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

2003
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Electronic Journal of Probability
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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.

doi:10.1214/ejp.v8-178
fatcat:4vlzdq4oqfaq7hr5dslwidvlii