 Differentiable modification of a stationary Gaussian process

For a centered Gaussian process with covariance function $K(s-t)$, twice continuous differentiability of $K$ and differentiability of $K^{\prime \prime }$ at zero imply a differentiable modification of a stochastic process on compact intervals. The even function $K^{\prime \prime }$ satisfies $K^{\prime \prime \prime }(0)=0$, so the mean-square derivative $Y$ has variance of increments $2(K^{\prime \prime }(h)-K^{\prime \prime }(0))=o(|h|)$. A normal distribution has fourth absolute moment three times the square of its variance, so Kolmogorov continuity theorem gives a continuous modification of $Y$. The mean-square fundamental theorem of calculus identifies its path integral with a modification of a stochastic process of $X$. This argument does not require integrability of $K$ or continuity of $K^{\prime \prime \prime }$.