Supercritical branching Brownian motion with catalytic branching at the origin

We consider a catalytic branching Brownian motion with general branching which takes place only when particles are at the origin at a rate β>0 on the local time scale. We first establish a spine decomposition for the case wherein the particles have a positive probability of having no children. Then using this tool, we obtain results regarding the asymptotic behavior of the number of particles above λt at time t for λ>0. Under an L log L condition, we prove a strong law of large numbers for this catalytic branching Brownian motion.     

半流的原像熵

本文对紧致度量空间上的连续半流引入了几类原像熵的定义,并对它们的性质进行了研究,证明了对于无不动点的连续半流而言,这些熵具有一定程度的拓扑共轭不变性,对这些熵的关系进行了研究并得到了联系这些熵的不等式,还证明了连续半流与其时刻1映射具有相同的拓扑熵和原像熵。     

Independence Inequality of Entropies

This paper provides an investigation of the independence inequalityfor more than ten entropy measures. It also provides the effectof linear transformations on this inequality. Various channelshave been considered in the article.     

Entropies and Flux-Splittings for the Isentropic Euler Equations

51. IntroductionThe Euler equations for an iselitropic compressible fluid readwhere p 2 0 denotes the density, v the velocity) and p(p) 2 0 the pressure. The equstiope(1.1) form a nonlinear hyperbolic system of conserVation laws. By definition, a msthematicalentropy n = n(p, v) and its corresponding elltropy flux-function q = q(p, v) satisfyfor any smooth solution (p,m) of (1.1). A weak entropy3 by definition, vanishes on thevacuum p = 0. Following Laxlll'lz], we are interested in measuxable…     

Discrete Entropies of Orthogonal Polynomials

Given a nontrivial Borel measure on ℝ, let p _n be the corresponding orthonormal polynomial of degree n whose zeros are λ _j ⁽ⁿ⁾, j=1,…,n. Then for each j=1,…,n,

On a property related to convergence in probability and some applications to branching processes

It is shown that any real-valued sequence of random variables {X_n} converging in probability to a non-degenerate, not necessarily a.s. finite limit X possesses the following property: for any c with P(X? (c ? δ, c + δ)) > 0 for all δ > 0, there exists a sequence {c_n} with lim_n→∞ c_n = c such that for any ε > 0, lim_n→∞ P(Xδ (c ? ε, c + ε) |X_n = c_n) = 1. This property is applied to various types of branching processes where $\text{X}{}_{\mathrm{n}}\text{=}\text{Z}{}_{\mathrm{n}}\text{C}{}_{\mathrm{n}}$ or $\text{X}{}_{\mathrm{n}}\text{=}\text{U(Z}{}_{\mathrm{n}}\text{)}\text{C}{}_{\mathrm{n}}\text{{C}{}_{\mathrm{n}}\text{}}$ being a sequence of constants or random variables and U a slowly varying function. If {Z_n} is a supercritical branching process in varying or random environment, X is shown to have a continuous and strictly increasing distribution function on (0, ∞). Characterizations of the tail of the liniting distribution of the finite mean and the infinite mean supercritical Galton-Watson processes are also obtained.     

非自治动力系统的原像熵

本文对紧致度量空间上的连续自映射序列应用生成集和分离集引入了点原像熵、原像分枝熵以及原像关系熵等几类原像熵的定义并进行了研究.主要结果是:(1) 证明了这些熵都是等度拓扑共轭不变量.(2)讨论了这些原像熵之间及它们与拓扑熵之间的关系,得到了联系这些熵的不等式.(3)证明了对正向可扩的连续自映射序列而言, 两类点原像熵相等,原像分枝熵与原像关系熵也相等.(4)证明了对(a).由闭Riemann 流形上的一个扩张映射经充分小的C1-扰动生成的自映射序列,以及(b).有限图上等度连续的自映射序列,有零原像分枝熵.     

The branching chain with drift in space-time random environment (I): Model,Markov property,moments

There are three parts in this article. In Section 1, we establish the model of branching chain with drift in space-time random environment (BCDSTRE), i.e., the coupling of branching chain and random walk. In Section 2, we prove that any BCDSTRE must be a Markov chain in time random environment when we consider the distribution of the particles in space as a random element. In Section 3, we calculate the first-order moments and the second-order moments of BCDSTRE.     

