SMOOTHNESS OF THE DENSITY FOR SOLUTIONS TO GAUSSIAN ROUGH DIFFERENTIAL EQUATIONS

• 成果类型: Article
• 署名作者: Cass, Thomas; Hairer, Martin; Litterer, Christian; Tindel, Samy
• 署名单位: Imperial College London; University of Warwick; Institut Polytechnique de Paris; Ecole Polytechnique; ENSTA Paris; Universite de Lorraine
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/13-AOP896
• 发表日期: 2015
• 页码: 188-239
• 关键词: fractional brownian motions; hypoelliptic sdes driven; local-times; hormanders theorem; small values; ergodicity; continuity; signals; paths; LAWS

摘要

We consider stochastic differential equations of the form dY(t) = V(Y-t)dX(t) + V-0(Y-t)dt driven by a multi-dimensional Gaussian process. Under the assumption that the vector fields V-0 and V = (V-1,...,V-d) satisfy Hormander's bracket condition, we demonstrate that Y-t admits a smooth density for any t is an element of (0, T], provided the driving noise satisfies certain nondegeneracy assumptions. Our analysis relies on relies on an interplay of rough path theory, Malliavin calculus and the theory of Gaussian processes. Our result applies to a broad range of examples including fractional Brownian motion with Hurst parameter H > 1/4, the Ornstein-Uhlenbeck process and the Brownian bridge returning after time T.