Quenched local central limit theorem for random walks in a time-dependent balanced random environment

Central Limit Theorem for Branching Random Walks in Random Environment

We consider branching random walks in d-dimensional integer lattice with time-space i.i.d. offspring distributions. When d ≥ 3 and the fluctuation of the environment is well moderated by the random walk, we prove a central limit theorem for the density of the population, together with upper bounds for the density of the most populated site and the replica overlap. We also discuss the phase tran...

A Local Limit Theorem for Random Walks in Balanced Environments

Central limit theorems for random walks in quenched random environments have attracted plenty of attention in the past years. More recently still, finer local limit theorems — yielding a Gaussian density multiplied by a highly oscillatory modulating factor — for such models have been obtained. In the one-dimensional nearest-neighbor case with i.i.d. transition probabilities, local limits of uni...

Quenched invariance principle for random walks in balanced random environment

We consider random walks in a balanced random environment in Z , d ≥ 2. We first prove an invariance principle (for d ≥ 2) and the transience of the random walks when d ≥ 3 (recurrence when d = 2) in an ergodic environment which is not uniformly elliptic but satisfies certain moment condition. Then, using percolation arguments, we show that under mere ellipticity, the above results hold for ran...

Central and Local Limit Theories in Markov Dependent Random Variables

Quenched Central Limit Theorems for Random Walks in Random Scenery

When the support of X1 is a subset of N , (Sn)n≥0 is called a renewal process. Each time the random walk is said to evolve in Z, it implies that the walk is truly d-dimensional, i.e. the linear space generated by the elements in the support of X1 is d-dimensional. Institut Camille Jordan, CNRS UMR 5208, Université de Lyon, Université Lyon 1, 43, Boulevard du 11 novembre 1918, 69622 Villeurbanne...