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 $X$.