First hitting time and place for pseudo-processes driven by the equation subject to a linear drift

Consider the high-order heat-type equation [not partial differential]u/[not partial differential]t=(-1)1+N/2[not partial differential]Nu/[not partial differential]xN for an even integer N>2, and introduce the related Markov pseudo-process (X(t))t[greater-or-equal, slanted]0. Let us define the drifted pseudo-process (Xb(t))t[greater-or-equal, slanted]0 by Xb(t)=X(t)+bt. In this paper, we study the following functionals related to (Xb(t))t[greater-or-equal, slanted]0: the maximum Mb(t) up to time t; the first hitting time of the half line (a,+[infinity]); and the hitting place at this time. We provide explicit expressions for the Laplace-Fourier transforms of the distributions of the vectors (Xb(t),Mb(t)) and , from which we deduce explicit expressions for the distribution of as well as for the escape pseudo-probability: . We also provide some boundary value problems satisfied by these distributions.