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 $(τ_{ab},X(τ_{ab}))$ 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 $τ_{ab}$, namely $X(τ_{ab})$, admits a fine expression involving famous Hermite interpolating polynomials.