What is the mean of time when the trajectory of the wiener process, $W_t$, is over the line $y=t$?
We need to find $\Bbb{E}\tau$, 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 do we find its mean?
We start from $$\tau=\int_0^\infty\mathbf 1_{W_t\gt t}\,\mathrm dt,\quad \Bbb E(\tau)=\int_0^\infty P(W_t\gt t)\,\mathrm dt$$
Since $W_t$ is normal centered with ariance $t$, $$P(W_t\gt t)=P(\sqrt{t}Z\gt t)=P(Z\gt\sqrt{t})$$ where $Z$ is standard normal. By symmetry, $P(Z\lt-\sqrt{t})=P(Z\gt\sqrt{t})$ hence $$2P(Z\gt\sqrt{t})=P(Z\gt\sqrt{t})+P(Z\lt-\sqrt{t})=P(Z^2\gt t)$$ which gives $$\Bbb E(\tau)=\frac12\left(\int_0^\infty P(Z^2\gt t)\,\mathrm dt\right)=\frac12 \Bbb E(Z^2)=\frac12$$
We don't need $\mathbb{E}(\tau)=\int\limits_{0}^{\infty}P\left(\tau>t\right)dt$.