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

Abstract

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 ) / n , 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 → ∞ . 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.