First hitting time and place, monopoles and multipoles for pseudo-processes driven by the equation $\partial/\partial t = \pm\partial^N/\partial x^N$

Abstract

Consider the high-order heat-type equation $\partial u/\partial t=\pm\partial^N u/\partial x^N$ for an integer $N>2$ and introduce the related Markov pseudo-process $(X(t))_{t\ge 0}$. In this paper, we study several functionals related to $(X(t))_{t\ge 0}$: the maximum $M(t)$ and minimum $m(t)$ up to time $t$; the hitting times $τ_a^+$ and $τ_a^-$ of the half lines $(a,+\infty)$ and $(-\infty,a)$ respectively. We provide explicit expressions for the distributions of the vectors $(X(t),M(t))$ and $(X(t),m(t))$, as well as those of the vectors $(τ_a^+,X(τ_a^+))$ and $(τ_a^-,X(τ_a^-))$.