Property of Wiener process sample path What is a mean of time, when the trajectory of wiener process $W_t$ is over the line $y=t$?
We need to find $\mathbb{E}\tau$, where $\tau=\sum\limits_{a,b:\forall t\in(a,b)
; W_t>t}\left(b-a\right)$.
By the law of iterated logarithm $P\left(sup\{t: W_t>t\}<\infty\right)=1$. Then we could find a sequence $t_i$ of points where $W_{t_i}=t_i$.
By local modulus of continuity there is infinite set of such points near $0$.
Then $\tau=\sum\limits_{n=1}^{\infty}\left(t_n-t_{n-1}\right)$. How to find it's mean?

Your definition of $\tau$ is flawed, actually,
$$\tau=\int_0^\infty\mathbf 1_{W_t\gt t}\,\mathrm dt,$$ hence $$E(\tau)=\int_0^\infty P(W_t\gt t)\,\mathrm dt.$$
For every $t$, $W_t$ is normal centered with variance $t$ hence $$P(W_t\gt t)=P(Z\gt\sqrt{t}),$$ where $Z$ is standard normal. By symmetry, $P(Z\gt\sqrt{t})=P(Z\lt-\sqrt{t})$ hence
$$
2P(Z\gt\sqrt{t})=P(Z\gt\sqrt{t})+P(Z\lt-\sqrt{t})=P(Z^2\gt t),$$ which implies finally
$$E(\tau)=\frac12\int_0^\infty P(Z^2\gt t)\,\mathrm dt=\frac12E(Z^2)=\frac12.$$
