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T. E. Harris was a pioneer par excellence in many fields of probability theory. In this paper, we give a brief survey of the many fundamental contributions of Harris to the theory of branching processes, starting with his doctoral work at Princeton in the late forties and culminating in his fundamental book The Theory of Branching Processes, published in 1963.
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Interacting particle systems and percolation have been among the most active areas of probability theory over the past half century. Ted Harris played an important role in the early development of both fields. This paper is a birds eye view of his work in these fields, and of its impact on later research in probability theory and mathematical physics.
We consider $mathbb{R}^d$-valued diffusion processes of type begin{align*} dX_t = b(X_t)dt, +, dB_t. end{align*} Assuming a geometric drift condition, we establish contractions of the transitions kernels in Kantorovich ($L^1$ Wasserstein) distances with explicit constants. Our results are in the spirit of Hairer and Mattinglys extension of Harris Theorem. In particular, they do not rely on a small set condition. Instead we combine Lyapunov functions with reflection coupling and concave distance functions. We retrieve constants that are explicit in parameters which can be computed with little effort from one-sided Lipschitz conditions for the drift coefficient and the growth of a chosen Lyapunov function. Consequences include exponential convergence in weighted total variation norms, gradient bounds, bounds for ergodic averages, and Kantorovich contractions for nonlinear McKean-Vlasov diffusions in the case of sufficiently weak but not necessarily bounded nonlinearities. We also establish quantitative bounds for sub-geometric ergodicity assuming a sub-geometric drift condition.
We introduce and study the dynamics of an emph{immortal} critical branching process. In the classic, critical branching process, particles give birth to a single offspring or die at the same rates. Even though the average population is constant in time, the ultimate fate of the population is extinction. We augment this branching process with immortality by positing that either: (a) a single particle cannot die, or (b) there exists an immortal stem cell that gives birth to ordinary cells that can subsequently undergo critical branching. We discuss the new dynamical aspects of this immortal branching process.
The decomposable branching processes are relatively less studied objects, particularly in the continuous time framework. In this paper, we consider various variants of decomposable continuous time branching processes. As usual practice in the theory of decomposable branching processes, we group various types into irreducible classes. These irreducible classes evolve according to the well-studied nondecomposable/ irreducible branching processes. And we investigate the time evolution of the population of various classes when the process is initiated by the other class particle(s). We obtained class-wise extinction probability and the time evolution of the population in the different classes. We then studied another peculiar type of decomposable branching process where any parent at the transition epoch either produces a random number of offspring, or its type gets changed (which may or may not be regarded as new offspring produced depending on the application). Such processes arise in modeling the content propagation of competing contents in online social networks. Here also, we obtain various performance measures. Additionally, we conjecture that the time evolution of the expected number of shares (different from the total progeny in irreducible branching processes) is given by the sum of two exponential curves corresponding to the two different classes.
Crump-Mode-Jagers (CMJ) trees generalize Galton-Watson trees by allowing individuals to live for an arbitrary duration and give birth at arbitrary times during their life-time. In this paper, we exhibit a simple condition under which the height and contour processes of CMJ forests belong to the universality class of Bellman-Harris processes. This condition formalizes an asymptotic independence between the chronological and genealogical structures. We show that it is satisfied by a large class of CMJ processes and in particular, quite surprisingly, by CMJ processes with a finite variance offspring distribution. Along the way, we prove a general tightness result.
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