A Survey on the Pseudo-process Driven by the High-order Heat-type Equation $boldsymbol{partial/partial t=pmpartial^N!/partial x^N}$ Concerning the Hitting and Sojourn Times

摘要

Fix an integer N 2 and let X = (X(t)) t ≥ 0 be the pseudo-process driven by the high-order heat-type equation $partial/partial t=pmpartial^N!/partial x^N$ . The denomination “pseudo-process” means that X is related to a signed measure (which is not a probability measure) with total mass equal to one. In this survey, we present several explicit results and discuss some problems concerning the pseudo-distributions of various functionals of the pseudo-process X: the first or last overshooting times of a single barrier {a} or a double barrier {a, b} by X; the sojourn times of X in the intervals [a, + ∞ ) and [a, b] up to a fixed time; the maximum or minimum of X up to a fixed time.