  Fix an integer n>2 and let $(X(t))_{t\ge 0}$ be the pseudo-process driven by
the high-order heat-type equation $\partial/\partial t=\pm\partial^N/\partial
x^N$. The denomination "pseudo-process" means that $(X(t))_{t\ge 0}$ is related
to a signed measure (which is not a probability measure) with total mass equal
to 1. In this note, we present some results and discuss some problems
concerning the pseudo-distributions of the first overshooting times of a single
barrier $\{a\}$ or a double barrier $\{a,b\}$ by $(X(t))_{t\ge 0}$, as well as
those of the sojourn times of $(X(t))_{t\ge 0}$ in the intervals $[a,+\infty)$
and $[a,b]$ up to a fixed time.
