Construction of immigration superprocesses with dependent spatial motion from one-dimensional excursions

A superprocess with dependent spatial motion and interactive immigration is constructed as the pathwise unique solution of a stochastic integral equation carried by a stochastic flow and driven by Poisson processes of one-dimensional excursions. Supported by an NSERC Research Grant and a Max Planck Award.Supported by the NSFC (No. 10121101 and No. 10131040).Mathematics Subject Classification (2000): Primary 60J80; Secondary 60G57 60H20     

A superprocess involving both branching and coalescing

We consider a superprocess with coalescing Brownian spatial motion. We first point out a dual relationship between two systems of coalescing Brownian motions. In consequence we can express the Laplace functionals for the superprocess in terms of coalescing Brownian motions, which allows us to obtain some explicit results. We also point out several connections between such a superprocess and the Arratia flow. A more general model is discussed at the end of this paper.     

A note on superprocesses

Summary Subject to a mild restriction onA, generator of the one-particle motion, we show theA-Fleming-Viot superprocess can be obtained from theA-Dawson-Watanabe superprocess by conditioning the latter to have constant total mass.Research conducted while a Sir Christopher Cox Junior Research FellowResearch supported in part by the National Science Foundation grant NSF-DMS-89-3474     

带移民的非临界分枝相依空间运动超过程

通过对一列带正跳跃的超过程取极限,本文构造了带移民的相依空间运动超过程.在此基础上,利用 Dawson型的Girsanov变换得到了相应的非临界分枝,此变换同时给出依赖于整体状态的空间漂移.     

Large scale localization of a spatial version of Neveu’s branching process

Recently a spatial version of Neveu’s (1992) continuous-state branching process was constructed by Fleischmann and Sturm (2004). This superprocess with infinite mean branching behaves quite differently from usual supercritical spatial branching processes. In fact, at macroscopic scales, the mass renormalized to a (random) probability measure is concentrated in a single space point which randomly fluctuates according to the underlying symmetric stable motion process.     

Continuous Local Time of a Purely Atomic Immigration Superprocess with Dependent Spatial Motion

Abstract

A purely atomic immigration superprocess with dependent spatial motion in the space of tempered measures is constructed as the unique strong solution of a stochastic integral equation driven by Poisson processes based on the excursion law of a Feller branching diffusion, which generalizes the work of Dawson and Li [3 Dawson , D.A. , and Li , Z.H. 2003 . Construction of immigration superprocesses with dependent spatial motion from one-dimensional excursions . Probability Theory and Related Fields 127 : 37 – 61 . [Google Scholar]]. As an application of the stochastic equation, it is proved that the superprocess possesses a local time which is Hölder continuous of order α for every α < 1/2. We establish two scaling limit theorems for the immigration superprocess, from which we derive scaling limits for the corresponding local time.     

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.     

Conditional Log-Laplace Functionals of Immigration Superprocesses with Dependent Spatial Motion

A non-critical branching immigration superprocess with dependent spatial motion is constructed and characterized as the solution of a stochastic equation driven by a time-space white noise and an orthogonal martingale measure. A representation of its conditional log-Laplace functionals is established, which gives the uniqueness of the solution and hence its Markov property. Some properties of the superprocess including an ergodic theorem are also obtained.Mathematics Subject Classifications (2000) 60J80, 60G57, 60J35.Zenghu Li: Supported by the NSFC (No. 10121101 and No. 10131040).Hao Wang: Supported by the research grant of UO.Jie Xiong: Research supported partially by NSA and by Alexander von Humboldt Foundation.     

Convergence to equilibrium of critical branching particle systems and superprocesses,and related nonlinear partial differential equations

We consider a class of multitype particle systems in ^d undergoing spatial diffusion and critical stable multitype branching, and their limits known as critical stable multitype Dawson-Watanabe processes, or superprocesses. We show that for large classes of initial states, the particle process and the superprocess converge in distribution towards known equilibrium states as time tends to infinity. As an application we obtain the asymptotic behavior of a system of nonlinear partial differential equations whose solution is related to the distribution of both the particle process and the superprocess.Research partially supported by CONACyT (Mexico), CNRS (France) and BMfWuF (Austria).     

Central Limit Theorems for a Super-Diffusion over a Stochastic Flow

Central limit theorems of the occupation time of a superprocess over a stochastic flow are proved. For the critical and higher dimensions d≥4, the limits are Gaussian variables. For d=3, the limit is conditional Gaussian. When the stochastic flow disappears, the results degenerate to those for the ordinary super-Brownian motion.     

