The distribution of the local time for "pseudoprocesses" and its connection with fractional diffusion equations

We prove that the pseudoprocesses governed by heat-type equations of order n >= 2 have a local time in zero (denoted by L-0(n)(t)) whose distribution coincides with the folded fundamental solution of a fractional diffusion equation of order 2(n - 1)1n, n >= 2. The distribution of L-0(n)(t) is also expressed in terms of stable laws of order n/(n - 1) and their form is analyzed. Furthermore, it is proved that the distribution of L-0(n)(t) is connected with a wave equation as n -> infinity. The distribution of the local time in zero for the pseudoprocess related to the Myiamoto's equation is also derived and examined together with the corresponding telegraph-type fractional equation. (c) 2005 Elsevier B.V. All rights reserved.