Self-repelling diffusions on a Riemannian manifold

• 成果类型: Article
• 署名作者: Benaim, Michel; Gauthier, Carl-Erik
• 署名单位: University of Neuchatel
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-016-0717-1
• 发表日期: 2017
• 页码: 63-104
• 关键词: interacting diffusions; CONVERGENCE

摘要

Let M be a compact connected oriented Riemannian manifold. The purpose of this paper is to investigate the long time behavior of a degenerate stochastic differential equation on the state space M x R-n; which is obtained via a natural change of variable from a self-repelling diffusion taking the form dX(t) = sigma dB(t)(X-t) - integral(t)(0) del V-Xs (X-t)dsdt, X-0 = x where {B-t} is a Brownian vector field on M, sigma > 0 and V-x(y) = V(x, y) is a diagonal Mercer kernel. We prove that the induced semi-group enjoys the strong Feller property and has a unique invariant probability mu given as the product of the normalized Riemannian measure on M and a Gaussian measure on R-n. We then prove an exponential decay to this invariant probability L-2(mu) in and in total variation.