Via a Dirichlet form extension theorem and making full use of two-sided heat kernel estimates, we establish quenched invariance principles for ran- dom walks in random environments with a boundary. In particular, we prove that the random walk on a supercritical percolation cluster or among random conductances bounded uniformly from below in a half-space, quarter-space, etc., converges when rescaled diffusively to a reflecting Brownian motion, which has been one of the important open problems in this area. We establish a similar result for the random conductance model in a box, which allows us to improve existing asymptotic estimates for the relevant mixing time. Furthermore, in the uniformly elliptic case, we present quenched invariance prin- ciples for domains with more general boundaries.
Ruihan GuoSchool of Mathematics and Statistics, Zhengzhou UniversityYan XuSchool of Mathematical Sciences, University of Science and Technology of ChinaZhengfu XuDepartment of Mathematical, Michigan Technological University
Consider an age-dependent, single-species branching process defined by a progeny number distribution, and a lifetime distribution associated with each independent particle. In this paper, we focus on the associated inverse problem where one wishes to formally solve for the progeny number distribution or the lifetime distribution that defines the Bellman-Harris branching process. We derive results for the existence and uniqueness (the identifiability) of these two distributions given one of two types of information: the extinction time probability of the entire process (extinction time distribution), or the distribution of the total number of particles at one fixed time. We demonstrate that perfect knowledge of the distribution of extinction times allows us to formally determine either the progeny number distribution or the lifetime distribution. Furthermore, we show that these constructions are unique. We then consider “data” consisting of a perfectly known total number distribution given at one specific time. For a process with known progeny number distribution and exponentially distributed lifetimes, we show that the rate parameter is identifiable. For general lifetime distributions, we also show that the progeny distribution is globally unique. Our results are presented through four theorems, each describing the constructions in the four distinct cases.
We develop mathematical models describing the evolution of stochastic
age-structured populations. After reviewing existing approaches, we
formulate a complete kinetic framework for age-structured interacting
populations undergoing birth, death and fission processes in
spatially dependent environments. We define the full probability
density for the population-size age chart and find results under
specific conditions. Connections with more classical models are also
explicitly derived. In particular, we show that factorial moments for
non-interacting processes are described by a natural generalization of
the McKendrick-von Foerster equation, which describes mean-field
deterministic behavior. Our approach utilizes mixed-type,
multidimensional probability distributions similar to those employed
in the study of gas kinetics and with terms that satisfy BBGKY-like