）ｎ（ｄｕ）的一般超过程的灭绝性问题，给出了超过程灭绝时的分布。