 Exponential covariance function

The exponential covariance function on the real line isHere $R$ is the marginal variance and $\tau$ the correlation time. For $R>0$, the centered stationary Gaussian process with this covariance function is the Ornstein-Uhlenbeck process solvingwith the initial value independent of future Brownian motion increments. The explicit stochastic-integral solution gives $\mathbb{E}[X_t{X}_s]=R{e}^{-|t-s|/\tau }$. Its conditional future equals the exponentially decayed present plus independent Gaussian noise, giving the Markov property.

The cusp at zero determines its roughness. For $R>0$,which diverges. Thus a continuous version exists, but it has no mean-square derivative of a Gaussian process. A smooth-looking covariance away from the diagonal does not imply differentiable paths.^[ref]