= Exponential covariance function
The <exponential covariance function> on the real line is
$$
k(s,t)=R\exp(-|t-s|/\tau),\qquad R\geq0,\quad\tau>0.
$$
Here $R$ 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> solving
$$
dX_t=-\tau^{-1}X_t\,dt+\sqrt{2R/\tau}\,dW_t,
\qquad X_0\sim N(0,R)
$$
with the initial value <independent> of future <Brownian motion> increments. The explicit stochastic-integral solution gives $\mathbb E[X_tX_s]=R e^{-|t-s|/\tau}$. Its conditional future equals the exponentially decayed present plus <independent> <Gaussian noise>, giving the <Markov property>.
The cusp at zero determines its roughness. For $R>0$,
$$
\mathbb E\left|\frac{X_{t+h}-X_t}{h}\right|^2
=\frac{2R(1-e^{-|h|/\tau})}{h^2}\sim\frac{2R}{\tau|h|},
$$
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.<Gaussian Processes for Machine Learning by Carl Edward Rasmussen and Christopher K. I. Williams>[Chapter 4, Exponential Covariance Function and Ornstein-Uhlenbeck Process, printed pp. 84–85]{ref}
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