Joint Distribution of the Process and its Sojourn Time on the Positive Half-Line for Pseudo-Processes Governed by High-Order Heat Equation

Abstract

Consider the high-order heat-type equation $\partial_t u=\pm \partial^n_x u$ for an integer $n>2$ and introduce the related Markov pseudo-process $(X(t))_{t\geq0}$. In this paper, we study the sojourn time $T(t)$ in the interval $[0,+\infty)$ up to a fixed time $t$ for this pseudo-process. We provide explicit expressions for the joint distribution of the couple $(T(t),X(t))$.