Semimartingales and geometric inequalities on locally symmetric manifolds

• 成果类型: Article
• 署名作者: Le, H.; Barden, D.
• 署名单位: University of Nottingham; University of Cambridge
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-007-0105-y
• 发表日期: 2008
• 页码: 285-311
• 关键词: stochastic differential geometry

摘要

We generalise, to complete, connected and locally symmetric Riemannian manifolds, the construction of coupled semimartingales X and Y given in Le and Barden (J Lond Math Soc 75: 522 - 544, 2007). When such amanifold has non-negative curvature, this makes it possible for the stochastic anti-development of the corresponding semimartingale exp(Xt) (alpha exp(Xt)(-1)(Y-t)) to be a time-changed Brownian motion with drift when X and Y are. As an application, we use the latter result to strengthen, and extend to locally symmetric spaces, the results of Le and Barden (J Lond Math Soc 75: 522-544, 2007) concerning an inequality involving the solutions of the parabolic equation partial derivative Psi/partial derivative t = 1/2 Delta Psi - h Psi with Dirichlet boundary condition and an inequality involving the first eigenvalues of the Laplacian, both on three related convex sets.