## The Annals of Probability

### White noise indexed by loops

Ognian B. Enchev

#### Abstract

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 $X$.

#### Article information

Source
Ann. Probab., Volume 26, Number 3 (1998), 985-999.

Dates
First available in Project Euclid: 31 May 2002

https://projecteuclid.org/euclid.aop/1022855741

Digital Object Identifier
doi:10.1214/aop/1022855741

Mathematical Reviews number (MathSciNet)
MR1634411

Zentralblatt MATH identifier
0937.60024

#### Citation

Enchev, Ognian B. White noise indexed by loops. Ann. Probab. 26 (1998), no. 3, 985--999. doi:10.1214/aop/1022855741. https://projecteuclid.org/euclid.aop/1022855741

#### References

• 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. ´
• 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.
• 3 NORRIS, J. 1996. Ornstein Uhlenbeck processes indexed by the circle. Preprint.
• 4 RECOULES, R. 1991. Gaussian reciprocal processes revisited. Statist. Probab. Lett. 12 297 303.
• BOSTON, MASSACHUSETTS 02215 E-MAIL: ogi@math.bu.edu