2013/12/08 by Mathias Rousset, Rousset, Mathias
Mathematics · #Analysis of PDEs (math.AP) #FOS: Mathematics #Gas Dynamics and Kinetic Theory #Geometric Analysis and Curvature Flows #Markov Chains and Monte Carlo Methods #Probability (math.PR)
paper · pdf · doi:10.48550/arxiv.1312.2253
openalex publication_date 2013/12/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 directions. 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+α-moments (α>0) of velocity distributions; upon N-uniform propagation of moments of the particle system, it yields a N-scalable α-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 contractivity are provided. The paper, (mostly) self-contained, does not require any propagation of chaos property and uses only elementary tools.