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

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