2019/01/20 by Zimo Hao, Xuhui Peng, Hao, Zimo +3
Computer Science · Economics, Econometrics and Finance · Mathematics · #60H07 #60H10 #60H30 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Numerical methods in inverse problems #Probability (math.PR) #Stochastic processes and financial applications
paper · pdf · doi:10.48550/arxiv.1901.06621
openalex publication_date 2019/01/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper we show the Hörmander hypoelliptic theorem for nonlocal operators by a purely probabilistic method: the Malliavin calculus. Roughly speaking, under general Hörmander's Lie bracket conditions, we show the regularization effect of discontinuous Lévy noises for possibly degenerate stochastic differential equations with jumps. To treat the large jumps, we use the perturbation argument together with interpolation techniques and some short time asymptotic estimates of the semigroup. As an application, we show the existence of fundamental solutions for operator ∂t-\mathscrK, where \mathscrK is the nonlocal kinetic operator: \mathscrK f(x,\rm v):=\rm p.v∫ℝd(f(x,\rm v+w)-f(x,\rm v))\fracκ(x,\rm v,w)|w|d+α\rm d w +\rm v⋅∇x f(x,\rm v)+b(x,\rm v)⋅∇\rm v f(x,\rm v). Here κ0-1≤ κ(x,\rm v,w)≤κ0 belongs to C^∞b(ℝ3d) and is symmetric in w, p.v. stands for the Cauchy principal value, and b∈ C^∞b(ℝ2d;ℝd).