CENTRAL LIMIT THEOREMS FOR A BRANCHING RANDOM WALK WITH A RANDOM ENVIRONMENT IN TIME

We consider a branching random walk with a random environment m time, in which the offspring distribution of a particle of generation n and the distribution of the displacements of its children depend on an environment indexed by the time n. The envi- ronment is supposed to be independent and identically distributed. For A C R, let Zn（A） be the number of particles of generation n located in A. We show central limit theorems for the counting measure Zn （-） with appropriate normalization.

A NOTE ON EXACT CONVERGENCE RATE IN THE LOCAL LIMIT THEOREM FOR A LATTICE BRANCHING RANDOM WALK

Consider a branching random walk, where the underlying branching mechanism is governed by a Galton-Watson process and the moving law of particles by a discrete random variable on the integer lattice Z. Denote by Zn（z） the number of particles in the n-th generation in the model for each z ∈ Z. We derive the exact convergence rate in the local limit theorem for Zn（z） assuming a condition like ＂EN（logN）1＋λ 〈 ∞＂ for the offspring distribution and a finite moment condition on the motion law. This complements the known results for the strongly non-lattice branching random walk on the real line and for the simple symmetric branching random walk on the integer lattice.

The Spread Speed of Multiple Catalytic Branching Random Walks

In this paper we study the asymptotic behavior of the maximal position of a supercritical multiple catalytic branching random walk(X_(n))on Z.If M_(n) is its maximal position at time n,we prove that there is a constantα>0 such that M_(n)/n converges toαalmost surely on the set of infinite number of visits to the set of catalysts.We also derive the asymptotic law of the centered process M_(n)-αn as n→∞.Our results are similar to those in[13].However,our results are proved under the assumption of finite L log L moment instead of finite second moment.We also study the limit of(X_(n))as a measure-valued Markov process.For any function f with compact support,we prove a strong law of large numbers for the process X_(n)(f).

Branching random walks with random environments in time

We consider a branching random walk on N with a random environment in time （denoted by ξ）. Let Zn be the counting measure of particles of generation n, and let Zn（t） be its Laplace transform. We show the convergence of the free energy n-llog Zn（t）, large deviation principles, and central limit theorems for the sequence of measures {Zn}, and a necessary and sufficient condition for the existence of moments of the limit of the martingale Zn（t）/E[Zn（t）ξ].

A Branching Random Walk Method for Many-Body Wigner Quantum Dynamics

A branching random walk algorithm for many-body Wigner equations and its numerical applications for quantum dynamics in phase space are proposed and ana-lyzed in this paper.Using an auxiliary function,the truncated Wigner equation and its adjoint form are cast into integral formulations,which can be then reformulated into renewal-type equations with probabilistic interpretations.We prove that the first mo-ment of a branching random walk is the solution for the adjoint equation.With the help of the additional degree of freedom offered by the auxiliary function,we are able to produce a weighted-particle implementation of the branching random walk.In contrast to existing signed-particle implementations,this weighted-particle one shows a key ca-pacity of variance reduction by increasing the constant auxiliary function and has no time discretization errors.Several canonical numerical experiments on the 2D Gaussian barrier scattering and a 4D Helium-like system validate our theoretical findings,and demonstrate the accuracy,the efficiency,and thus the computability of the proposed weighted-particle Wigner branching random walk algorithm.

Convergence of complex martingale for a branching random walk in an independent and identically distributed environment

We consider anR^(d)-valued discrete time branching random walk in an independent and identically distributed environment indexed by time n∈N.Let W_(n)(z)(z∈C^(d))be the natural complex martingale of the process.We show necessary and sufficient conditions for the L^(α)-convergence of W_(n)(z)forα>1,as well as its uniform convergence region.

Extremum of a time-inhomogeneous branching random walk

Consider a time-inhomogeneous branching random walk, generated by the point process Ln which composed by two independent parts: ‘branching’offspring Xn with the mean 1+B(1+n)−β for β∈(0,1) and ‘displacement’ ξn with a drift A(1+n)^(−2α) for α∈(0,1/2), where the ‘branching’ process is supercritical for B>0 but ‘asymptotically critical’ and the drift of the ‘displacement’ ξn is strictly positive or negative for |A|>0 but ‘asymptotically’ goes to zero as time goes to infinity. We find that the limit behavior of the minimal (or maximal) position of the branching random walk is sensitive to the ‘asymptotical’ parameter β and α.

Critical survival barrier for branching random walk

We consider a branching random walk with an absorbing barrier,where the associated one-dimensional random walk is in the domain of attraction of an a-stable law.We shall prove that there is a barrier and a critical value such that the process dies under the critical barrier,and survives above it.This generalizes previous result in the case that the associated random walk has finite variance.

Fixed points of smoothing transformation in random environment

At each time n∈N,let Y^(n)(ξ)=(y^(n)_(1)(ξ),y^(n)_(2)(ξ),…)be a random sequence of non-negative numbers that are ultimately zero in a random environmentξ=(ξ_(n))n∈N.The existence and uniqueness of the non-negative fixed points of the associated smoothing transformation in random environment are considered.These fixed points are solutions to the distributional equation for a.e.ξ,Z(ξ)=dΣ_(i∈N_(+))y^(0)_(i)(ξ)Z^(1)_(i)(ξ),where{Z^(1)_(i):i∈N_(+)}are random variables in random environment which satisfy that for any environmentξ,under P_(ξ),{Z^(1)_(i)(ξ):i∈N_(+)}are independent of each other and Y^(0)(ξ),and have the same conditional distribution P_(ξ)(Z^(1)_(i)(ξ)∈·)=P_(Tξ)(Z(Tξ)∈·),where T is the shift operator.This extends the classical results of J.D.Biggins[J.Appl.Probab.,1977,14:25-37]to the random environment case.As an application,the martingale convergence of the branching random walk in random environment is given as well.