Results 1  10
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25
ON SOME RANDOM WALKS ON Z IN RANDOM MEDIUM
, 2002
"... We consider random walks on Z in a stationary random medium, defined by an ergodic dynamical system, in the case when the possible jumps are {−L,...,−1, 0, +1} for some fixed integer L. Weprovidearecurrence criterion expressed in terms of the sign of the maximal Liapounov exponent of a certain rando ..."
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Cited by 21 (3 self)
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We consider random walks on Z in a stationary random medium, defined by an ergodic dynamical system, in the case when the possible jumps are {−L,...,−1, 0, +1} for some fixed integer L. Weprovidearecurrence criterion expressed in terms of the sign of the maximal Liapounov exponent of a certain random matrix and give an algorithm of calculation of that exponent. Next, we characterize the existence of the absolutely continuous invariant measure for the Markov chain of “the environments viewed from the particle ” and also characterize, in the transient cases, the existence of a nonzero drift. To study the validity of the central limit theorem, we consider the notion of harmonic coordinates introduced by Kozlov. We characterize the existence of both the invariant measure and the harmonic coordinates and show in the recurrent case that the existence of those two objects is equivalent to the validity of an invariance principle. We give sufficient conditions for the validity of the central limit theorem in the transient cases. Finally, we consider the previous results in the context of a random medium defined
Survival of branching random walks in random environment
, 811
"... We study survival of nearestneighbour branching random walks in random environment (BRWRE) on Z. A priori there are three different regimes of survival: global survival, local survival, and strong local survival. We show that local and strong local survival regimes coincide for BRWRE and that they ..."
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Cited by 17 (2 self)
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We study survival of nearestneighbour branching random walks in random environment (BRWRE) on Z. A priori there are three different regimes of survival: global survival, local survival, and strong local survival. We show that local and strong local survival regimes coincide for BRWRE and that they can be characterized with the spectral radius of the first moment matrix of the process. These results are generalizations of the classification of BRWRE in recurrent and transient regimes. Our main result is a characterization of global survival that is given in terms of Lyapunov exponents of an infinite product of i.i.d. 2 × 2 random matrices. 1 1
Almost Sure Rates of Mixing for I.i.d. Unimodal Maps
, 1999
"... . It has been known since the pioneering work of Jakobson and subsequent work by BenedicksCarleson and others that a positive measure set of quadratic maps admit an absolutely continuous invariant measure. Young and KellerNowicki proved exponential decay of its correlation functions. BenedicksYou ..."
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Cited by 16 (3 self)
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. It has been known since the pioneering work of Jakobson and subsequent work by BenedicksCarleson and others that a positive measure set of quadratic maps admit an absolutely continuous invariant measure. Young and KellerNowicki proved exponential decay of its correlation functions. BenedicksYoung [BeY] and BaladiViana [BV] studied stability of the density and exponential rate of decay of the Markov chain associated to i.i.d. small perturbations. The almost sure statistical properties of the sample measures of i.i.d. itineraries are more dicult to estimate than the \averaged statistics." Adapting to random systems, on the one hand the notion of hyperbolic times due to Alves [A], and on the other a probabilistic coupling method introduced by Young [Yo2] to study rates of mixing, we prove stretched exponential upper bounds for the almost sure rates of mixing. 1. Introduction An important class of discretetime deterministic dynamical systems (given by a transformation f on a Rieman...
Consensus and products of random stochastic matrices: Exact rate for convergence in probability
 Carnegie Mellon University (CMU
, 2013
"... Abstract—We find the exact rate for convergence in probability of products of independent, identically distributed symmetric, stochastic matrices. It is wellknown that if the matrices have positive diagonals almost surely and the support graph of the mean or expected value of the random matrices ..."
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Cited by 9 (6 self)
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Abstract—We find the exact rate for convergence in probability of products of independent, identically distributed symmetric, stochastic matrices. It is wellknown that if the matrices have positive diagonals almost surely and the support graph of the mean or expected value of the random matrices is connected, the products of the matrices converge almost surely to the average consensus matrix, and thus in probability. In this paper, we show that the convergence in probability is exponentially fast, and we explicitly characterize the exponential rate of this convergence. Our analysis reveals that the exponential rate of convergence in probability depends only on the statistics of the support graphs of the randommatrices. Further, we show how to compute this rate for commonly used randommodels: gossip and link failure.With thesemodels, the rate is found by solving a mincut problem, and hence it is easily computable. Finally, as an illustration, we apply our results to solving power allocation among networked sensors in a consensus+innovations distributed detection problem. Index Terms—Consensus, consensus innovations, convergence in probability, exponential rate, performance analysis, random network. I.
Onedimensional finite range random walk in random medium and
, 2007
"... invariant measure equation ..."
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On fixed points of a generalized multidimensional affine recursion. arXiv:1111.1756v1
, 2011
"... ar ..."
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Series Expansions of Lyapunov Exponents and Forgetful Monoids
, 2000
"... We consider Lyapunov exponents of random iterates of monotone homogeneous maps. We assume that the images of some iterates are lines, with positive probability. Using this memoryloss property which holds generically for random products of matrices over the maxplus semiring, and in particular, for ..."
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Cited by 5 (0 self)
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We consider Lyapunov exponents of random iterates of monotone homogeneous maps. We assume that the images of some iterates are lines, with positive probability. Using this memoryloss property which holds generically for random products of matrices over the maxplus semiring, and in particular, for Tetrislike heaps of pieces models, we give a series expansion formula for the Lyapunov exponent, as a function of the probability law. In the case of rational probability laws, we show that the Lyapunov exponent is an analytic function of the parameters of the law, in a domain that contains the absolute convergence domain of a partition function associated to a special "forgetful" monoid, defined by generators and relations.
Does dormancy increase fitness of bacterial populations in timevarying environments
, 2008
"... A simple family of models of a bacterial population in a time varying environment in which cells can transit between dormant and active states is constructed. It consists of a linear system of ordinary differential equations for active and dormant cells with timedependent coefficients reflecting ..."
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A simple family of models of a bacterial population in a time varying environment in which cells can transit between dormant and active states is constructed. It consists of a linear system of ordinary differential equations for active and dormant cells with timedependent coefficients reflecting an environment which may be periodic or random, with alternate periods of low and high resource levels. The focus is on computing/estimating the dominant Lyapunov exponent, the fitness, and determining its dependence on various parameters and the two strategies – responsive and stochastic – by which organisms switch between dormant and active states. A responsive switcher responds to good and bad times by making timely and appropriate transitions while a stochastic switcher switches continuously without regard to the environmental state. The fitness of a responsive switcher is examined and compared with fitness of a stochastic switcher, and with the fitness of a dormancyincapable organism. Analytical methods show that both switching strategists have higher fitness than a dormancyincapable organism when good times are rare and that responsive switcher has higher fitness than stochastic switcher when good times are either rare or common. Numerical calculations show that stochastic switcher can be most fit when good times are neither too rare or too common.