First exit time from a bounded interval for pseudo-processes driven by the equation $\partial /\partial t=(-1{)}^{N-1}{\partial }^{2N}/\partial x^{2N}$

Abstract

Let $N$ be a positive integer. We consider pseudo-Brownian motion $X=(X(t){)}_{t\ge 0}$ driven by the high-order heat-type equation $\partial /\partial t=(-1{)}^{N-1}{\partial }^{2N}/\partial x^{2N}$. Let us introduce the first exit time τab from a bounded interval $(a,b)$ by $X$ ($a,b\in \mathbb{R}$). In this paper, we provide a representation of the joint pseudo-distribution of the vector $({\tau }_{ab},X({\tau }_{ab}))$ by means of Vandermonde-like determinants. The method we use is based on the Feynman-Kac functional related to pseudo-Brownian motion which leads to a boundary value problem. In particular, the pseudo-distribution of the location of $X$ at time ${\tau }_{ab}$, namely $X({\tau }_{ab})$, admits a fine expression involving famous Hermite interpolating polynomials.

28 pages

References in corpus (2)

• 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

Cited by in corpus (1)

• From pseudo-random walk to pseudo-Brownian motion: first exit time from a one-sided or a two-sided interval