Random walks and Lorentz processes serve as fundamental models for Brownian motion. The study of random walks is a favorite object of probability theory, whereas that of Lorentz processes belongs to the theory of hyperbolic dynamical systems. Here we first present an example where the method based on the probabilistic approach led to new results for the Lorentz process: concretely, the recurrence of the planar periodic Lorentz process with a finite horizon. Afterwards, an unsolved problem—related to a 1981 question of Sinai on locally perturbed periodic Lorentz processes—is formulated as an analogous problem in the language of random walks.

We survey applications of the theory of hyperbolic (and to a lesser extent non hyperbolic) billiards to some fundamental problems of statistical physics and their mathematically rigorous derivations in the framework of classical Hamiltonian systems.

Markov partitions designed by Sinai(1968) and Bowen(1970) proved to be an efficient tool for descibing statistical properties of uniformly hyperbolic systems. For hyperbolic systems with singularities, in particular, for hyperbolic billiards the construction of a Markov partition by Bunimovich and Sinai(1980) was a delicate and hard task. Therefore later more and more flexible and simple variants of Markov partitions appeared: Markov sieves (Bunimovich-Chernov-Sinai, 1990), Markov towers (Young, 1998), standard pairs (Dolgopyat). This remarkable evolution of Sinai's original idea is surveyed in this paper.

We prove exponential correlation decay in dispersing billiard flows on the 2-torus assuming finite horizon and lack of corner points. With applications aimed at describing heat conduction, the highly singular initial measures are concentrated here on 1-dimensional submanifolds (given by standard pairs) and the observables are supposed to satisfy a generalized Hölder continuity property. The result is based on the exponential correlation decay bound of Baladi et al. (Invent Math, 211:39–117, 2018. https://doi.org/10.1007/s00222-017-0745-1 ) obtained for Hölder continuous observables in these billiards. The model dependence of the bounds is also discussed.

We analyse the process of energy exchanges generated by the elastic collisions between a point-particle, confined to a two-dimensional cell with convex boundaries, and a ‘piston’, i.e. a line-segment, which moves back and forth along a one-dimensional interval partially intersecting the cell. This model can be considered as the elementary building block of a spatially extended high-dimensional billiard modeling heat transport in a class of hybrid materials exhibiting the kinetics of gases and spatial structure of solids. Using heuristic arguments and numerical analysis, we argue that, in a regime of rare interactions, the billiard process converges to a Markov jump process for the energy exchanges and obtain the expression of its generator.

The theory of planar hyperbolic billiards is already quite well developed by having also achieved spectacular successes. In addition there also exists an excellent monograph by Chernov and Markarian on the topic. In contrast, apart from a series of works culminating in Simanyi's remarkable result on the ergodicity of hard ball systems and other sporadic successes, the theory of hyperbolic billiards in dimension 3 or more is much less understood. The goal of this work is to survey the key results of their theory and highlight some central problems which deserve particular attention and efforts.

In the simplest case, consider a \({\mathbb{Z}^d}\)-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 > > 1 is \({\sim\frac{C}{t}}\), 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 \({\mathcal{L}}\)-periodic configuration of—possibly intersecting—convex bodies with \({\mathcal{L}}\) being a non-degenerate lattice. These questions are related to Pólya’s visibility problem (Arch Math Phys Ser 2:135–142, 1918), to theories of Bourgain et al. (Commun Math Phys 190:491–508,1998), and of Marklof–Strömbergsson (Ann Math 172:1949–2033,2010). The results also provide the asymptotic covariance of the periodic Lorentz process assuming it has a limit in the super-diffusive scaling, a fact if d = 2 and the horizon is infinite.

Let $\{X_n\}_{n\in\mathbb{N}}$ be a sequence of i.i.d. random variables in $\mathbb{Z}^d$. Let $S_k=X_1+...+X_k$ and $Y_n(t)$ be the continuous process on $[0,1]$ for which $Y_n(k/n)=S_k/\sqrt{n}$ $k=1,...,n$ and which is linearly interpolated elsewhere. The paper gives a generalization of results of Belkin, \cite{B72} on the weak limit laws of $Y_n(t)$ conditioned to stay away from some small sets. In particular, it is shown that the diffusive limit of the random walk meander on $\mathbb Z^d: d\ge 2$ is the Brownian motion.

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).

