论文标题
动力学fokker-planck方程的分数扩散极限，在半线中具有扩散边界条件
Fractional diffusion limit for a kinetic Fokker-Planck equation with diffusive boundary conditions in the half-line
论文作者
论文摘要
我们考虑一个生活在$ \ mathbb {r} _+$中的粒子，当粒子以$（0，\ infty）$固定时，其速度是一个正复发，并具有重尾不变分布。当它撞到边界$ x = 0 $时，粒子以随机的严格正速度重新启动。我们表明，正确重新缩放的位置过程薄弱地收敛到反映在其量中的稳定过程。从P.D.E.观点，$（x_t，v_t）_ {t \ geq0} $的时间 - 边界求解$（0，\ infty）\ times \ times \ times \ times \ mathbb {r} _+++ \+ \ \ \ \ \ \ \ times \ times \ mathbb {r {r} $的动力学fokker-planck方程。正确重新缩放，空间 - 划线收敛于$（0，\ infty）\ times \ times \ mathbb {r} _+$上的某些分数热方程的解决方案。
We consider a particle living in $\mathbb{R}_+$, whose velocity is a positive recurrent diffusion with heavy-tailed invariant distribution when the particle lives in $(0,\infty)$. When it hits the boundary $x=0$, the particle restarts with a random strictly positive velocity. We show that the properly rescaled position process converges weakly to a stable process reflected on its infimum. From a P.D.E. point of view, the time-marginals of $(X_t, V_t)_{t\geq0}$ solve a kinetic Fokker-Planck equation on $(0,\infty)\times\mathbb{R}_+ \times \mathbb{R}$ with diffusive boundary conditions. Properly rescaled, the space-marginal converges to the solution of some fractional heat equation on $(0,\infty)\times\mathbb{R}_+$.
