Ultrafast subordinators and their hitting times

Ultrafast subordinators are nondecreasing Levy processes obtained as the limit of suitably normalized sums of independent random variables with slowly varying probability tails. They occur in a physical model of ultraslow diffusion, where the inverse or hitting time process randomizes the time vari- able. In this paper, we use regular variation arguments to prove that a wide class of ultrafast subordinators generate holomorphic semigroups. We then use this fact to compute the density of the hitting times. The density formula is important in the physics application, since it is used to calculate the solutions of certain distributed-order fractional diffusion equations. Ultrafast subordinators are connected with certain random walk models in physics. In these models, waiting times with power-law probability tails are ran- domized in terms of the power law exponent. The renewal process with these waiting times is then the inverse or hitting time process of the random walk with these jumps. The probability tails of these random variables are slowly varying, so that the random walk grows vary fast, and the renewal (inverse) process very slow. The random walk limits form an interesting new class of subordinators. The paper (19) develops the limit theory for these ultrafast subordinators, together with some results on their hitting times. The basic approach is to study the asymptotic be- havior of the renewal process by first proving a limit theorem for the random walk, and then inverting. The random walk converges to a subordinator (a Levy process with nondecreasing sample paths) and the renewal process converges to the inverse or hitting time (or first passage time) process of the subordinator. The paper (19) imposes a technical condition which is difficult to check. In this paper, we remove