Inverse Stable Subordinator

• Inverse stable subordinator is defined as the first-hitting time process of a β-stable subordinator, pivotal in describing subdiffusive phenomena and anomalous transport.
• Its law is characterized by the Mittag–Leffler function and solutions to time-fractional PDEs, exhibiting self-similarity, strong pathwise regularity, and non-Markovian dynamics.
• Practical applications include fractional Poisson processes, renewal models, and tempered variants, which extend its utility in modeling fractional and subdiffusive systems.

An inverse stable subordinator is a stochastic process arising as the right-continuous inverse (or first-hitting-time process) of a strictly $0<\beta<1$ $\beta$-stable subordinator. It serves as a central random time-change mechanism in the theory of anomalous diffusion, time-fractional differential equations, fractional renewal processes, and a wide array of subdiffusive models. The inverse stable subordinator’s law is links to the Mittag–Leffler function, and the process is non-Markovian, non-Lévy, self-similar, and exhibits strong pathwise regularity and aging phenomena.

1. Definition and Analytical Properties

Let $S_\beta = \{S_\beta(u): u \ge 0\}$ be a strictly increasing (pure-jump) $\beta$-stable subordinator, a Lévy process with Laplace transform

$$\mathbb{E}\bigl[e^{-s S_\beta(u)}\bigr] = \exp(-u s^\beta), \qquad 0 < \beta < 1, \; s \ge 0.$$

The inverse stable subordinator $E_\beta$ is then defined as the first-hitting time process

$$E_\beta(t) = \inf\{u \ge 0 : S_\beta(u) > t\}, \qquad t \ge 0.$$

$E_\beta$ is continuous, non-decreasing, and non-Markovian. The increments generally are not stationary. By construction, for each fixed $t>0$, $E_\beta(t)$ is supported on $(0,\infty)$. The process is self-similar with index $\beta$: $E_\beta(ct)\overset{d}{=}c^\beta E_\beta(t)$ for any $c>0$, and all positive integer moments exist: $$\mathbb{E}\left[E_\beta(t)^k\right] = \frac{k!}{\Gamma(1+k\beta)}\, t^{k\beta}, \quad k=1,2,\ldots.$$

The Laplace transform of the marginal law of $E_\beta(t)$ is the classic Mittag–Leffler function: $$\mathbb{E}\bigl[e^{-s E_\beta(t)}\bigr] = E_\beta(-s t^\beta), \quad E_\beta(z) = \sum_{k=0}^\infty \frac{z^k}{\Gamma(\beta k +1)}.$$ The density $f_{E_\beta}(x,t)$, when it exists, admits the Laplace representation

$$\mathcal{L}_t\{f_{E_\beta}(x,t)\}(s) = s^{\beta-1}\, e^{-x s^\beta},$$

and an explicit representation in terms of the stable subordinator’s density: $$f_{E_\beta}(x,t) = \frac{t}{\beta} x^{-1-1/\beta} f_{S_\beta}\!\bigl(t x^{-1/\beta}, 1\bigr), \quad x>0, t>0.$$ The distribution function is linked to that of the stable subordinator via

$$\mathbb{P}[E_\beta(t) \leq x] = 1 - \mathbb{P}[S_\beta(x) \leq t] = \int_t^\infty f_{S_\beta}(u,x) du.$$

A Wright/M-Wright function expansion also exists: $$f_{E_\beta}(x,t) = t^{-\beta} M_\beta(x t^{-\beta}),$$ with $$M_\beta(z) = \sum_{k=0}^\infty \frac{(-z)^k}{k! \Gamma(-\beta k + 1 - \beta)}$$ for $0 < \beta < 1$ (Gorenflo et al., 2013, Chen et al., 2018).

2. Fractional Differential Equations and Governing Laws

The density $f_{E_\beta}(x,t)$ solves a time-space fractional PDE of Caputo type: $$\frac{\partial^\beta}{\partial t^\beta} f_{E_\beta}(x,t) = - \frac{\partial}{\partial x} f_{E_\beta}(x,t), \qquad x>0,\; t>0,$$ with

$$\frac{\partial^\beta}{\partial t^\beta}g(t) = \frac{1}{\Gamma(1-\beta)}\int_0^t (t-s)^{-\beta} g'(s) ds$$

and boundary/initial data

$$f_{E_\beta}(x,0) = \delta(x), \qquad f_{E_\beta}(0+,t) = t^{-\beta}/\Gamma(1-\beta), \qquad \lim_{x\to\infty}f_{E_\beta}(x,t)=0.$$

These equations encode the non-Markovian, “subdiffusive” time-change behavior seen in processes subordinated by $E_\beta$ (Kumar et al., 2011, 0705.0168).

