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July 1998 White noise indexed by loops
Ognian B. Enchev
Ann. Probab. 26(3): 985-999 (July 1998). DOI: 10.1214/aop/1022855741


Given a Riemannian manifold $M$ and loop $\phi: X^1 \mapsto M$, we construct a Gaussian random process $S^1 \ni \theta \leadsto X_{\theta} \epsilon T_{\phi(\theta)}M$, which is an analog of the Brownian motion process in the sense that the formal covariant derivative $\theta \leadsto \nabla_{\theta}X_{\theta}$ 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 $I: T_{\phi(0)}M \mapsto T_{\phi(0)}M$. 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 $X$.


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Ognian B. Enchev. "White noise indexed by loops." Ann. Probab. 26 (3) 985 - 999, July 1998.


Published: July 1998
First available in Project Euclid: 31 May 2002

zbMATH: 0937.60024
MathSciNet: MR1634411
Digital Object Identifier: 10.1214/aop/1022855741

Primary: 60G10 , 60J25 , 60J65

Keywords: diffusion , reciprocal process , stationary Gaussian process , White noise process

Rights: Copyright © 1998 Institute of Mathematical Statistics

Vol.26 • No. 3 • July 1998
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