Poisson Functional Representation

• Poisson Functional Representation is a method that characterizes random variables and processes as functionals of Poisson point processes, providing explicit simulation and distributional insights.
• It leverages Laplace transforms, Campbell’s theorem, and combinatorial identities to derive precise moment formulas and distributional results for complex stochastic systems.
• Applications range from analyzing sojourn times in Lévy processes to advances in statistical physics, information theory, and wireless network modeling.

A Poisson functional representation provides an explicit characterization of certain random variables or stochastic processes as sums or functionals of Poisson point processes, with far-reaching applications in probability theory, stochastic processes, statistical physics, and information theory. The methodology enables constructive simulations, structural probabilistic insights, and explicit calculations of distributional quantities, particularly in settings where direct computation is otherwise intractable.

1. Foundational Principles

The core principle underlying a Poisson functional representation is the identification of a (possibly random) quantity as a measurable function of a Poisson process: either as the sum of its points, as a more general functional, or as a structure induced by marked Poisson points. A central example is the positive sojourn time of a Lévy process $X=(X_t)_{t\geq 0}$ up to a random (typically exponential) time,

$$A_{E^{(q)}} = \int_0^{E^{(q)}} \mathbf{1}\{X_s > 0\}\,ds,$$

which is shown to have the same distribution as the sum of points of a Poisson process on $(0,\infty)$ with explicitly computable intensity determined by the positivity function $t \mapsto \mathbb{P}\{X_t > 0\}$ (Pitters, 15 Sep 2025). This identification is made rigorous via Laplace transform methods, Campbell’s theorem, and combinatorial arguments.

Key structural tools include:

• The Laplace functional of the Poisson process links exponential moments of functionals to intensity measures.
• Campbell’s theorem provides formulas for expectations of sums functionals over Poisson point processes.
• Markings and thinning expose further probabilistic structure, such as connections to Poisson–Dirichlet distributions.

2. Explicit Representation Formulas

Consider a Lévy process $X$ started at $0$ and an independent exponential time $E^{(q)}$ of rate $q>0$. Define the measure

$\nu(dt) = q\,\mathbb{P}\{X_t>0\}\,dt,\quad t > 0.$

Let $\Pi$ be a Poisson point process on $(0, \infty)$ with intensity $\nu$. The primary identity is

$A_{E^{(q)}} \stackrel{d}{=} \sum_{T \in \Pi} T,$

meaning the sojourn time up to $E^{(q)}$ is distributed as the sum of the points of $\Pi$ (Pitters, 15 Sep 2025).

Laplace transforms play a central role: for $\lambda > 0$,

$$\mathbb{E}[e^{-\lambda A_{E^{(q)}}}] = \exp\left(-\int_0^\infty (1-e^{-\lambda t})\,\nu(dt)\right),$$

matching the Laplace transform of the sum of points of a Poisson process.

The first two moments are straightforward consequences,

$$\mathbb{E}[A_{E^{(q)}}] = \int_0^\infty q t\,\mathbb{P}\{X_t > 0\}\,dt, \quad \text{Var}(A_{E^{(q)}}) = \int_0^\infty q t^2\,\mathbb{P}\{X_t > 0\}\,dt.$$

3. Special Cases and Connections

A variety of classical processes emerge as concrete cases:

• Brownian Motion: For $X_t = B_t$, standard Brownian motion, $\mathbb{P}\{B_t > 0\} = 1/2$. The corresponding sojourn time distribution recovers the Lévy arcsine law, with explicit density $\frac{1}{\pi\sqrt{x(1-x)}}$ for $A_t/t \in (0,1)$ (Pitters, 15 Sep 2025).
• Stable processes: A symmetric strictly $\alpha$-stable Lévy process also yields a constant positivity $\mathbb{P}\{X_t>0\}=1/2$, recovering generalized arcsine laws.
• Processes with Constant Positivity: By Getoor-Sharpe, all Lévy processes with $\mathbb{P}\{X_t > 0\} \equiv c$ for some constant $c\in [0,1]$ give rise to sojourn times whose normalized distribution is $\operatorname{Arcsin}(c)$, and the ranked normalized Poisson points of $\Pi$ have the Poisson–Dirichlet($c$) distribution.
• Stable Subordinator with Drift: For $S_t^{(-\mu)}=S_t-\mu t$, with $S_t$ a strictly $1/2$-stable subordinator, $$\mathbb{P}\{S_t^{(-\mu)}>0\} = \operatorname{erf}(\sqrt{t/\mu})$$. The associated sojourn time's density and Laplace transform are given explicitly in terms of convolution formulas involving special functions (Pitters, 15 Sep 2025).

