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.

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.

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.

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.

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.

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:

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.

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

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 {Xt, P μ∞ }. When d = 1, as t goes to infinity, Xt/√t converges to ηλ in weak sense under P μ∞ , where η is a nonnegative random variable and λ is the Lebesgue measure on R. When d 2, the conditioned binary super-Brownian motion is locally extinct under P μ∞ .

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.

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.

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-Reimers1 and Mytnik's SPDEs, and their related distribution-function-valued SPDEs.

Tanaka Formula and Local Time for a Class of Interacting Branching Measure-valued Diffusions

We construct superprocesses with dependent spatial motion(SDSMs)in Euclidean spaces R^(d)with d≥1 and show that,even when they start at some unbounded initial positive Radon measure such as Lebesgue measure on R^(d),their local times exist when d≤3.A Tanaka formula of the local time is also derived.

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

Stable Central Limit Theorems for Super Ornstein–Uhlenbeck Processes,Ⅱ

This paper is a continuation of our recent paper(Electron.J.Probab.,24(141),(2019))and is devoted to the asymptotic behavior of a class of supercritical super Ornstein-Uhlenbeck processes(X_(t))t≥0 with branching mechanisms of infinite second moments.In the aforementioned paper,we proved stable central limit theorems for X_(t)(f)for some functions f of polynomial growth in three different regimes.However,we were not able to prove central limit theorems for X_(t)(f)for all functions f of polynomial growth.In this note,we show that the limiting stable random variables in the three different regimes are independent,and as a consequence,we get stable central limit theorems for X_(t)(f)for all functions f of polynomial growth.

Regularity of the Density for the Total Weighted Occupation Measure of Super-Brownain Motion

Let （Xt） be a super-Brownian motion in a bounded domain D in R^d. The random measure Y^D（.） = ∫o^∞ Xt（.）dt is called the total weighted occupation time of （Xt）. We consider the regularity properties for the densities of a class of yD. When d = 1, the densities have continuous modifications. When d ≥ 2, the densities are locally unbounded on any open subset of D with positive y D （dx）-measure.

INTEGRAL REPRESENTATIONS OF CONTINUOUS FUNCTIONS

In this note, we give a necessary and sufficient condition for a continuous function σ(λ), λ≥0, to have the integral

Some problems on super-diffusions and one class of nonlinear differential equations

The historical superprocesses are considered on bounded regular domains with a complete branching form, as a probabilistic argument, the limit property of superprocesses is studied when the domains enlarge to the whole space. As an important application of superprocess, the representation of solutions of involved differential equations is used in term of historical superprocesses. The differential equations including the existence of nonnegative solution, the closeness of solutions and probabilistic representations to the maximal and minimal solutions are discussed, which helps develop the well-known results on nonlinear differential equations.

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

The oscillation of the occupation time process of super- Brownian motion on Sierpinski gasket

The occupation time process of super-Brownian motion on the Sierpinski gasket is studied. It is shown that this process does not possess stable property in the long run, but oscillates periodically in some sense. Other convergence properties are also studied.