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Random Walks in Random Environment.

We present an invariance principle for antisymmetric functions of a reversible Markov process which immediately implies convergence to Brownian motion for a wide class of random motions in random environments. We apply it to establish convergence to Brownian motion (i) for a walker moving in the infinite cluster of the two-dimensional bond percolation model, (ii) for ad-dimensional walker moving in a symmetric random environment under very mild assumptions on the distribution of the environment, (iii) for a tagged particle in ad-dimensional symmetric lattice gas which allows interchanges, (iv) for a tagged particle in ad-dimensional system of interacting Brownian particles. Our formulation also leads naturally to bounds on the diffusion constant.

An invariance principle for reversible Markov processes. Applications to random motions in random environments

/pdf/topical-review-cosmological-billiards-30k9pt2npd.pdf

TOPICAL REVIEW: Cosmological billiards

In this work we principally study random walk on the supercritical infinite cluster for bond percolation on ℤd. We prove a quenched functional central limit theorem for the walk when d≥4. We also prove a similar result for random walk among i.i.d. random conductances along nearest neighbor edges of ℤd, when d≥1.

Quenched invariance principles for walks on clusters of percolation or among random conductances

We study the long time evolution of a quantum particle in a Gaussian random environment. We show that in the weak coupling limit the Wigner distribution of the wave function converges to the solution of a linear Boltzmann equation globally in time. The Boltzmann collision kernel is given by the Born approximation of the quantum scattering cross section.

Linear Boltzmann equation as the weak coupling limit of a random Schrödinger equation

We consider the Boltzmann-Grad limit for the Lorentz, or wind-tree, model. We prove that if ω is a fixed configuration of scatterer centers belonging to a set of full measure with respect to the Poisson distribution with parameter λ>0, then the evolution of an initial a.c. particle density tends in the Boltzmann-Grad limit to the solution of the Boltzmann equation for the model. As an intermediate step we prove that the process of the free path lengths and impact parameters induced by the Lebesgue measure on a small region tends to a limiting independent process.

On the Boltzmann equation for the Lorentz gas

It was shown earlier by I. M. Lifshitz and two of us that the evolution of the relativistic cosmological models towards the singularity undergoes spontaneous stochastization.(1) In the present paper it is shown that the statistical parameters of this evolution can be calculated in an exact manner. From the point of view of the general ergodic theory we deal here with a specific mode of stochastization of a deterministic dynamical system with a five-dimensional phase space. The knowledge of the source of stochasticity makes it possible to develop a quantitative statistical theory with appreciable completeness.

On the Stochasticity in Relativistic Cosmology

A new proof of the M. Herman theorem on the smooth conjugacy of a circle map is presented here. It is based on the thermodynamic representation of dynamical systems and the study of the ergodic properties for the corresponding radom variables.

/pdf/a-new-proof-of-m-herman-s-theorem-qvlufvetr7.pdf

A new proof of M. Herman's theorem

We consider a random walk on thed-dimensional lattice ℤ
 d
 where the transition probabilitiesp(x,y) are symmetric,p(x,y)=p(y,x), different from zero only ify−x belongs to a finite symmetric set including the origin and are random. We prove the convergence of the finite-dimensional probability distributions of normalized random paths to the finite-dimensional probability distributions of a Wiener process and find our an explicit expression for the diffusion matrix.

/pdf/symmetric-random-walks-in-random-environments-53z7gy69cp.pdf

Symmetric random walks in random environments

This survey is a study of a dynamical system consisting of a massive piston in a cubic container of large size filled with an ideal gas. The piston has mass and undergoes elastic collisions with non-interacting gas particles of mass . It is found that under suitable initial conditions there is a scaling regime with time and space scaled by in which the motion of the piston and the one-particle distribution of the gas satisfy autonomous coupled equations (hydrodynamic equations) such that in the limit the mechanical trajectory of the piston converges in probability to the solution of the hydrodynamic equations for a certain period of time. There is also a heuristic discussion of the dynamics of the system on longer intervals of time.

https://people.cas.uab.edu/~mosya/papers/pist.pdf

Yakov Grigor'evich Sinai

Papers

On the Boltzmann equation for the Lorentz gas

On the Stochasticity in Relativistic Cosmology

A new proof of M. Herman's theorem

Symmetric random walks in random environments

Dynamics of a massive piston in an ideal gas