Joint distribution of the process and its sojourn time in a half-line [a,+∞)[a,+∞) for pseudo-processes driven by a high-order heat-type equation

Let (X(t))t≥0(X(t))t≥0 be the pseudo-process driven by the high-order heat-type equation ∂u∂t=±∂Nu∂xN, where NN is an integer greater than 2. We consider the sojourn time spent by (X(t))t≥0(X(t))t≥0 in [a,+∞)[a,+∞) (a∈Ra∈R), up to a fixed time t>0t>0: Ta(t)=∫0t1[a,+∞)(X(s))ds. The purpose of this paper is to provide an explicit expression for the joint pseudo-distribution of the vector (Ta(t),X(t))(Ta(t),X(t)) when the pseudo-process starts at a point x∈Rx∈R at time 00. The method consists in solving a boundary value problem satisfied by the Laplace transform of the aforementioned distribution.