On fundamental solutions, transition probabilities and fractional derivatives

The aim of this note is to clarify the connection between different notions of fundamental solution and to outline the interplay between transitional probabilities of stochastic processes, evolution semigroups, evolution equations and their fundamental solutions. We discuss different notions of the fundamental solution for Levy processes with infinitely smooth symbol and for stable subordinators. In the case of Levy processes with infinitely smooth symbol we find the fundamental solution of the corresponding forward evolution equation and recover the Duhamel formula for the solution of the Cauchy problem for this equation. In the case of the 1/2-stable subordinator, we find the transition density by solving an evolution equation with the (weak) Riemann | Liouville fractional derivative and show that the Weyl fractional derivative is the negative of the adjoint to the Riemann | Liuoville (weak) fractional derivative