 Mean-square derivative of a Gaussian process

A Gaussian process has a mean-square derivative at $t$ when its difference quotients converge in L2 space. The mean-square convergence need not give differentiable sample paths. For a centered stationary process with twice continuously differentiable covariance function $K(t-s)$, the difference quotients are Cauchy in L2 space, since their covariance is a rectangular average of $-K^{\prime \prime }$. Their limits form a Gaussian process with covariance function $\mathbb{E}[Y(s)Y(t)]=-K^{\prime \prime }(s-t)$.