Distributions of Sojourn Time, Maximum and Minimum for Pseudo-Processes Governed by Higher-Order Heat-Type Equations
Aime Lachal, Institut National des Sciences Appliquèes de Lyon, France
Abstract
The higher-order heat-type equation
$
partial u/partial t=pmpartial^{n} u/ partial x^{n}
$
has been investigated by many authors. With this equation is associated
a pseudo-process $(X_t)_{tge 0}$ which is governed by a signed measure.
In the even-order case, Krylov, cite{kry}, proved that the classical
arc-sine law of Paul L'evy for standard
Brownian motion holds for the pseudo-process $(X_t)_{tge 0}$, that is,
if $T_t$ is the sojourn time of $(X_t)_{tge 0}$ in the half line
$(0,+infty)$ up to time $t$, then $P(T_tin,ds)=frac{ds}{pisqrt{s(t-s)}}$, $0<s<t$. Orsingher, cite{ors}, and next HO, cite{hoch-ors1}, obtained a counterpart to that law in the odd cases $n=3,5,7.$ Actually HO 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, cite{nik-ors}, in the cases $n=3,4.$
In this paper, we prove that the conjecture of HO 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 $(X_t)_{tge 0}$, as well as the distribution of the last time before becoming definitively negative up to time $t$.
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