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

Abstract

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.