Monte-Carlo Approximations and Fluctuations for 2D Boltzmann Equations without Cutoff

#### N. Fournier, S. Meleard

2001, v.7, №1, 159-191

ABSTRACT

Using the main ideas of Tanaka [Probabilistic treatment of the Boltzmann equation of Maxwellian molecules, Z. Wahrsch. verw. Geb., 1978, v.46, pp.67-105], the measure solution $\{P_t\}_t$ of a $2$-dimensional spatially homogeneous Boltzmann equation of Maxwellian molecules without cutoff is related to a Poisson-driven nonlinear stochastic differential equation. Using this tool and a generalized law of large numbers, we present two ways to prove the convergence of the empirical measure associated with an interacting particle system to this measure solution of the Boltzmann equation. Then we give numerical results. We finally discuss about a central limit theorem associated with the above law of large numbers.

Keywords: Boltzmann equations without cutoff,stochastic differentialequations,jump measures,interacting particle systems,fluctuation theorems