In the pepper "Chaoticity and invariant measures for a cell population model" page 162, I have some problems
-*)Here are the data of the problem:
Let $\omega_{t},t\geq0,$ be a Brownian motion starting from $0$ defined on a probability space $(\Omega,\Sigma,P).$ Assume that the sample functions of $\omega_{t}$ are continuous. Set $\xi_{x}=\mathrm{e}^{x}\omega_{\mathrm{e}^{-2x}}$ for $x\in\mathbb{R}.$ Then $\xi_{x}$ is a stationary Gaussian process with mean value $\mathbb{E}\xi_{x}=0$ and correlation function $\mathbb{E}\xi_{x}\xi_{x+h}=\mathrm{e}^{-\lvert h\rvert}.$ The sample function of $\xi_{x}$ are continuous functions. From the law of iterated logarithms it follows that \begin{eqnarray}\label{eq1.25} \limsup_{\lvert x\rvert\to\infty}\dfrac{\lvert \xi_{x}\rvert}{\sqrt{2\ln\lvert 2x\rvert}}=1 \end{eqnarray} with probability 1
*)The questions :
1) do the sample functions of $ \omega_ {t} $ mean that the functions $ t\to\ \omega_ {t}(w) \ \forall w $?
2)I need the law of the iterated logarithms statement that he used to have the limit above, because I found it just for the discrete case in this site "https://en.wikipedia.org/wiki/Law_of_the_iterated_logarithm"
1) Yes.
2) Review the section 8.8 of Durrett's book Probability Theory and Examples. https://www.cambridge.org/core/books/probability/81949AABAA8B3A8411CB88402F0F05C9