First hitting time and place for pseudo-processes driven by the equation ∂∂t=±∂N∂xN subject to a linear drift

Consider the high-order heat-type equation ∂u/∂t=(−1)1+N/2∂Nu/∂xN∂u/∂t=(−1)1+N/2∂Nu/∂xN for an even integer N>2N>2, and introduce the related Markov pseudo-process (X(t))t⩾0(X(t))t⩾0. Let us define the drifted pseudo-process (Xb(t))t⩾0(Xb(t))t⩾0 by Xb(t)=X(t)+btXb(t)=X(t)+bt. In this paper, we study the following functionals related to (Xb(t))t⩾0(Xb(t))t⩾0: the maximum Mb(t)Mb(t) up to time tt; the first hitting time τab of the half line (a,+∞)(a,+∞); and the hitting place Xb(τab) at this time. We provide explicit expressions for the Laplace–Fourier transforms of the distributions of the vectors (Xb(t),Mb(t))(Xb(t),Mb(t)) and (τab,Xb(τab)), from which we deduce explicit expressions for the distribution of Xb(τab) as well as for the escape pseudo-probability: P{τab=+∞}. We also provide some boundary value problems satisfied by these distributions.