First exit time from a bounded interval for pseudo-processes driven by the equation

Let N be an integer greater than 1. We consider the pseudo-process X = ( X t ) t ≥ 0 driven by the high-order heat-type equation ∂ / ∂ t = ( − 1 ) N − 1 ∂ 2 N / ∂ x 2 N . Let us introduce the first exit time τ a b from a bounded interval ( a , b ) by X ( a , b ∈ R ) together with the related location, namely X τ a b . s paper, we provide a representation of the joint pseudo-distribution of the vector ( τ a b , X τ a b ) by means of some determinants. The method we use is based on a Feynman–Kac-like functional related to the pseudo-process X which leads to a boundary value problem. In particular, the pseudo-distribution of X τ a b admits a fine expression involving famous Hermite interpolating polynomials.