Nonlinear branching processes with immigration

The nonlinear branching process with immigration is constructed as the pathwise unique solution of a stochastic integral equation driven by Poisson random measures. Some criteria for the regularity, recurrence, ergodicity and strong ergodicity of the process are then established.     

Smooth quasiregular mappings with branching

We give an example of a -smooth quasiregular mapping in 3-space with nonempty branch set. Moreover, we show that the branch set of an arbitrary quasiregular mapping in n-space has Hausdorff dimension quantitatively bounded away from n. By using the second result, we establish a new, qualitatively sharp relation between smoothness and branching.     

Entropies and weak solutions of the compressible isentropic Euler equations

In this Note, we study the system of isentropic Euler equations for compressible fluids, with a general equation of state. We establish the existence of the fundamental kernel that generates the family of weak entropies, and study its singularities. The kernel is the solution of an equation of Euler-Poisson-Darboux type, and its partial derivative with respect to the density variable tends to a Dirac measure as the density approaches zero. We prove a new reduction theorem for the Young measures associated with the compressible Euler system. From these results, we deduce the existence, compactness, and asymptotic decay of measurable and bounded entropy solutions.     

Multifractal Analysis of Local Entropies for Gibbs Measures

Recently in the series of papers L. Barreira, Ya.B. Pesin, J. Schmeling and H. Weiss performed a complete multifractal analysis of local dimensions, entropies and Lyapunov exponents of conformal expanding maps and surface Axion A diffeomorphisms for Gibbs measures. The main goal of these papers was primarily the analysis of the local (pointwise) dimensions. This is an extremely difficult problem and, for example, similar results for hyperbolic systems in dimensions 3 and higher have not been yet obtained. In the present work we concentrate our attention on the multifractal analysis of the local (pointwise) entropies. We are able to obtain results, which are similar to those mentioned above, for Gibbs measures of the expansive homeomorphisms with specification property. Note that such homeomorphisms may not have Markov partitions, which is a crucial condition in the previous works. However, due to the fact that less is known about thermodynamical properties of these dynamical systems we were able to obtain only the continuous differentiability of the multifractal spectrum of local entropies (compare: the same spectra for the dynamical systems with Markov partitions are analytic). We believe that the smoothness of the multifractal spectrum in our case can be improved. We have related the multifractal spectrum of the local entropies to the the spectrum of correlation entropies. These correlation entropies serve as entopy-like analogs of the Hentshel-Procaccia and Rényi spectra of generalized dimensions. This allows us to complete the duality between the multifractal analyses of local dimensions and entropies. Complete proofs can be found in [TV98] and will appear elsewhere.     

Scenery reconstruction with branching random walk

We study the problem of scenery reconstruction in arbitrary dimension using observations registered in boxes of size k (for k fixed), seen along a branching random walk. We prove that, using a large enough k for almost all the realizations of the branching random walk, almost all sceneries can be reconstructed up to equivalence.     

出生率与年龄段有关的生灭分枝树

本文基于生灭过程的生灭演化机理, 将生物繁衍过程描述为有向随机图过程-随机分枝树, 建立了出生率与年龄段有关的生灭分枝树演化模型. 本文研究了任一节点在不同年龄及临死时刻的出度分布、虚出度分布和拟出度分布, 并证明了拟出度过程是随机时刻终止的Poisson过程, 讨论了首生年龄及相对出生年龄的分布, 给出了任一节点成为孤立节点的概率.     

Measure-valued branching diffusions with spatial interactions

Summary A strong equation driven by a historical Brownian motion is used to construct and characterize measure-valued branching diffusions in which the spatial motions obey an Itô equation with drift and diffusion depending on the position of an individual and the entire population.     

Markov branching processes with instantaneous immigration

Summary Markov branching processes with instantaneous immigration possess the property that immigration occurs immediately the number of particles reaches zero, i.e. the conditional expectation of sojourn time at zero is zero. In this paper we consider the existence and uniqueness of such a structure. We prove that if the sum of the immigration rates is finite then no such structure can exist, and we provide a necessary and sufficient condition for existence for the case in which this sum is infinite. Study of the uniqueness problem shows that for honest processes the solution is unique.     

Supercritical age-dependent branching processes with immigration

This paper considers a population process where individuals reproduce according to an age-dependent branching process and immigrants enter the population at the event epochs of an ergodic point process. A limit theorem is proven for what corresponds to the supercritical case, and the limit random variable is investigated.