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 { \tau}ab 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.
Categories: math.PR