On total progeny of multitype Galton-Watson process and the first passage time of random walk on lattice

In this paper,we form a method to calculate the probability generating function of the total progeny of multitype branching process.As examples,we calculate probability generating function of the total progeny of the multitype branching processes within random walk which could stay at its position and(2-1) random walk.Consequently,we could give the probability generating functions and the distributions of the first passage time of corresponding random walks.Especially,for recurrent random walk which could stay at its position with probability 0 r 1,we show that the tail probability of the first passage time decays as 2/(π(1-r)~(1/2)) n~(1/1)= when n →∞.     

Some results on first passage times in one dimensional random walks

Analytic expressions are presented for the characteristic function of the first passage time distribution for biased random walk on a finite chain (and diffusion with drift on a finite line); of the first passage time distribution for a random walk on a chain, in which the events (jumps) are governed by an arbitrary renewal process; and of the distribution of the time of escape from a bounded set of points in the latter case. A fundamental relation between the first passage time distribution and the conditional probability for random walk (or diffusion) in one dimension is analyzed and generalized.     

带形上随机环境中随机游动的内蕴分枝结构

揭示了带形上随机环境中随机游动的内蕴分枝结构一带移民的多物种分枝过程.利用内蕴分枝结构,可精确表达游动的首次击中时.给出了内蕴分枝结构的如下两个应用:(1)计算出首次击中时的均值,给出游动大数定律速度的显示表达,(2)得到从粒子角度看环境的马氏链不变测度的密度函数的显示表达,进而可用另一种"站在粒子看环境"的方法直接证明游动的大数定律.     

Almost sure convergence of continuous time branching processes

We prove necessary and sufficient conditions for the transience of the non-zero states in a non-homogeneous, continuous time Markov branching process. The result is obtained by passing from results about the discrete time skeleton of the continuous time chain to the continuous time chain itself. An alternative proof of a result for continuous time Markov branching processes in random environments is then given, showing that earlier moment conditions were not necessary.     

Approximation for the Normal Inverse Gaussian Process Using Random Sums

Abstract

We approximate the normal inverse Gaussian (NIG) process with random summations. The random sum we introduce is a random walk subordinated to the first passage time of another independent random walk; the model is interpreted as an internal mechanism at small scale that generates the NIG process. The main result is a functional limit theorem of weak convergence in the Skorohod topology.     

On the first passage time of a simple random walk on a tree

We consider a simple random walk on a tree. Exact expressions are obtained for the expectation and the variance of the first passage time, thereby recovering the known result that these are integers. A relationship of the mean first passage matrix with the distance matrix is established and used to derive a formula for the inverse of the mean first passage matrix.     

Quenched Large Deviations for Multidimensional Random Walk in Random Environment with Holding Times

We consider a random walk in random environment with random holding times, that is, the random walk jumping to one of its nearest neighbors with some transition probability after a random holding time. Both the transition probabilities and the laws of the holding times are randomly distributed over the integer lattice. Our main result is a quenched large deviation principle for the position of the random walk. The rate function is given by the Legendre transform of the so-called Lyapunov exponents for the Laplace transform of the first passage time. By using this representation, we derive some asymptotics of the rate function in some special cases.     

A random walk with a branching system in random environments

We consider a branching random walk in random environments, where the particles are reproduced as a branching process with a random environment (in time), and move independently as a random walk on Z with a random environment (in locations). We obtain the asymptotic properties on the position of the rightmost particle at time n, revealing a phase transition phenomenon of the system.     

On Exit Times of a Multivariate Random Walk and Its Embedding in a Quasi Poisson Process

Abstract

We introduce and analyze a delayed renewal process  = {τ₀,τ₁,…} marked by a multivariate random walk (,) and its behavior about fixed levels to be crossed by one of the components of (,). We derive the joint distribution of first passage time τ_ρ, pre-exit time τ_ρ?1 (i.e., the instant one phase prior to the first passage time), and the respective values of (,) at τ_ρ and τ_ρ?1 in a closed form. The results obtained are then applied to a multivariate quasi Poisson process Π, forming a random walk ((Π),) embedded in Π over . Processes like these can model various phenomena including stock market and option trading.

One of the central issues in the investigation of ((Π),) is to obtain the information about Π at any moment of time in random vicinities of τ_ρ and τ_ρ?1 previously available only upon . The results offer, again, closed form functionals. Numerous examples throughout the paper illustrate introduced constructions and connect the results with real-world applications, most prominently the stock market.     

