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The space-fractional and the time-fractional Poisson processes are twowell-known models of fractional evolution. They can be constructed as standardPoisson processes with the time variable replaced by a stable subordinator andits inverse, respectively. The aim of this paper is to study non-homogeneousversions of such models, which can be defined by means of the so-calledmultistable subordinator (a jump process with non-stationary increments),denoted by H. Firstly, we consider the Poisson process time-changed by H and weobtain its explicit distribution and governing equation. Then, by using theright-continuous inverse of H, we define an inhomogeneous analogue of thetime-fractional Poisson process.

Time-inhomogeneous fractional Poisson processes defined by the multistable subordinator