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The inverse tempered stable subordinator is a stochastic process that models power law
waiting times between particle movements, with an exponential tempering that allows
all moments to exist.This paper shows that the probability density function of an inverse
tempered stable subordinator solves a tempered time-fractional diffusion equation, and
its ''folded'' density solves a tempered time-fractional telegraph equation.
The inverse tempered stable subordinator is a stochastic process that models power law
waiting times between particle movements, with an exponential tempering that allows
all moments to exist. This paper shows that the probability density function of an inverse
tempered stable subordinator solves a tempered time-fractional diffusion equation, and
its ‘‘folded’’ density solves a tempered time-fractional telegraph equation. Two explicit
formulae for the density function are developed, and applied to compute explicit solutions
to tempered fractional Cauchy problems, where a tempered fractional derivative replaces
the first derivative in time. Several examples are given, including tempered fractional
diffusion equations on bounded or unbounded domains, and the probability distribution of
a tempered fractional Poisson process. It is shown that solutions to the tempered fractional
diffusion equation have a cusp at the origin.
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