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

Let ( X ( t ) ) t ≥ 0 be the pseudo-process driven by the high-order heat-type equation ∂ u ∂ t = ± ∂ N u ∂ x N , where N is an integer greater than 2. We consider the sojourn time spent by ( X ( t ) ) t ≥ 0 in [ a , + ∞ ) ( a ∈ R ), up to a fixed time t > 0 : T a ( t ) = ∫ 0 t 1 [ a , + ∞ ) ( X ( s ) ) d s . The purpose of this paper is to provide an explicit expression for the joint pseudo-distribution of the vector ( T a ( t ) , X ( t ) ) when the pseudo-process starts at a point x ∈ R at time 0 . The method consists in solving a boundary value problem satisfied by the Laplace transform of the aforementioned distribution.