The distribution of the local time for "pseudo-processes" and its connections with fractional diffusion equations

We prove that the pseudoprocesses governed by heat-type equations of order $n\geq2$ 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 $2(n-1)/n,n\geq2$: 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 $n\rightarrow\infty$. 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.