157 research outputs found
Motion in a Random Force Field
We consider the motion of a particle in a random isotropic force field.
Assuming that the force field arises from a Poisson field in , , and the initial velocity of the particle is sufficiently large, we
describe the asymptotic behavior of the particle
Energy transfer in a fast-slow Hamiltonian system
We consider a finite region of a lattice of weakly interacting geodesic flows
on manifolds of negative curvature and we show that, when rescaling the
interactions and the time appropriately, the energies of the flows evolve
according to a non linear diffusion equation. This is a first step toward the
derivation of macroscopic equations from a Hamiltonian microscopic dynamics in
the case of weakly coupled systems
Constructive approach to limit theorems for recurrent diffusive random walks on a strip.
We consider recurrent diffusive random walks on a strip. We present
constructive conditions on Green functions of finite sub-domains which imply a
Central Limit Theorem with polynomial error bound, a Local Limit Theorem, and
mixing of environment viewed by the particle process. Our conditions can be
verified for a wide class of environments including independent environments,
quasiperiodic environments, and environments which are asymptotically constant
at infinity. The conditions presented deal with a fixed environment, in
particular, no stationarity conditions are imposed
On second order elliptic equations with a small parameter
The Neumann problem with a small parameter
is
considered in this paper. The operators and are self-adjoint second
order operators. We assume that has a non-negative characteristic form
and is strictly elliptic. The reflection is with respect to inward
co-normal unit vector . The behavior of
is effectively described via
the solution of an ordinary differential equation on a tree. We calculate the
differential operators inside the edges of this tree and the gluing condition
at the root. Our approach is based on an analysis of the corresponding
diffusion processes.Comment: 28 pages, 1 figure, revised versio
LOCAL LIMIT THEOREMS FOR RANDOM WALKS IN A RANDOM ENVIRONMENT ON A STRIP
The paper consists of two parts. In the first part we review recent work on
limit theorems for random walks in random environment (RWRE) on a strip with
jumps to the nearest layers. In the second part, we prove the quenched Local
Limit Theorem (LLT) for the position of the walk in the transient diffusive
regime. This fills an important gap in the literature. We then obtain two
corollaries of the quenched LLT. The first one is the annealed version of the
LLT on a strip. The second one is the proof of the fact that the distribution
of the environment viewed from the particle (EVFP) has a limit for a. e.
environment. In the case of the random walk with jumps to nearest neighbours in
dimension one, the latter result is a theorem of Lally \cite{L}. Since the
strip model incorporates the walks with bounded jumps on a one-dimensional
lattice, the second corollary also solves the long standing problem of
extending Lalley's result to this case
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An Error Term in the Central Limit Theorem for Sums of Discrete Random Variables
The file accessible on this institutional repository is the preprint, arXiv:2303.10235v1 [math.PR] (for this version) available under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License (https://creativecommons.org/licenses/by-nc-sa/4.0/) at: https://doi.org/10.48550/arXiv.2303.10235 -- it has not been certified by peer review. The embargoed file is the authors accepted manuscript available online from the author's website at: https://www.math.umd.edu/~dolgop/papers.html . The final, peer reviewed version of record is available online at: https://doi.org/10.1093/imrn/rnad088 .Copyright © The Author(s) 2023. We consider sums of independent identically distributed random variables whose distributions have d + 1 atoms. Such distributions never admit an Edgeworth expansion of order d but we show that for almost all parameters the Edgeworth expansion of order d − 1 is valid and the error of the order d − 1 Edgeworth expansion is typically of order n^{-d/2}.DD was partially supported by the NSF
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