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 $\mathbb{R}^d$, $d \geq 4$, 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 $$(\dfrac{1}{\epsilon}L_0+L_1)u^\epsilon(x)=f(x) \text{for} x\in G, .\dfrac{\partial u^\epsilon}{\partial \gamma^\epsilon}(x)|_{\partial G}=0$$ is considered in this paper. The operators $L_0$ and $L_1$ are self-adjoint second order operators. We assume that $L_0$ has a non-negative characteristic form and $L_1$ is strictly elliptic. The reflection is with respect to inward co-normal unit vector $\gamma^\epsilon(x)$. The behavior of $\lim\limits_{\epsilon\downarrow 0}u^\epsilon(x)$ 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

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