Scalable and Quasi-Contractive Markov Coupling of Maxwell Collision
Résumé
This paper considers space homogenous Boltzmann kinetic equations in dimension d with Maxwell collisions (and without Grad's cut-off). An explicit Markov coupling of the associated conservative (Nanbu) stochastic N-particle system is constructed, using plain parallel coupling of isotropic random walks on the sphere of two-body collisional direc- tions. The resulting coupling is almost surely decreasing, and the L2-coupling creation is computed explicitly. Some quasi-contractive and uniform in N coupling / coupling creation inequalities are then proved, relying on 2 + ®-moments (® > 0) of velocity dis- tributions; upon N-uniform propagation of moments of the particle system, it yields a N-scalable ®-power law trend to equilibrium. The latter are based on an original sharp inequality, which bounds from above the coupling distance of two centered and normalized random variables (U, V ) in Rd, with the average square parallelogram area spanned by (U − U¤, V − V¤), (U¤, V¤) denoting an independent copy. Two counter-examples proving the necessity of the dependance on > 2-moments and the impossibility of strict contrac- tivity are provided. The paper, (mostly) self-contained, does not require any propagation of chaos property and uses only elementary tools.
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