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

摘要

Consider the high-order heat-type equation $partial u/partial t=pmpartial^N u/partial x^N$ for an integer $N>2$ and introduce the related Markov pseudo-process $(X(t))_{tge 0}$. In this paper, we study several functionals related to $(X(t))_{tge 0}$: the maximum $M(t)$ and minimum $m(t)$ up to time $t$; the hitting times $au_a^+$ and $au_a^-$ 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 $(au_a^+,X(au_a^+))$ and $(au_a^-,X(au_a^-))$.