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 ${\tau }_a^{+}$ 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 $({\tau }_a^{-},X({\tau }_a^{-}))$.

51 pages

Cited by in corpus (7)

• First passage leapovers of Lévy flights and the proper formulation of absorbing boundary conditions
• Vibrations and fractional vibrations of rods, plates and Fresnel pseudo-processes
• From pseudo-random walk to pseudo-Brownian motion: first exit time from a one-sided or a two-sided interval
• 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
• Bessel processes and hyperbolic Brownian motions stopped at different random times
• First exit time from a bounded interval for pseudo-processes driven by the equation $\partial /\partial t=(-1{)}^{N-1}{\partial }^{2N}/\partial x^{2N}$
• Joint distribution of the process and its sojourn time for pseudo-processes governed by high-order heat equation