[Paper Review] Laplace-Laplace analysis of the fractional Poisson process

Rudolf Gorenflo, Francesco Mainardi|arXiv (Cornell University)|May 23, 2013

Fractional Differential Equations Solutions15 references16 citations

TL;DR

This paper introduces a Laplace-Laplace transform analysis of the fractional Poisson process, showing it arises via time change of the standard Poisson process through subordination to an inverse stable subordinator. The key contribution is a rigorous derivation of the probability mass function using double Laplace transforms, leading to a subordination integral representation that confirms the fractional Poisson process as a renewal process with Mittag-Leffler waiting times.

ABSTRACT

We generate the fractional Poisson process by subordinating the standard Poisson process to the inverse stable subordinator. Our analysis is based on application of the Laplace transform with respect to both arguments of the evolving probability densities.

Motivation & Objective

• To provide a rigorous, transform-based derivation of the fractional Poisson process using Laplace transforms in both time and space.
• To establish the equivalence between the fractional Poisson process and a time-changed standard Poisson process via subordination to the inverse stable subordinator.
• To reformulate the Continuous Time Random Walk (CTRW) framework using Laplace transforms instead of Fourier transforms, suitable for strictly positive increments.
• To derive an explicit integral representation for the probability mass function of the fractional Poisson process through double Laplace inversion.

Proposed method

• Apply the Laplace transform to both time and space variables in the CTRW framework, replacing the standard Fourier transform with Laplace due to positive unit jumps.
• Use the Cox-Weiss formula in Laplace-Laplace form to express the characteristic function of the counting process.
• Introduce an operational time variable $ t_* $ via an improper integral representation of the reciprocal denominator in the transform domain.
• Invert the double Laplace transform to derive a subordination integral: $ p_\beta(x,t) = \int_0^\infty p_1(x,t_*) q_\beta(t_*,t) \, dt_* $.
• Utilize known results on the inverse stable subordinator, including its density $ q_\beta(t_*,t) = t^{-\beta} M_\beta(t_* t^{-\beta}) $, where $ M_\beta $ is the M-Wright function.
• Establish the stochastic interpretation: $ x(t) = y(t_*(t)) $, where $ y $ is the standard Poisson process and $ t_*(t) $ is the inverse stable subordinator.

Experimental results

Research questions

• RQ1 How can the fractional Poisson process be rigorously derived using double Laplace transforms instead of Fourier transforms?
• RQ2 What is the precise stochastic mechanism that generates the fractional Poisson process from the standard Poisson process?
• RQ3 How does the use of the Mittag-Leffler waiting time distribution relate to subordination in the context of renewal processes?
• RQ4 Can the CTRW framework be adapted to use Laplace transforms in both time and space when jumps are positive and of unit size?
• RQ5 What is the exact integral representation of the fractional Poisson probability mass function in terms of the inverse stable subordinator?

Key findings

• The fractional Poisson process is rigorously derived as a time-changed standard Poisson process via subordination to the inverse stable subordinator of index $ \beta \in (0,1] $.
• The probability mass function $ p_n(t) $ is expressed as a subordination integral: $ p_n(t) = \frac{1}{n!} \int_0^\infty t_*^n e^{-t_*} q_\beta(t_*,t) \, dt_* $, with $ q_\beta(t_*,t) = t^{-\beta} M_\beta(t_* t^{-\beta}) $.
• The Laplace-Laplace transform method allows exact inversion to yield a variable-separable solution, confirming the role of the inverse stable subordinator as the operational time process.
• The M-Wright function $ M_\beta $ appears naturally as the density of the inverse stable subordinator, linking the fractional process to stable distributions.
• The result confirms that the fractional Poisson process is a renewal process with Mittag-Leffler waiting time distribution $ E_\beta(-t^\beta) $, consistent with prior definitions.
• The analysis provides a direct, accessible derivation that avoids advanced stochastic process terminology, making it suitable for applied scientists.