First Hitting Time and Place, Monopoles and Multipoles for Pseudo-Processes Driven by the Equation $\partial u/\partial t=\pm\partial^N u/\partial x^N$

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 $\tau_a^+$ and $\tau_a^-$ 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 $(\tau_a^+,X(\tau_a^+))$ and $(\tau_a^-,X(\tau_a^-))$.