Joint distribution of the process and its sojourn time in a half-line $[a,+\infty)$ for pseudo-processes governed by higher-order heat-type equations

25 pagesInternational audienceLet $(X(t))_{t \ge 0}$ be the pseudo-process driven by the high-order heat-type equation $\frac{\partial u}{\partial t}=\pm \frac{\partial^N\!u}{\partial x^N}$, where $N$ is an integer greater than 2. We consider the sojourn time spent by $(X(t))_{t \ge 0}$ in $[a,+\infty)$ ($a\in \mathbb{R}$), up to a fixed time $t>0$: $T_a(t)=\int_0^t\mathbf{1}_{[a,+\infty)}(X(s))\,\mathrm{d}s$. The purpose of this paper is to explicit the joint pseudo-distribution of the vector $(T_a(t),X(t))$ when the pseudo-process starts at a point $x\in \mathbb{R}$ at time $0$. The method consists in solving a boundary value problem satisfied by the Laplace transform of the aforementioned distribution