Distributions of Sojourn Time, Maximum and Minimum for Pseudo-Processes Governed by Higher-Order Heat-Type Equations

Abstract

The higher-order heat-type equation $ \partial u/\partial t=\pm\partial^{n} u/ \partial x^{n} $ has been investigated by many authors. With this equation is associated a pseudo-process $(X_t)_{t\ge 0}$ which is governed by a signed measure. In the even-order case, Krylov (1960) proved that the classical arc-sine law of Paul Levy for standard Brownian motion holds for the pseudo-process $(X_t)_{t\ge 0}$, that is, if $T_t$ is the sojourn time of $(X_t)_{t\ge 0}$ in the half line $(0,+\infty)$ up to time $t$, then $P(T_t\in ds)=\frac{ds}{\pi\sqrt{s(t-s)}}$, $0 \lt s \lt t$. Orsingher (1991) and next Hochberg and Orsingher (1994) obtained a counterpart to that law in the odd cases $n=3,5,7.$ Actually Hochberg and Orsingher (1994) proposed a more or less explicit expression for that new law in the odd-order general case and conjectured a quite simple formula for it. The distribution of $T_t$ subject to some conditioning has also been studied by Nikitin & Orsingher (2000) in the cases $n=3,4.$ In this paper, we prove that the conjecture of Hochberg and Orsingher (1994) is true and we extend the results of Nikitin & Orsingher for any integer $n$. We also investigate the distributions of maximal and minimal functionals of $(X_t)_{t\ge 0}$, as well as the distribution of the last time before becoming definitively negative up to time $t$.