Estimates on the tail probabilities of subordinators and applications to general time fractional equations

In this paper, we study estimates on tail probabilities $$\mathbb{P}$(S_r \ge t)$ of several classes of subordinators under mild assumptions on the tail of its L'evy measure. As an application of that result, we obtain two-sided estimates for fundamental solutions of general homogeneous time fraction equations including those with Dirichlet boundary conditions.