The monotonicity of the probability of return into the source in models of branching random walks

The paper discusses two models of a branching random walk on a many-dimensional lattice with birth and death of particles at a single node being the source of branching. The random walk in the first model is assumed to be symmetric. In the second model an additional parameter is introduced which enables “artificial” intensification of the prevalence of branching or walk at the source and, as the result, violating the symmetry of the random walk. The monotonicity of the return probability into the source is proved for the second model, which is a key property in the analysis of branching random walks.     

Hitting times with taboo for a random walk

For a symmetric homogeneous and irreducible random walk on the d-dimensional integer lattice, which have a finite variance of jumps, we study passage times (taking values in [0,??]) determined by a starting point x, a hitting state y, and a taboo state z. We find the probability that these passage times are finite, and study the distribution tail. In particular, it turns out that, for the above-mentioned random walks on ? ^d except for a simple random walk on ?, the order of the distribution tail decrease is specified by dimension d only. In contrast, for a simple random walk on ?, the asymptotic properties of hitting times with taboo essentially depend on mutual location of the points x, y, and z. These problems originated in recent study of a branching random walk on ? ^d with a single source of branching.     

组合分布在求解直线上n步随机游动首次通过问题上的应用

基于组合过程模型给出其轨道对目标集的首次通过概率及首中点的分布函数 ,并由此给出直线上n步随机游动的首次通过概率及首中点分布函数的一类显式 .     

A model of longitudinal dispersion in non–homogeneous porous media

This paper is concerned with a model of the longitudinal dispersion of a set of tagged particles in a flow through nonhomogeneous stratified porous media. The model is a particular homogeneous Markov jump process, the successive states visited forming a non-homogeneous Poisson process. The passage time of alevel x of this Markov process is an additive stochastic process.This property is used to evaluate the general form of the longitudinal concentration function of the tagged particles in the flow.     

解几何微扰问题的一种无分枝蒙特卡罗方法:共域变换方法

§1.引言对于任何一个核增殖系统,当系统的条件发生微小变化时,系统的有效增殖因子将随     

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We study a birth and death process $\{N_t\}_{t\ge0}$ in i.i.d. random environment, for which at each discontinuity, one particle might be born or at most $L$ particles might be dead. Along with investigating the existence and the recurrence criterion, we also study the law of large numbers of $\{N_t\}$. We show that the first passage time can be written as a functional of an $L$-type branching process in random environment and a sequence of independent and exponentially distributed random variables. Consequently, an explicit velocity of the law of large numbers can be given.     

Subcritical branching processes in a random environment without the Cramer condition

A subcritical branching process in random environment (BPRE) is considered whose associated random walk does not satisfy the Cramer condition. The asymptotics for the survival probability of the process is investigated, and a Yaglom type conditional limit theorem is proved for the number of particles up to moment $n$ given survival to this moment. Contrary to other types of subcritical BPRE, the limiting distribution is not discrete. We also show that the process survives for a long time owing to a single big jump of the associate random walk accompanied by a population explosion at the beginning of the process.     

The first passage time density of Ornstein–Uhlenbeck process with continuous and impulsive excitations

The first passage time of the Ornstein–Uhlenbeck process plays a prototype role in various noise-induced escape problems. In order to calculate the first passage time density of the Ornstein–Uhlenbeck process modulated by continuous and impulsive periodic excitations using the second kind Volterra integral equation method, we adopt an approximation scheme of approaching Dirac delta function by alpha function to transform the involved discontinuous dynamical threshold into a smooth one. It is proven that the first passage time of the approximate model converges to the first passage time of the original model in probability as the approximation exponent alpha tends to infinity. For given parameters, our numerical realizations further demonstrate that good approximation effect can be achieved when the approximation exponent alpha is 10.     

Limit distributions for the number of particles in branching random walks

We study branching random walks with continuous time. Particles performing a random walk on ?², are allowed to be born and die only at the origin. It is assumed that the offspring reproduction law at the branching source is critical and the random walk outside the source is homogeneous and symmetric. Given particles at the origin, we prove a conditional limit theorem for the joint distribution of suitably normalized numbers of particles at the source and outside it as time unboundedly increases. As a consequence, we establish the asymptotic independence of such random variables.     

A weakly supercritical mode in a branching random walk

The case of weakly supercritical branching random walks is considered. A theorem on asymptotic behavior of the eigenvalue of the operator defining the process is obtained for this case. Analogues of the theorems on asymptotic behavior of the Green function under large deviations of a branching random walk and asymptotic behavior of the spread front of population of particles are established for the case of a simple symmetric branching random walk over a many-dimensional lattice. The constants for these theorems are exactly determined in terms of parameters of walking and branching.