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Harmonic measures versus quasiconformal measures for hyperbolic groups

Sébastien Blachère, Peter Haïssinsky, Pierre Mathieu (2011)

Annales scientifiques de l'École Normale Supérieure

We establish a dimension formula for the harmonic measure of a finitely supported and symmetric random walk on a hyperbolic group. We also characterize random walks for which this dimension is maximal. Our approach is based on the Green metric, a metric which provides a geometric point of view on random walks and, in particular, which allows us to interpret harmonic measures as quasiconformal measures on the boundary of the group.

Heat kernel for random walk trace on ℤ3 and ℤ4

Daisuke Shiraishi (2010)

Annales de l'I.H.P. Probabilités et statistiques

We study the simple random walk X on the range of simple random walk on ℤ3 and ℤ4. In dimension four, we establish quenched bounds for the heat kernel of X and max0≤k≤n|Xk| which require extra logarithmic correction terms to the higher-dimensional case. In dimension three, we demonstrate anomalous behavior of X at the quenched level. In order to establish these estimates, we obtain several asymptotic estimates for cut times of simple random walk and asymptotic estimates for loop-erased random walk,...

Homogenization and diffusion asymptotics of the linear Boltzmann equation

Thierry Goudon, Antoine Mellet (2003)

ESAIM: Control, Optimisation and Calculus of Variations

We investigate the diffusion limit for general conservative Boltzmann equations with oscillating coefficients. Oscillations have a frequency of the same order as the inverse of the mean free path, and the coefficients may depend on both slow and fast variables. Passing to the limit, we are led to an effective drift-diffusion equation. We also describe the diffusive behaviour when the equilibrium function has a non-vanishing flux.

Homogenization and Diffusion Asymptotics of the Linear Boltzmann Equation

Thierry Goudon, Antoine Mellet (2010)

ESAIM: Control, Optimisation and Calculus of Variations

We investigate the diffusion limit for general conservative Boltzmann equations with oscillating coefficients. Oscillations have a frequency of the same order as the inverse of the mean free path, and the coefficients may depend on both slow and fast variables. Passing to the limit, we are led to an effective drift-diffusion equation. We also describe the diffusive behaviour when the equilibrium function has a non-vanishing flux.

Homogenization results for a linear dynamics in random Glauber type environment

Cédric Bernardin (2012)

Annales de l'I.H.P. Probabilités et statistiques

We consider an energy conserving linear dynamics that we perturb by a Glauber dynamics with random site dependent intensity. We prove hydrodynamic limits for this non-reversible system in random media. The diffusion coefficient turns out to depend on the random field only by its statistics. The diffusion coefficient defined through the Green–Kubo formula is also studied and its convergence to some homogenized diffusion coefficient is proved.

Hydrodynamics of Inelastic Maxwell Models

V. Garzó, A. Santos (2011)

Mathematical Modelling of Natural Phenomena

An overview of recent results pertaining to the hydrodynamic description (both Newtonian and non-Newtonian) of granular gases described by the Boltzmann equation for inelastic Maxwell models is presented. The use of this mathematical model allows us to get exact results for different problems. First, the Navier–Stokes constitutive equations with explicit expressions for the corresponding transport coefficients are derived by applying the Chapman–Enskog...

Hypoelliptic estimates for some linear diffusive kinetic equations

Frédéric Hérau (2010)

Journées Équations aux dérivées partielles

This note is an announcement of a forthcoming paper [13] in collaboration with K. Pravda-Starov on global hypoelliptic estimates for Fokker-Planck and linear Landau-type operators. Linear Landau-type equations are a class of inhomogeneous kinetic equations with anisotropic diffusion whose study is motivated by the linearization of the Landau equation near the Maxwellian distribution. By introducing a microlocal method by multiplier which can be adapted to various hypoelliptic kinetic equations,...

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