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Register a new userOn the derivative martingale in a branching random walk
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We work under the A{i}d{e}kon-Chen conditions which ensure that the derivative martingale in a supercritical branching random walk on the line converges almost surely to a nondegenerate nonnegative random variable that we denote by $Z$. It is shown that $mathbb{E} Zmathbf{1}_{{Zle x}}=log x+o(log x)$ as $xtoinfty$. Also, we provide necessary and sufficient conditions under which $mathbb{E} Zmathbf{1}_{{Zle x}}=log x+{rm const}+o(1)$ as $xtoinfty$. This more precise asymptotics is a key tool for proving distributional limit theorems which quantify the rate of convergence of the derivative martingale to its limit $Z$. The methodological novelty of the present paper is a three terms representation of a subharmonic function of at most linear growth for a killed centered random walk of finite variance. This yields the aforementioned asymptotics and should also be applicable to other models.
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A continuous-time particle system on the real line verifying the branching property and an exponential integrability condition is called a branching Levy process, and its law is characterized by a triplet $(sigma^2,a,Lambda)$. We obtain a necessary and sufficient condition for the convergence of the derivative martingale of such a process to a non-trivial limit in terms of $(sigma^2,a,Lambda)$. This extends previously known results on branching Brownian motions and branching random walks. To obtain this result, we rely on the spinal decomposition and establish a novel zero-one law on the perpetual integrals of centred Levy processes conditioned to stay positive.
In this paper, we reveal the branching structure for a non-homogeneous random walk with bounded jumps. The ladder time $T_1,$ the first hitting time of $[1,infty)$ by the walk starting from $0,$ could be expressed in terms of a non-homogeneous multitype branching process. As an application of the branching structure, we prove a law of large numbers of random walk in random environment with bounded jumps and specify the explicit invariant density for the Markov chain of ``the environment viewed from the particle .The invariant density and the limit velocity could be expressed explicitly in terms of the environment.
Consider a discrete-time one-dimensional supercritical branching random walk. We study the probability that there exists an infinite ray in the branching random walk that always lies above the line of slope $gamma-epsilon$, where $gamma$ denotes the asymptotic speed of the right-most position in the branching random walk. Under mild general assumptions upon the distribution of the branching random walk, we prove that when $epsilonto 0$, the probability in question decays like $exp{- {beta + o(1)over epsilon^{1/2}}}$, where $beta$ is a positive constant depending on the distribution of the branching random walk. In the special case of i.i.d. Bernoulli$(p)$ random variables (with $0<p<{1over 2}$) assigned on a rooted binary tree, this answers an open question of Robin Pemantle.
Using a high performance computer cluster, we run simulations regarding an open problem about d-dimensional critical branching random walks in a random IID environment The environment is given by the rule that at every site independently, with probability p>0, there is a cookie, completely suppressing the branching of any particle located there. Abstract. The simulations suggest self averaging: the asymptotic survival probability in n steps is the same in the annealed and the quenched case; it is frac{2}{qn}, where q:=1-p. This particular asymptotics indicates a non-trivial phenomenon: the tail of the survival probability (both in the annealed and the quenched case) is the same as in the case of non-spatial unit time critical branching, where the branching rule is modified: branching only takes place with probability q for every particle at every iteration.
The stochastic solutions to the Wigner equation, which explain the nonlocal oscillatory integral operator $Theta_V$ with an anti-symmetric kernel as {the generator of two branches of jump processes}, are analyzed. All existing branching random walk solutions are formulated based on the Hahn-Jordan decomposition $Theta_V=Theta^+_V-Theta^-_V$, i.e., treating $Theta_V$ as the difference of two positive operators $Theta^pm_V$, each of which characterizes the transition of states for one branch of particles. Despite the fact that the first moments of such models solve the Wigner equation, we prove that the bounds of corresponding variances grow exponentially in time with the rate depending on the upper bound of $Theta^pm_V$, instead of $Theta_V$. In other words, the decay of high-frequency components is totally ignored, resulting in a severe {numerical sign problem}. {To fully utilize such decay property}, we have recourse to the stationary phase approximation for $Theta_V$, which captures essential contributions from the stationary phase points as well as the near-cancelation of positive and negative weights. The resulting branching random walk solutions are then proved to asymptotically solve the Wigner equation, but {gain} a substantial reduction in variances, thereby ameliorating the sign problem. Numerical experiments in 4-D phase space validate our theoretical findings.
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