  Be $X_t$ a random process starting at $x \in [0,1]$ with absorbing boundary
conditions at both ends of the interval. Denote $P_1(x)$ the probability to
first exit at the upper boundary. For Brownian motion, $P_1(x)=x$, equivalent
to $P_1'(x)=1$. For fractional Brownian motion with Hurst exponent $H$, we
establish that $P_1'(x) = {\cal N} [x(1-x)]^{\frac1H -2} e^{\epsilon {\cal
F}(x)+ {\cal O}(\epsilon^2)}$, where $\epsilon=H-\frac12$. The function ${\cal
F}(x)$ is analytic, and well approximated by its Taylor expansion, ${\cal
F}(x)\simeq 16 (C-1) (x-1/2)^2 +{\cal O}(x-1/2)^4$, where $C= 0.915...$ is the
Catalan-constant. A similar result holds for moments of the exit time starting
at $x$. We then consider the span of $X_t$, i.e. the size of the (compact)
domain visited up to time $t$. For Brownian motion, we derive an analytic
expression for the probability that the span reaches 1 for the first time, then
generalized to fBm. Using large-scale numerical simulations with system sizes
up to $N=2^{24}$ and a broad range of $H$, we confirm our analytic results.
There are important finite-discretization corrections which we quantify. They
are most severe for small $H$, necessitating to go to the large systems
mentioned above.
