We consider a branching random walk on the lattice, where the branching rates are given by an i.i.d. Pareto random potential. We describe the process, including a detailed shape theorem, in terms of a system of growing lilypads. As an application we show that the branching random walk is intermittent, in the sense that most particles are concentrated on one very small island with large potential. Moreover, we compare the branching random walk to the parabolic Anderson model and observe that although the two systems show similarities, the mechanisms that control the growth are fundamentally different.
Closely related to our model is a branching random walk on @math in discrete time with a spatial i.i.d. offspring distribution. Here, much more is known about the number of particles. Early work includes @cite_15 @cite_7 for @math , who start with an infinite population and describe the local and global growth rates in terms of a variational problem (depending on a drift in the underlying random walk). Many other authors address the question of survival (see e.g. @cite_12 @cite_14 ) and recurrence vs. transience (see e.g. @cite_2 @cite_1 @cite_24 @cite_13 ).
@cite_14 @cite_7 @cite_1 @cite_24 @cite_2 @cite_15 @cite_13 @cite_12
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We study survival of nearest-neighbor 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. We study branching random walks in random i.i.d. environment in Z d , d ≥ 1. For this model, the population size cannot decrease, and a natural definition of recurrence is introduced. We prove a dichotomy for recurrence transience, depending only on the support of the environmental law. We give sufficient conditions for recurrence and for transience. In the recurrent case, we study the asymptotics of the tail of the distribution of the hitting times and prove a shape theorem for the set of lattice sites which are visited up to a large time. This paper considers an infinite system of particles on the integers @math that: (1) step to the right with a random delay, and (2) split or die along the way according to a random law depending on their position. The exponential growth rate of the particle density is computed in the long time limit in the form of a variational formula that can be solved explicitly. The result reveals two phase transitions associated with localization vs. delocalization and survival vs. extinction. In addition, the system exhibits an intermittency effect. Greven and den Hollander considered the more difficult situation where the particles may step both to the left and right, but the analysis of the phase diagram was less complete. Let (η n ) be the infinite particle system on ℤ whose evolution is as follows. At each unit of time each particle independently is replaced by a new generation. The size of a new generation descending from a particle at sitex has distributionF x and each of its members independently jumps to sitex±1 with probability (1±h) 2,h∈[0, 1]. The sequence F x is i.i.d. with uniformly bounded second moment and is kept fixed during the evolution. The initial configurationη0 is shift invariant and ergodic. We develop a criterion for transience for a general model of branching Markov chains. In the case of multi-dimensional branching random walk in random environment (BRWRE) this criterion becomes explicit. In particular, we show that Condition L of Comets and Popov [3] is necessary and sufficient for transience as conjectured. Furthermore, the criterion applies to two important classes of branching random walks and implies that the critical branching random walk is transient resp. dies out locally. We consider a particular Branching Random Walk in Random Environment (BRWRE) on @math started with one particle at the origin. Particles reproduce according to an offspring distribution (which depends on the location) and move either one step to the right (with a probability in @math which also depends on the location) or stay in the same place. We give criteria for local and global survival and show that global survival is equivalent to exponential growth of the moments. Further, on the event of survival the number of particles grows almost surely exponentially fast with the same growth rate as the moments.