Iterated inverses: Composing $n$ independent copies yields another inverse stable subordinator $E_\beta^{(n)}$ of index $\beta^n$; the corresponding density solves a fractional PDE with order $\beta^n$ (Kumar et al., 2011). More generally, the extension to non-homogeneous (multistable) cases results in variable-order (Riemann–Liouville) fractional drift equations (Beghin et al., 2016).

3. Connections to Fractional Renewal and Counting Processes

The inverse stable subordinator underlies the construction of the fractional Poisson process, as well as more general fractional renewal processes. Given a Poisson process $N(t)$ independent of $E_\beta$, the time-changed process $N(E_\beta(t))$—or “fractional Poisson process”—has the same one-dimensional law as the renewal process with IID Mittag–Leffler waiting times: $$P_k(t) = \mathbb{P}\{N(E_\beta(t))=k\} = \int_0^\infty \frac{(\lambda x)^k}{k!} e^{-\lambda x} f_{E_\beta}(x,t) dx.$$ The pmf $P_k(t)$ solves a fractional difference-differential equation,

$$\frac{d^\beta}{dt^\beta} P_k(t) = -\lambda (P_k(t) - P_{k-1}(t)), \qquad P_k(0) = \delta_{k,0}.$$

This unifies the subordination approach with the renewal view, covering the transition from Markovian to non-Markovian dynamics in a rigorous framework (Meerschaert et al., 2010, Kumar et al., 2011, Gorenflo et al., 2013).

Analogous results, with appropriate modifications, also hold for inhomogeneous (multistable) subordinators and their inverses, yielding time-inhomogeneous fractional Poisson models governed by variable-order fractional evolution equations. The moments of the inverse subordinator in these cases can often only be written in integral forms depending on the variable stability function $a(t)$ (Beghin et al., 2016).

4. Tempered and Generalized Inverse Stable Subordinators

Tempered stable subordinators and their inverses generalize classical models by introducing a tempering parameter $\lambda \geq 0$. The tempered stable subordinator $S_{\beta,\lambda}(t)$ has Laplace transform

$$\mathbb{E}[e^{-s S_{\beta,\lambda}(t)}] = \exp\{-t[(s+\lambda)^\beta - \lambda^\beta]\},$$

and its inverse $E_{\beta,\lambda}(t)$, called the inverse tempered stable subordinator, possesses a density with Laplace transform in $t$: $$\mathcal{L}_t[h_{\beta,\lambda}(x,t)](s) = \frac{1}{s}\{(s+\lambda)^\beta - \lambda^\beta\} \exp\left[-x((s+\lambda)^\beta - \lambda^\beta)\right].$$ The governing equations become tempered fractional PDEs involving the tempered Caputo derivative. Explicit representations and series expansions for $h_{\beta,\lambda}$ involve incomplete gamma functions and enable computational analysis. The asymptotic behavior of the moments interpolates from $t^{q\beta}$ as $t \rightarrow 0$ to $t^q$ as $t \rightarrow \infty$, demonstrating a crossover from sublinear to linear growth (Kumar et al., 2014, Gupta et al., 2021, Tang et al., 2022).

Distributional and asymptotic properties: For $x \to 0$,

$$h_{\beta,\lambda}(0+,t) \sim e^{-\lambda t}\, t^{-\beta} \Gamma(1-\beta),$$

while for $x \to \infty$,

$$h_{\beta,\lambda}(x, t) \sim \frac{\lambda^\beta}{\Gamma(1-\beta)} x^{-1-\beta} e^{-\lambda x}.$$

For $\lambda \to 0$, one recovers the purely stable case.

5. Functionals, Limit Theorems, and Path Properties

The process $E_\beta$ and its functionals are characterized by rich stochastic structure:

