Non-perturbative approach to random walk in markovian environment
We prove an averaged CLT for a random walk in a dynamical environment where the states of the environment at different sites are independent Markov chains.
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arXiv subjects
Dmitry Dolgopyat
Publications and source records attributed to Dmitry Dolgopyat.
Random walk in Markovian environment
We prove a quenched central limit theorem for random walks with bounded increments in a randomly evolving environment on $\mathbb{Z}^d$. We assume that the transition probabilities of the walk depend not too strongly on the environment and that the evolution of the environment is Markovian with strong spatial and temporal mixing properties.
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Random Walk in deterministically changing environment
We consider a random walk with transition probabilities weakly dependent on an environment with a deterministic, but strongly chaotic, evolution. We prove that for almost all initial conditions of the environment the walk satisfies the CLT.
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Sample path properties of the stochastic flows
We consider a stochastic flow driven by a finite dimensional Brownian motion. We show that almost every realization of such a flow exhibits strong statistical properties such as the exponential convergence of an initial measure to the equilibrium state and the central limit theorem. The proof uses new estimates of the mixing rates of the multipoint motion.
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Hausdorff dimension in stochastic dispersion
We consider the evolution of a connected set in Euclidean space carried by a periodic incompressible stochastic flow. While for almost every realization of the random flow at time t most of the particles are at a distance of order sqrt{t} away from the origin [DKK1], there is an uncountable set of measure zero of points, which escape to infinity at the linear rate [CSS1]. In this paper we prove that this set of linear escape points has full Hausdorff dimension.
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A limit shape theorem for periodic stochastic dispersion
We consider the evolution of a connected set on the plane carried by a periodic incompressible stochastic flow. While for almost every realization of the random flow at time t most of the particles are at a distance of order sqrt{t} away from the origin, there is a measure zero set of points, which escape to infinity at the linear rate. We study the set of points visited by the original set by time t, and show that such a set, when scaled down by the factor of t, has a limiting non random shape.
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