White noise indexed by loops

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 $X$.