Geodesic
On the Structure of the Infinitesimal $\sigma$-Algebra of a Gaussian Process
Teoriâ veroâtnostej i ee primeneniâ, Tome 7 (1962) no. 2, pp. 204-208

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Let $x(t)$ be a Gaussian stationary process $\mathfrak{M}_{+0}=\bigcap _{t>0}\mathfrak{M}_t$, where $\mathfrak{M}_t$ is the $\sigma $-algebra generated by $x(s),0\leq s\leq t$. It is proved that if the spectral density $f(\lambda)$ of the process satisfies the condition $f(\lambda)\geq{1}/{\lambda^p}$ for all $|\lambda|>\lambda_0$ and some $p>0$, the $\sigma $-algebra $\mathfrak{M}_{+0}$ is generated by $x(0),{dx(0)}/{dt},\dots,{dx^{(k)}{(0)}}/{dt^k}$, where $k$ is the order of the derivative the sample functions admit. 
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     author = {V. N. Tutubalin and M. I. Freidlin},
     title = {On the {Structure} of the {Infinitesimal} $\sigma${-Algebra} of a {Gaussian} {Process}},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
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     number = {2},
     year = {1962},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TVP_1962_7_2_a7/}
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V. N. Tutubalin; M. I. Freidlin. On the Structure of the Infinitesimal $\sigma$-Algebra of a Gaussian Process. Teoriâ veroâtnostej i ee primeneniâ, Tome 7 (1962) no. 2, pp. 204-208. http://geodesic.mathdoc.fr/item/TVP_1962_7_2_a7/