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

methodology and computing in applied probability

A survey on the pseudo-process driven by the high-order heat-type  equation $\partial/\partial t=\pm\partial^N/\partial x^N$ concerning the  first hitting times and sojourn times