On functionals of excursions for Bessel processes with negative index

Closed formula for the integral of functionals on the space of excursions from the point $z\geq 0$ of a Bessel process $R^{(\mu)}$ with index $\mu \in (-1,0)$ with respect to the characteristic (It^o) measure is presented. This integral is equal to an integral with respect to some kernel $K$. The kernel $K$ is established due to the explicit form of the joint distribution of $(\tau_z, R_t^{(\mu)})$ at fixed time $t$ and for $\tau_z$ being the first hitting time of the point $z$ by the process $R^{(\mu)}$. The expected time spent by excursion in an interval with respect to the It^o measure is calculated.