An alternative approach to super-Brownian motion with a locally infinite branching mass

Fleischmann and Mueller (Probab. Theory Related Fields 107 (1997) 325) constructed a super-Brownian motion in R¹ with a locally infinite branching rate function, and they showed that this super-Brownian motion has a strong killing property in the critical case. In this paper, we first construct, via a limiting procedure, a super-Brownian motion which is equivalent to the super-Brownian motion with a locally infinite branching. From this construction, one can easily see the connection between the superprocess and a killed Brownian motion. Next, by taking advantage of the new construction, we give a new proof of the strong killing property of the process.     

Limit Processes for Age-Dependent Branching Particle Systems

We consider systems of spatially distributed branching particles in R ^d . The particle lifelengths are of general form, hence the time propagation of the system is typically not Markov. A natural time-space-mass scaling is applied to a sequence of particle systems and we derive limit results for the corresponding sequence of measure-valued processes. The limit is identified as the projection on R ^d of a superprocess in R ₊×R ^d . The additive functional characterizing the superprocess is the scaling limit of certain point processes, which count generations along a line of descent for the branching particles.     

A super-Brownian motion with a locally infinite catalytic mass

Summary. A super-Brownian motion in with “hyperbolic” branching rate , is constructed, which symbolically could be described by the formal stochastic equation (with a space-time white noise ). Starting at this superprocess will never hit the catalytic center: There is an increasing sequence of Brownian stopping times strictly smaller than the hitting time of such that with probability one Dynkin's stopped measures vanish except for finitely many Received: 27 November 1995 / In revised form: 24 July 1996     

Measure diffusions and related explosion problems

For random measure-valued stochastic partial differential equations for biological processes, growth represented by scalar partial differential equations at each point of the support and spread being a diffusion on R ^d, solutions are constructed by smearing the growth processes at each spatial point and composing the resulting generator with the generator for the spread. If these solutions are unique the equation is called solvable. We find conditions for the noise term of a solvable equations to have trivial effect and we identify some non-solvable equations, for example the diffusion-free bilinear equation. The search led to an investigation of explosion and the effect of point barriers for scalar stochastic differential equations with linear drift; this is used to explain the clustering effect in the usual superprocess.     

Local time and Tanaka formula for a multitype Dawson–Watanabe superprocess

In [3] Dynkin defined the local time of a continuous superprocess as a stochastic integral and gave a criterion for existence of local time. Here we prove that the conditions in Dynkin's existence criterion are satisfied by the multitype Dawson–Watanabe superprocess, and give a Tanaka formula‐like representation of the local time which is used to show that the occupation measure of the multitype superprocess is absolutely continuous with respect to an appropriate reference measure, and that the corresponding density coincides a.s. with the local time. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

CONDITIONED SUPERPROCESSES

ＣＯＮＤＩＴＩＯＮＥＤＳＵＰＥＲＰＲＯＣＥＳＳＥＳＬＩＣＵＮＨＡＮＧＷＵＲＯＮＧＡｂｓｔｒａｃｔＡｃｌａｓｓｏｆｓｕｐｅｒｐｒｏｃｅｓｅｓｗｈｉｃｈｄｉｅｓｏｕｔｉｓｉｎｖｅｓｔｉｇａｔｅｄ．Ｕｎｄｅｒｔｈｅｃｏｎｄｉｔｉｏｎｏｆｎｏｒｅｘｔｉ...     

On Changes of Measure for Super Brownian Motion

Let be a superprocess under the measure P. We show the existence of probability measures which are absolutely continuous with respect to P, and whose Randon–Nikodym derivatives are suitably normalized functions of the self intersection local time of . These measures correspond to measure valued processes exhibiting a certain amount of (reinforcing) self interaction in terms of both the particle motion and the branching mechanism.     

Supercritical superprocesses: Proper normalization and non-degenerate strong limit

Suppose that X = {X_t, t≥0; P_μ} is a supercritical superprocess in a locally compact separable metric space E. Let φ0 be a positive eigenfunction corresponding to the first eigenvalue λ_0 of the generator of the mean semigroup of X. Then Mt := e~(-λ_0t)〈φ0,X_t〉 is a positive martingale. Let M_∞ be the limit of M_t. 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 γ_t on [0,∞) and a non-degenerate random variable W such that for any finite nonzero Borel measure μ on E,lim/t→∞γ_t〈φ0,X_t〉=W, a.s.-P_μ~.We also give the almost sure limit of γ_t〈f, X_t〉for a class of general test functions f.