4. Analytical and Probabilistic Structure

The Poisson functional representation enables comprehensive analytical treatment via Laplace and double Laplace transforms. The structure links stochastic path properties to explicit Poissonian summations, facilitating inversion to density formulas and the study of fine properties, such as path decompositions and partition structures.

For cases with constant positivity, the normalization

$$\frac{A_{E^{(q)}}}{E^{(q)}} \stackrel{d}{=} \frac{\Gamma(q, c)}{\Gamma(q, c) + \Gamma(q, 1-c)}$$

shows sojourn fractions as generalized Beta-distributed, and ranks of ordered interval lengths correspond to Poisson–Dirichlet($c$) partitions. Thinning arguments relate these to the beta-GEM construction for Poisson–Dirichlet distributions.

The connection to Campbell’s theorem and Spitzer’s combinatorial lemma enables rigorous derivations and combinatorial interpretations of distributional identities.

5. Methodological Implications and Extensions

The Poisson functional representation paradigm extends beyond sojourn times in Lévy processes. It encompasses:

• Random coding and information theory: Variable-length source coding via marked Poisson processes and explicit geometric tail bounds (see, e.g., Poisson functional representation in one-shot lossy source coding (Li, 2024)).
• Stochastic geometry: Exact calculation of interference functionals and joint moments in wireless Poisson networks using explicit sum-product representations and generalized Campbell-Mecke theorems (Schilcher et al., 2014).
• Measure-valued processes and branching structures: Particle-level representations provide a means to recover and characterize measure-valued processes as conditional Poisson states, with applications to analysis of extinction probabilities, martingale techniques, and diffusion limits (Kurtz et al., 2011).
• Malliavin calculus and stochastic analysis: Pathwise Clark-Ocone-type formulas for Poisson functionals extend the Poisson imbedding, enabling $L^1$ expansions and canonical pseudo-chaotic representations for wide classes of functionals (Hillairet et al., 2024).

6. Representation Table: Special Processes

Process Type	Intensity Measure $\nu$	Distributional Result
Brownian Motion	$q/2\ dt$	$A_t/t\sim$ Arcsine$(1/2)$ law
Symm. Strictly $\alpha$-Stable	$q/2\ dt$	$A_t/t\sim$ generalized arcsine law
Constant Positivity $\mathbb{P}\{X_t>0\}=c$	$q c\ dt$	$A_t/t\sim$ Arcsine$(c)$, Poisson–Dirichlet($c$)
$1/2$-Stable Subordinator with Drift	$q\,\operatorname{erf}(\sqrt{t/\mu})\,dt$	Density $h(x)g(t-x)$ on $(0, t)$

7. Analytical Tools and Proof Techniques

Derivations of Poisson functional representations utilize:

• Campbell’s theorem: Relates expectations of functionals of the Poisson process to integrals over the intensity measure, leading to moment and transform formulas.
• Laplace transform methods: Provide explicit characterization of nontrivial distributions, enabling direct matching with Poisson functionals.
• Combinatorial identities: Bell polynomials, Spitzer’s lemma, and sum-product expansions allow for transition from moment calculations to full distributional results.
• Connections to partition structures: Ranked points and normalized sums naturally extend to Poisson–Dirichlet and GEM representations.

These methods facilitate constructive simulation, allow explicit evaluation of otherwise implicit path properties, and reveal deeper connections to intersection regimes of stochastic process theory, information theory, and mathematical physics.

---

References:

(Pitters, 15 Sep 2025, Li, 2024, Schilcher et al., 2014, Hillairet et al., 2024, Kurtz et al., 2011, G. et al., 2015)