White noise indexed by loops
成果类型：
Article
署名作者：
Enchev, OB
署名单位：
Boston University
刊物名称：
ANNALS OF PROBABILITY
ISSN/ISSBN：
0091-1798
发表日期：
1998
页码：
985-999
关键词：
摘要：
Given a Riemannian manifold M and loop phi: S-1 bar right arrow M, we construct a Gaussian random process S-1 There Exists theta curved right arrow X-theta epsilon T-phi(theta)M, which is an analog of the Brownian motion process in the sense that the formal covariant derivative theta curved right arrow del(theta)X(theta) appears as a stationary process whose spectral measure is uniformly distributed over some discrete set. We show that X satisfies the two-point Markov property (reciprocal process) if the holonomy along the loop phi is nontrivial. The covariance function of X is calculated and the associated abstract Wiener space is described. We also characterize X as a solution of a special (nondiffusion type) stochastic differential equation. Somewhat surprisingly, the nature of X turns out to be very different if the holonomy along phi is the identity map I: T-phi(0)M bar right arrow T-phi(0)M. In this case, we show that the usual periodic Ornstein-Uhlenbeck process, associated with a harmonic oscillator at nonzero temperature, may be viewed as a standard velocity process in which the driving Brownian motion is replaced by the process X.