The diffusion approximation of the spatially homogeneous Boltzmann equation (Q1088312)

The article continues the history of papers on the stochastic spatially homogeneous caricature of the Boltzmann equation, which leads from M. Kac via H. McKean to H. Tanaka, A. S. Sznitman, and so on. On \({\mathbb{R}}^ 3\), its formulation as a martingale problem runs: \[ f(\chi_ t)-\int^{t}_{0}ds\int_{[0,2\pi]*[0,\pi]\times {\mathbb{R}}^ 3}[f(X^*(\chi_ s,y,\delta,\rho))-f(X_ s)]k(X_ s,y)d\rho Q(d\delta)P(X_ s\in d_ y) \] is a P-martingale for all nice f, where \(\chi^*(x,y,\delta,\rho)\) is the postcollision velocity of the first particle after collision with another one, both having x and y as precollision velocities, and \(\delta\) and \(\rho\) are the colatitude and the longitude angles, respectively, in the sphere with north pole x and south pole y. For so-called soft potentials without cutoff, i.e. assuming \(0\leq k(x,y)\leq C(1+| x-y|^{\alpha}\}\) for some \(\alpha\in [0,1)\), some additional smoothness properties on k, and \(\int_{(0,\pi]}\delta Q(d\delta)<+\infty\), the author proves that the above Boltzmann martingale problem has a unique solution P. Moreover, replacing Q by a family \(Q_{\epsilon}\) which satisfies in addition \(Q_{\epsilon}((\epsilon \pi,\pi])=0\) and \(0<\lim_{\epsilon \searrow 0}\int (\delta /\epsilon)^ 2Q_{\epsilon}(d\delta)<\infty,it\) is shown that the processes \(X^{\epsilon}(t)=\chi_{Q_{\epsilon}}(t/\epsilon^ 2)\) converge to a nonlinear diffusion process whose density solves the so-called spatially homogeneous Landau equation.