How can one prove that with probability 1 $X(t)=Y(t)$ for infinitely many values of $t\in [0, \infty)$ , where $\{X (t); t \geq 0\}$ and $\{Y (t); t \geq 0\}$ are independent, standard Brownian motions? In a previous step we are asked to show that $Z(t) = X(t) - Y(t)$ is a Brownian motions with zero drift and volatility parameter 2t.
This is just a practice problem as I prepare for my exam.
You know that $Z(t) = X(t) - Y(t)$ and hence $\hat{W}(t) = \frac{Z(t)}{\sqrt{2}}$ is a standard Brownian motion. It follows from the law of iterated logarithm that
$$\overline{lim}_{t \to \infty} \frac{ \hat{W}_t}{ \sqrt{2t \ln \ln t} } = 1$$ and $$\underline{lim}_{t \to \infty} \frac{\hat{W}_t}{ \sqrt{2t \ln \ln t} } = -1.$$ Hence $\hat{W}(t) = 0$ for infinitely many values of $t$, q.e.d.