Zenghu Li

Measure-valued branching Markov processes

Preface.- 1. Random Measures on Metric Spaces.- 2. Measure-valued Branching Processes.- 3. One-dimensional Branching Processes.- 4. Branching Particle Systems.- 5. Basic Regularities of Superprocesses.- 6. Constructions by Transformations.- 7. Martingale Problems of Superprocesses.- 8. Entrance Laws and Excursion Laws.- 9. Structures of Independent Immigration.- 10. State-dependent Immigration Structures.- 11. Generalized Ornstein-Uhlenbeck Processes.- 12. Small Branching Fluctuation Limits.- 13. Appendix: Markov Processes.- Bibliography.- Index.

Skew convolution semigroups and affine Markov processes

A general affine Markov semigroup is formulated as the convolution of a homogeneous one with a skew convolution semigroup. We provide some sufficient conditions for the regularities of the homogeneous affine semigroup and the skew convolution semigroup. The corresponding affine Markov process is constructed as the strong solution of a system of stochastic equations with non-Lipschitz coefficients and Poisson-type integrals over some random sets. Based on this characterization, it is proved that the affine process arises naturally in a limit theorem for the difference of a pair of reactant processes in a catalytic branching system with immigration.

Skew Convolution Semigroups and Related Immigration Processes

A special type of immigration associated with measure-valued branching processes is formulated by using skew convolution semigroups. We give a characterization for a general inhomogeneous skew convolution semigroup in terms of probability entrance laws. The related immigration process is constructed by summing up measure-valued paths in the Kuznetsov process determined by an entrance rule. The behavior of the Kuznetsov process is then studied, which provides insight into trajectory structures of the immigration process. Some well-known results on excessive measures are formulated in terms of stationary immigration processes.

Measure-valued branching processes with immigration

Starting from the cumulant semigroup of a measure-valued branching process, we construct the transition probabilities of some Markov process Y([beta])=(Y([beta])t, t [epsilon] , which we call a measure-valued branching process with discrete immigration of unit[beta]. The immigration of Y([beta]) is governed by a Poisson random measure [rho] on the time-distribution space and a probability generating function h, both depending on [beta]. It is shown that, under suitable hypotheses, Y([beta]) approximates to a Markov process Y=(Yt, t [epsilon] as [beta]-->0+. The latter is the one we call a measure-valued branching process with immigration. The convergence of branching particle systems with immigration is also studied.

Immigration structures associated with Dawson-Watanabe superprocesses

The immigration structure associated with a measure-valued branching process may be described by a skew convolution semigroup. For the special type of measure-valued branching process, the Dawson-Watanabe superprocess, we show that a skew convolution semigroup corresponds uniquely to an infinitely divisible probability measure on the space of entrance laws for the underlying process. An immigration process associated with a Borel right superprocess does not always have a right continuous realization, but it can always be obtained by transformation from a Borel right one in an enlarged state space.

Non-local 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.

Stationarity and ergodicity for an affine two-factor model

We study the existence of a unique stationary distribution and ergodicity for a two-dimensional affine process. Its first coordinate process is supposed to be a so-called α-root process with α ∈ (1, 2]. We prove the existence of a unique stationary distribution for the affine process in the α ∈ (1, 2] case; furthermore, we show ergodicity in the α = 2 case.

On parameter estimation for critical affine processes

First we provide a simple set of sufficient conditions for the weak convergence of scaled affine processes with state space

Measure-valued diffusions and stochastic equations with Poisson process

R_+ \times R^d

Yamada-Watanabe Results for Stochastic Differential Equations with Jumps

. We specialize our result to one-dimensional continuous state branching processes with immigration. As an application, we study the asymptotic behavior of least squares estimators of some parameters of a two-dimensional critical affine diffusion process.