= 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 $K''$ at zero imply a differentiable <modification of a stochastic process> on compact intervals. The <even function> $K''$ satisfies $K'''(0)=0$, so the mean-square derivative $Y$ has <variance> of increments $2(K''(h)-K''(0))=o(|h|)$. 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 $K'''$.
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