Péter Nándori

A Central Limit Theorem for Time-Dependent Dynamical Systems

The work by Ott et al. (Math. Res. Lett. 16:463–475, 2009) established memory loss in the time-dependent (non-random) case of uniformly expanding maps of the interval. Here we find conditions under which we have convergence to the normal distribution of the appropriately scaled Birkhoff-like partial sums of appropriate test functions. A substantial part of the problem is to ensure that the variances of the partial sums tend to infinity (cf. the zero-cohomology condition in the autonomous case). In fact, the present paper is the first one where non-random examples are also found, which are not small perturbations of a given map. Our approach uses martingale approximation technique in the form of Sethuraman and Varadhan (Electron. J. Probab. 10:121–1235, 2005).

Tail Asymptotics of Free Path Lengths for the Periodic Lorentz Process: On Dettmann’s Geometric Conjectures

In the simplest case, consider a

Number of distinct sites visited by a random walk with internal states

{mathbb{Z}^d}

Lorentz process with shrinking holes in a wall.

Zd-periodic (d ≥ 3) arrangement of balls of radii < 1/2, and select a random direction and point (outside the balls). According to Dettmann’s first conjecture, the probability that the so determined free flight (until the first hitting of a ball) is larger than t >xa0xa0>xa01 is

Local Thermal Equilibrium for Certain Stochastic Models of Heat Transport

{simfrac{C}{t}}

Logarithmic scaling of planar random walk's local times

∼Ct, where C is explicitly given by the geometry of the model. In its simplest form, Dettmann’s second conjecture is related to the previous case with tangent balls (of radii 1/2). The conjectures are established in a more general setup: for