# The Fractional Poisson Process and the Inverse Stable Subordinator

TL;DR: In this paper, it was shown that a traditional Poisson process, with the time variable replaced by an independent inverse stable subordinator, is also a fractional poisson process with Mittag-Leffler waiting times.

Abstract: The fractional Poisson process is a renewal process with Mittag-Leffler waiting times. Its distributions solve a time-fractional analogue of the Kolmogorov forward equation for a Poisson process. This paper shows that a traditional Poisson process, with the time variable replaced by an independent inverse stable subordinator, is also a fractional Poisson process. This result unifies the two main approaches in the stochastic theory of time-fractional diffusion equations. The equivalence extends to a broad class of renewal processes that include models for tempered fractional diffusion, and distributed-order (e.g., ultraslow) fractional diffusion. The paper also {discusses the relation between} the fractional Poisson process and Brownian time.

...read more

##### Citations

202 citations

55 citations

48 citations

### Cites background from "The Fractional Poisson Process and ..."

...7 in [55] shows that the pdfw(t) of the waiting times Jn has Laplace transform ∞...

[...]

...1 in [55] shows that one can also write N(t) = max{n ≥ 0 : Tn ≤ t} (7....

[...]

...2 in [55] shows that one can also construct the fractional Poisson process by replacing the time t in the traditional Poisson process N1(t)with an independent inverse stable subordinator....

[...]

47 citations

### Cites background from "The Fractional Poisson Process and ..."

...Then the density qk(t) = P ( N(E(t)) = k ) solves (see Meerschaert et al. (2011)). dβ dtβ qk(t) = −λ(1−▽)qk(t)....

[...]

...There is an interesting connection between continuous time random walks and fractional Cauchy problems, see [20, 18, 19] It is well known that the Poisson process N(t) with parameter λ > 0 solves the following difference-differential equation (DDE) d dt pk(t) = −λpk(t) + λpk−1(t), (1....

[...]

...Also, the density of Jn is fJn(x) = g(x)e −axη + a β η , (2.12) with g(x) = d dx [1− Eβ(−ηxβ)] and η = λ− aβ (2.13) (see Example 5.7 of Meerschaert et al. (2011))....

[...]

39 citations

### Cites background from "The Fractional Poisson Process and ..."

...[23] showed that the same fractional Poisson process can also be obtained via an inverse stable time change....

[...]

...It is also proven in [23] that the definition of the fractional Poisson process as a renewal process with Mittag-Leffler distribution of inter-arrival times is equivalent to the time change definition Nα(t) = N1(E(t)), where N1(t), t ≥ 0 is a homogeneous Poisson process with parameter λ > 0 and E(t), t ≥ 0 is the inverse stable subordinator independent of N1(t)....

[...]

##### References

55 citations

48 citations

47 citations

39 citations

35 citations