DENSITIES FOR SDES DRIVEN BY DEGENERATE α-STABLE PROCESSES

• 成果类型: Article
• 署名作者: Zhang, Xicheng
• 署名单位: Wuhan University
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/13-AOP900
• 发表日期: 2014
• 页码: 1885-1910
• 关键词: STOCHASTIC DIFFERENTIAL-EQUATIONS; malliavin calculus; smooth densities; jumps

摘要

In this work, by using the Malliavin calculus, under Hormander's condition, we prove the existence of distributional densities for the solutions of stochastic differential equations driven by degenerate subordinated Brownian motions. Moreover, in a special degenerate case, we also obtain the smoothness of the density. In particular, we obtain the existence of smooth heat kernels for the following fractional kinetic Fokker Planck (nonlocal) operator: L-b((alpha)) := Delta(alpha/2)(v) + v . del(x) + b(x, v) . del v, x, v is an element of R-d, where a is an element of(0, 2) and b :R-d x R-d -> R-d is smooth and has bounded derivatives of all orders.