• Weak convergence: Time-changed Lévy processes $S(E_\beta(t))$ satisfy non-classical large deviation principles and noncentral moderate deviation theorems. Rate functions and scaling exponents are non-quadratic and encode anomalous diffusive behavior. Specifically, the moment generating function is governed by the Mittag–Leffler transform, and asymptotic laws depart from the Gaussian, except when $\beta = 1$ (Iuliano et al., 2 Jan 2024).
• Fractional integration: Fractionally integrated inverse stable subordinators $$Y_{\beta, \gamma}(t) = \int_0^t (t-s)^{\gamma-1}/\Gamma(\gamma) E_\beta(s)\,ds$$ arise as scaling limits of shot-noise processes with heavy-tailed input and obey self-similarity of index $\beta+\gamma$. Pathwise regularity admits tight two-sided laws of the iterated logarithm, and local Hölder continuity is controlled by $\beta+\gamma$ (Iksanov et al., 2016).
• Sampling and numerical simulation: Efficient algorithms exist for simulating paths and joint laws of $E_\beta$ and for Monte Carlo evaluation of functionals of time-changed diffusions. Complexity is bounded, and the discretization error in subdiffusive SDEs is explicitly controlled by the Hölder regularity (Biočić et al., 20 Dec 2024, Gorenflo et al., 2013).
• Long-range dependence: When used as a random clock for mixed fractional Brownian motion, time-change by $E_\beta$ yields processes exhibiting long-range dependence, with correlation decay rates $\sim t^{-(1-\beta H_2)}$ for $H_2$ the higher Hurst index, whenever $0 < \beta H_2 < 1$ (Mliki, 2023).

6. Applications and Extensions

The inverse stable subordinator is foundational in time-fractional Cauchy problems. If $X(u)$ is a Markov process with generator $L_x$, then the time-changed process $X(E_\beta(t))$ solves: $$\frac{\partial^\beta}{\partial t^\beta} u(t,x) = L_x u(t,x), \qquad u(0,x) = f(x),$$ naturally capturing subdiffusive behavior (0705.0168).

Beyond standard Lévy inputs, recent research considers:

• Time-inhomogeneous (multistable) inverse subordinators, yielding variable-order time-fractional equations and associated time-inhomogeneous fractional Poisson processes (Beghin et al., 2016).
• Tempered and distributed-order generalizations, relevant for modeling crossover diffusive regimes and ultraslow diffusion, where the inverse process underlies space-time PDEs with distributed or tempered fractional derivatives (Kumar et al., 2014, Chen et al., 2018).
• Fractional renewal equations, where the inverse stable random time-change encodes heavy-tailed waiting structures and memory (Meerschaert et al., 2010, Gorenflo et al., 2013).

7. Infinite Divisibility, Series Representations, and Limitations

Crucially, the one-dimensional laws of $E_\beta(t)$ are not infinitely divisible for any $t>0$ (except $\beta=1$). This property distinguishes the process from subordinators and has implications for the structure of processes subordinated by $E_\beta$, such as the fractional Poisson process (Kumar et al., 2018).

A selection of analytic representations for the density and distribution exploits:

• Series expansions in terms of Mittag–Leffler and M-Wright functions.
• Explicit Laplace transforms and Mellin transforms, yielding closed or semi-closed forms for marginals, products, and quotients, and precise asymptotics as $x \to 0$ or $x \to \infty$ (Gupta et al., 2021).

These explicit formulas facilitate numerical evaluation, simulation, and rigorous asymptotic analysis, while also revealing the central mathematical mechanism—random heavy-tailed time-scaling—by which $E_\beta$ interpolates between standard and anomalous diffusions.

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Table: Key Analytical Aspects of the Inverse Stable Subordinator

Aspect	Formula/Result	Reference
Laplace transform	$$\mathbb{E}\bigl[e^{-s E_\beta(t)}\bigr]=E_\beta(-s t^\beta)$$	(Meerschaert et al., 2010)
Fractional PDE	$$\frac{\partial^\beta}{\partial t^\beta} f_{E_\beta} = -\frac{\partial}{\partial x} f_{E_\beta}$$	(Kumar et al., 2011)
Moments	$$\mathbb{E}[E_\beta(t)^k] = k!/ \Gamma(1+k\beta) \, t^{k\beta}$$	(Chen et al., 2018)
Infinite divisibility	Not infinitely divisible for $\beta\in(0,1)$	(Kumar et al., 2018)
Self-similarity	$E_\beta(c t) \overset{d}{=} c^\beta E_\beta(t)$	(Iksanov et al., 2016)

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In summary, the inverse stable subordinator is a canonical non-Markovian, non-Levy, self-similar process encoding fractional, heavy-tailed time-change dynamics, foundational for time-fractional stochastic models. Its governing equations, distributional properties, and analytic representations underpin extensive applications in fractional PDEs, subdiffusive phenomena, anomalous transport, and generalized renewal theory (Meerschaert et al., 2010, Kumar et al., 2011, 0705.0168, Gorenflo et al., 2013).