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Register a new userHeight and contour processes of Crump-Mode-Jagers forests (II): The Bellman-Harris universality class
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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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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.
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 prove the universality for the eigenvalue gap statistics in the bulk of the spectrum for band matrices, in the regime where the band width is comparable with the dimension of the matrix, $Wsim N$. All previous results concerning universality of non-Gaussian random matrices are for mean-field models. By relying on a new mean-field reduction technique, we deduce universality from quantum unique ergodicity for band matrices.
This paper contributes to the study of class $(Sigma^{r})$ as well as the c`adl`ag semi-martingales of class $(Sigma)$, whose finite variational part is c`adl`ag instead of continuous. The two above-mentioned classes of stochastic processes are extensions of the family of c`adl`ag semi-martingales of class $(Sigma)$ considered by Nikeghbali cite{nik} and Cheridito et al. cite{pat}; i.e., they are processes of the class $(Sigma)$, whose finite variational part is continuous. The two main contributions of this paper are as follows. First, we present a new characterization result for the stochastic processes of class $(Sigma^{r})$. More precisely, we extend a known characterization result that Nikeghbali established for the non-negative sub-martingales of class $(Sigma)$, whose finite variational part is continuous (see Theorem 2.4 of cite{nik}). Second, we provide a framework for unifying the studies of classes $(Sigma)$ and $(Sigma^{r})$. More precisely, we define and study a new larger class that we call class $(Sigma^{g})$. In particular, we establish two characterization results for the stochastic processes of the said class. The first one characterizes all the elements of class $(Sigma^{g})$. Hence, we derive two corollaries based on this result, which provides new ways to characterize classes $(Sigma)$ and $(Sigma^{r})$. The second characterization result is, at the same time, an extension of the above mentioned characterization result for class $(Sigma^{r})$ and of a known characterization result of class $(Sigma)$ (see Theorem 2 of cite{fjo}). In addition, we explore and extend the general properties obtained for classes $(Sigma)$ and $(Sigma^{r})$ in cite{nik,pat,mult, Akdim}.
We prove fluctuation bounds for the particle current in totally asymmetric zero range processes in one dimension with nondecreasing, concave jump rates whose slope decays exponentially. Fluctuations in the characteristic directions have order of magnitude $t^{1/3}$. This is in agreement with the expectation that these systems lie in the same KPZ universality class as the asymmetric simple exclusion process. The result is via a robust argument formulated for a broad class of deposition-type processes. Besides this class of zero range processes, hypotheses of this argument have also been verified in the authors earlier papers for the asymmetric simple exclusion and the constant rate zero range processes, and are currently under development for a bricklayers process with exponentially increasing jump rates.
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