Scalable and Quasi-Contractive Markov Coupling of Maxwell Collision

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 $L_2$-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 $\R^d$, with the average square parallelogram area spanned by $(U-U_\ast,V-V_\ast)$, $(U_\ast,V_\ast)$ 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.

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