White noise indexed by loops. (English) Zbl 0937.60024

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 \(X\).


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


[1] ALBEVERIO, S., LEANDRE, R. and ROCKNER, M., 1993. Construction of a rotational invariant \' \" diffusion on the free loop space. C. R. Acad. Sci. Paris Ser. I 316 287 292. \' · Zbl 0776.58041
[2] KLEIN, A. and LANDAU, L. J., 1981. Periodic Gaussian Osterwalder Schrader positive processes and the two-sided Markov property on the circle. Pacific J. Math. 94 341 367. · Zbl 0464.60041
[3] NORRIS, J. 1996. Ornstein Uhlenbeck processes indexed by the circle. · Zbl 0940.60052
[4] RECOULES, R. 1991. Gaussian reciprocal processes revisited. Statist. Probab. Lett. 12 297 303. · Zbl 0739.60029
[5] BOSTON, MASSACHUSETTS 02215 E-MAIL: ogi@math.bu.edu
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