A fundamental problem of non-equilibrium statistical mechanics is the derivation of macroscopic transport equations in the hydrodynamic limit. The rigorous study of such limits requires detailed information about rates of convergence to equilibrium for finite sized systems. In this paper, we consider the finite lattice {1, 2, …, N}, with an energy xi ∈ (0, ∞) associated with each site. The energies evolve according to a Markov jump process with nearest neighbour interaction such that the total energy is preserved. We prove that for an entire class of such models the spectral gap of the generator of the Markov process scales as . Furthermore, we provide a complete classification of reversible stationary distributions of product type. We demonstrate that our results apply to models similar to the billiard lattice model considered in Gaspard and Gilbert (2009 J. Stat. Mech.: Theory Exp. 2009 24), and hence provide a first step in the derivation of a macroscopic heat equation for a microscopic stochastic evolution of mechanical origin.

A paradigm model is suggested for describing the diffusive limit of trajectories of two Lorentz disks moving in a finite horizon periodic configuration of smooth, strictly convex scatterers and interacting with each other via elastic collisions. For this model the diffusive limit of the two trajectories is a mixture of joint Gaussian laws (analogous behavior is expected for the mechanical model of two Lorentz disks).

We ascertain the diffusively scaled limit of a periodic Lorentz process in a strip with an almost reflecting wall at the origin. Here, almost reflecting means that the wall contains a small hole waning in time. The limiting process is a quasi-reflected Brownian motion, which is Markovian, but not strong Markovian. Local time results for the periodic Lorentz process, having independent interest, are also found and used.

Imagine a poll to choose the best-known mathematician of the twentieth century. No doubt the winner would be John von Neumann. Reasons are seen, for instance, in the title of the excellent biography [M] by Macrae: John von Neumann. The Scientific Genius who Pioneered the Modern Computer, Game Theory, Nuclear Deterrence, and Much More. Indeed, he was a fundamental figure not only in designing modern computers but also in defining their place in society and envisioning their potential. His minimax theorem, the first theorem of game theory, and later his equilibrium model of economy, essentially inaugurated the new science of mathematical economics. He played an important role in the development of the atomic bomb. However, behind all these, he was a brilliant mathematician. My goal here is to concentrate on his development and achievements as a mathematician and the evolution of his mathematical interests.

Scaled type Markov renewal processes generalize classical renewal processes: renewal times come from a one parameter family of probability laws and the sequence of the parameters is the trajectory of an ergodic Markov chain. Our primary interest here is the asymptotic distribution of the Markovian parameter at time t \to \infty. The limit, of course, depends on the stationary distribution of the Markov chain. The results, however, are essentially different depending on whether the expectations of the renewals are finite or infinite. If the expectations are uniformly bounded, then we can provide the limit in general (beyond the class of scaled type processes), where the expectations of the probability laws in question appear, too. If the means are infinite, then - by assuming that the renewal times are rescaled versions of a regularly varying probability law with exponent 0 \leq alpha \leq 1 - it is the exponent a which emerges in the limits.

In \cite{SzT}, D. Sz\'asz and A. Telcs have shown that for the diffusively scaled, simple symmetric random walk, weak convergence to the Brownian motion holds even in the case of local impurities if $d \ge 2$. The extension of their result to finite range random walks is straightforward. Here, however, we are interested in the situation when the random walk has unbounded range. Concretely we generalize the statement of \cite{SzT} to unbounded random walks whose jump distribution belongs to the domain of attraction of the normal law. We do this first: for diffusively scaled random walks on $\mathbf Z^d$ $(d \ge 2)$ having finite variance; and second: for random walks with distribution belonging to the non-normal domain of attraction of the normal law. This result can be applied to random walks with tail behavior analogous to that of the infinite horizon Lorentz-process; these, in particular, have infinite variance, and convergence to Brownian motion holds with the superdiffusive $\sqrt{n \log n}$ scaling.

For two (or more) interacting classical particles the existing few results (for diffusion or energy transfer, for instance) assume that the mass of one of them as compared to the other mass becomes negligible in the limit (cf. [ChD 09]). Here two models are presented for energy transfer in systems with two or more identical hard disks. The first one, a stochastic paradigm for two Lorentz disks, suggests that the joint diffusive limit of two disks is the mixture of independent pairs of Wiener processes ( [P-GySz 09]. In the second one it is shown that in a quasi-1D mechanical chain of localized hard disks in the scaling of [GG 08] the limit for the energies of the disks is a n. n. interacting Markov process ( [SzT 09]. This latter result should open the way toward a rigorous derivation of Fourier law of heat conduction for a deterministic particle system.