We consider the properties of Green’s function for the nonlinear fractional differential equation boundary value problem: , where , ${\mathbf{D}}_{0+}^{\alpha }$ is the standard Riemann-Liouville derivative. Here our nonlinearity $f$ may be singular at $u=0$. As applications of Green’s function, we give some multiple positive solutions for singular boundary value problems by means of Schauder fixed-point theorem.

The joint law of the total local times at two levels for $h$-paths of symmetric Lévy processes is shown to admit an explicit representation in terms of the laws of the squared Bessel processes of dimensions two and zero. The law of the total local time at a single level for bridges is also discussed.