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

Aime Lachal
2003 Electronic Journal of Probability  
The higher-order heat-type equation ∂u/∂t = ±∂ n u/∂x n has been investigated by many authors. With this equation is associated a pseudo-process (X t ) t 0 which is governed by a signed measure. In the even-order case, Krylov, [9], proved that the classical arc-sine law of Paul Lévy for standard Brownian motion holds for the pseudo-process (X t ) t 0 , that is, if T t is the sojourn time of (X t ) t 0 in the half line (0, +∞) up to time t, then P(T t ∈ ds) = ds π √ s(t−s) , 0 < s < t.
more » ... < s < t. Orsingher, [13], and next Hochberg and Orsingher, [7], obtained a counterpart to that law in the odd cases n = 3, 5, 7. Actually Hochberg and Orsingher 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, [11] , in the cases n = 3, 4. In this paper, we prove that the conjecture of Hochberg and Orsingher 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 0 , as well as the distribution of the last time before becoming definitively negative up to time t.
doi:10.1214/ejp.v8-178 fatcat:4vlzdq4oqfaq7hr5dslwidvlii