White noise indexed by loops. (English) Zbl 0937.60024

Enchev, Ognian B.

Ann. Probab. 26, No. 3, 985-999 (1998).

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

MSC:

60G10	Stationary stochastic processes
60J25	Continuous-time Markov processes on general state spaces
60H40	White noise theory

Keywords:

white noise process; stationary Gaussian process; reciprocal process; diffusion

References:

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[5]	BOSTON, MASSACHUSETTS 02215 E-MAIL: [e-mail]