First exit time from a bounded interval for pseudo-processes driven by the equation ∂/∂t=(−1)N−1∂2N/∂x2N∂/∂t=(−1)N−1∂2N/∂x2N

Let NN be an integer greater than 1. We consider the pseudo-process X=(Xt)t≥0X=(Xt)t≥0 driven by the high-order heat-type equation ∂/∂t=(−1)N−1∂2N/∂x2N∂/∂t=(−1)N−1∂2N/∂x2N. Let us introduce the first exit time τabτab from a bounded interval (a,b)(a,b) by XX (a,b∈Ra,b∈R) together with the related location, namely XτabXτab.In this paper, we provide a representation of the joint pseudo-distribution of the vector (τab,Xτab)(τab,Xτab) by means of some determinants. The method we use is based on a Feynman–Kac-like functional related to the pseudo-process XX which leads to a boundary value problem. In particular, the pseudo-distribution of XτabXτab admits a fine expression involving famous Hermite interpolating polynomials.