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.

Cited by in corpus (3)

• Estimates for functionals of solutions to higher-order heat-type equations with random initial conditions
• From pseudo-random walk to pseudo-Brownian motion: first exit time from a one-sided or a two-sided interval
• First exit time from a bounded interval for pseudo-processes driven by the equation $\partial /\partial t=(-1{)}^{N-1}{\partial }^{2N}/\partial x^{2N}$