Let{Z_(n)}_(n)≥0 be a critical or subcritical d-dimensional branching random walk started from a Poisson random measure whose intensity measure is the Lebesugue measure on R^(d).Denote by R_(n):=sup{u>0:Z_(n)({x∈...Let{Z_(n)}_(n)≥0 be a critical or subcritical d-dimensional branching random walk started from a Poisson random measure whose intensity measure is the Lebesugue measure on R^(d).Denote by R_(n):=sup{u>0:Z_(n)({x∈R^(d):∣x∣<u})=0}the radius of the largest empty ball centered at the origin of Z_(n).In this work,we prove that after suitable renormalization,Rn converges in law to some non-degenerate distribution as n→∞.Furthermore,our work shows that the renormalization scales depend on the offspring law and the dimension of the branching random walk.This completes the results of Révész[13]for the critical binary branching Wiener process.展开更多
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 environmen...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.展开更多
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...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.展开更多
Consider a branching random walk with a random environment in time in the d-dimensional integer lattice.The branching mechanism is governed by a supercritical branching process,and the particles perform a lazy random ...Consider a branching random walk with a random environment in time in the d-dimensional integer lattice.The branching mechanism is governed by a supercritical branching process,and the particles perform a lazy random walk with an independent,non-identical increment distribution.For A■Z^(d),let Z_(n)(A)be the number of offsprings of generation n located in A.The exact convergence rate of the local limit theorem for the counting measure Z_(n)(·)is obtained.This partially extends the previous results for a simple branching random walk derived by Gao(2017,Stoch.Process Appl.).展开更多
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 converge...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].展开更多
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 Wi...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.展开更多
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 s...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.展开更多
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’ ξ...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.展开更多
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 constan...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).展开更多
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 va...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.展开更多
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-nega...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.展开更多
基金supported by the National Key R&D Program of China(2022YFA1006102).
文摘Let{Z_(n)}_(n)≥0 be a critical or subcritical d-dimensional branching random walk started from a Poisson random measure whose intensity measure is the Lebesugue measure on R^(d).Denote by R_(n):=sup{u>0:Z_(n)({x∈R^(d):∣x∣<u})=0}the radius of the largest empty ball centered at the origin of Z_(n).In this work,we prove that after suitable renormalization,Rn converges in law to some non-degenerate distribution as n→∞.Furthermore,our work shows that the renormalization scales depend on the offspring law and the dimension of the branching random walk.This completes the results of Révész[13]for the critical binary branching Wiener process.
基金partially supported by the National Natural Science Foundation of China(NSFC,11101039,11171044,11271045)a cooperation program between NSFC and CNRS of France(11311130103)+1 种基金the Fundamental Research Funds for the Central UniversitiesHunan Provincial Natural Science Foundation of China(11JJ2001)
文摘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.
文摘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.
基金supported by the National Natural Science Foundation of China(No.11971063)。
文摘Consider a branching random walk with a random environment in time in the d-dimensional integer lattice.The branching mechanism is governed by a supercritical branching process,and the particles perform a lazy random walk with an independent,non-identical increment distribution.For A■Z^(d),let Z_(n)(A)be the number of offsprings of generation n located in A.The exact convergence rate of the local limit theorem for the counting measure Z_(n)(·)is obtained.This partially extends the previous results for a simple branching random walk derived by Gao(2017,Stoch.Process Appl.).
基金Acknowledgements The authors would like to thank the anonymous referees for valuable comments and remarks. This work was partially supported by the Natural Scientific Research Innovation Foundation in Harbin Institute of Technology （HIT. NSRIF. 2015102）, the National Natural Science Foundation of China （Grant Nos. 11171044, 11101039）, and by the Natural Science Foundation of Hunan Province （Grant No. 11JJ2001）.
文摘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].
基金This research was supported by grants from the National Natural Science Foundation of China(Nos.11471025,11421101,11822102).
文摘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.
基金This work was supported in part by the National Natural Science Foundation of China(Nos.11601019,11971063,11501146)the Scientific Research Project of Beijing Municipal Education(Grant No.SQKM201610011006).
文摘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.
基金This work was supported by the National Key Research and Development Program of China(No.2020YFA0712900)the National Natural Science Foundation of China(Grant NO.11971062)the Fundamental Research Funds for the Central Universities Grant(No.N180503019).
文摘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.
基金supported in part by the National Natural Science Foundation of China (No.12271374)。
文摘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).
基金The authors are very grateful to Professor Mu-Fa Chen for his helpful comments and suggestions.This work was supported by the National Natural Science Foundation of China(Grant Nos.11871103,11371061)the Scientific and Technological Research Program of Chongqing Municipal Education Commission(Grant No.KJQN 201900514).
文摘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.
基金the National Key Research and Development Program of China(No.2020YFA0712900)the National Natural Science Foundation of China(Grant No.11971062)the Scientific Research Foundation for Young Teachers in Capital University of Economics and Business(NO.XRZ2021035).
文摘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.
