FUNDAMENTAL SOLUTIONS OF NONLOCAL HORMANDER'S OPERATORS II

• 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: 10.1214/16-AOP1102
• 发表日期: 2017
• 页码: 1799-1841
• 关键词: STOCHASTIC DIFFERENTIAL-EQUATIONS; malliavin calculus; smooth densities; jump-processes; sdes driven

摘要

Consider the following nonlocal integro-differential operator: for alpha is an element of (0, 2): L-sigma,b((alpha)) f(x) := p.v. integral(|z| < delta) f(x + sigma(x)z) - f(x)/|z|(d+alpha) dz + b(x) . del f(x) + Lf(x), where sigma :R-d -> R-d circle times R-d and b : R-d -> R-d are smooth functions and have bounded partial derivatives of all orders greater than 1, delta is a small positive number, p.v. stands for the Cauchy principal value and L is a bounded linear operator in Sobolev spaces. Let B-1(x) := sigma(x) and Bj+1(x) := b(x) . del B-j (x) - del b(x) . B-j (x) for j is an element of N. Suppose B-j is an element of C-b(infinity) (R-d ; R-d circle times R-d) each j is an element of N. Under the following uniform Hormander's type condition: for some j(0) is an element of N, inf x is an element of R-d inf |u| = 1 Sigma(j=1) (j0) |uB(j)(x)|(2) > 0, by using Bismut's approach to the Malliavin calculus with jumps, we prove the existence of fundamental solutions to operator L-sigma,b((alpha)). In particular, we answer a question proposed by Nualart [Sankhy (a) over bar A 73 (2011) 46-49] and Varadhan [Sankhy (a) over bar A 73 (2011) 50-51].