Lina Ji

We study the martingale problem for two mutually competing superprocesses in independent Brownian random environments. The competition rates depend on the spatial position and both population measures. Under suitable assumptions, we establish well-posedness for every pair of finite initial measures and derive moment estimates for the total masses. Existence is proved by constructing branching particle systems and establishing the tightness of their empirical measure processes. Uniqueness in law follows from pathwise uniqueness for the system of stochastic partial differential equations satisfied by the associated distribution-function-valued processes.

We study the martingale problem for two mutually competing superprocesses in independent Brownian random environments. The competition rates depend on the spatial position and both population measures. Under suitable assumptions, we establish well-posedness for every pair of finite initial measures and derive moment estimates for the total masses. Existence is proved by constructing branching particle systems and establishing tightness of their empirical measure processes. Uniqueness in law follows from pathwise uniqueness for the system of stochastic partial differential equations satisfied by the associated distribution-function-valued processes.

We propose threshold diffusion processes as unique solutions to stochastic differential equations with step-function coefficients, and obtain explicit expressions for the conditional Laplace transform of the hitting times and the potential measures. Applying these results, we further discuss their asymptotic behaviors such as the stationary distributions and the escape probabilities.

We establish the exponential ergodic property in a weighted total variation distance of continuous-state branching processes with immigration in random environments with competition and catastrophes, under a Lyapunov-type condition and other mild assumptions. The proof is based on a Markov coupling process along with some delicate estimates for the associated coupling generator. In particular, the main result indicates whether and how the competition mechanism, the environment and the catastrophe could balance the branching mechanism respectively to guarantee the exponential ergodicity of the process.

We prove a scaling limit theorem for two-type Galton-Waston branching processes with interaction. The limit theorem gives rise to a class of mixed state branching processes with interaction using to simulate the evolution for cell division affected by parasites. Such process can also be obtained by the pathwise unique solution to a stochastic equation system. Moreover, we present sufficient conditions for extinction with probability one and the exponential ergodicity in the total variation distance of such process.

In this paper a martingale problem for super-Brownian motion with interactive branching is derived. The uniqueness of the solution to the martingale problem is obtained by using the pathwise uniqueness of the solution to a corresponding system of SPDEs with proper boundary conditions. The existence of the solution to the martingale problem and the local Hölder continuity of the density process are also studied.

A system of mutually interacting superprocesses with migration is constructed as the limit of a sequence of branching particle systems arising from population models. The uniqueness in law of the superprocesses is established using the pathwise uniqueness of a system of stochastic partial differential equations, which is satisfied by the corresponding system of distribution function-valued processes.

We provide constructions of age-structured branching processes without or with immigration as pathwise-unique solutions to stochastic integral equations. A necessary and sufficient condition for the ergodicity of the model with immigration is also given.