Inverse Subordinator

• Inverse subordinators are the right-continuous inverses of non-decreasing Lévy processes, providing a rigorous framework for time-change in stochastic models.
• They generate non-local, convolution-type time derivatives in fractional and generalized PDEs, linking analytic properties of Bernstein functions to anomalous diffusion.
• Applications span anomalous diffusion, network traffic, finance, and simulation methods, underpinned by detailed regularity, scaling limits, and numerical techniques.

An inverse subordinator is the right-continuous inverse (or first-passage time/hitting-time process) of a real-valued, non-decreasing Lévy process called a subordinator. Inverse subordinators arise as time-changes in stochastic processes, providing a rigorous analytic and probabilistic framework for modeling phenomena such as anomalous diffusion, heavy-tailed renewal processes, and the solution theory of fractional and generalized fractional partial differential equations (PDEs). The theory of inverse subordinators intimately connects with Bernstein functions, convolution-type operators, path regularity, functional limit theorems, infinite divisibility questions, simulation techniques, and a wide range of applied models.

1. Definition and Basic Properties

Let $\sigma = \{\sigma(x), x \geq 0\}$ be a subordinator—a non-decreasing Lévy process—whose law is characterized by its Laplace exponent (a Bernstein function) $\phi(\lambda)$ via

$$-\log \mathbb{E}[e^{-\lambda \sigma(1)}] = \phi(\lambda) = q + b \lambda + \int_0^\infty (1 - e^{-\lambda y}) \mu(dy),$$

where $q \geq 0$ (killing rate), $b \geq 0$ (drift), and $\mu$ (Lévy measure) satisfies $\int_0^\infty (1 \wedge y) \mu(dy) < \infty$. The inverse subordinator $L = \{L(t), t \geq 0\}$ is defined as

$L(t) = \inf \{ x > 0 : \sigma(x) > t \},$

with almost surely continuous, non-decreasing sample paths under mild conditions (e.g., positive drift or infinite activity). The distribution of $L(t)$ has a Lebesgue density $f_\phi(x,t)$ whenever $\sigma$ has strictly increasing sample paths or positive drift (Ascione et al., 2023). The path structure of $L$ is determined by the jump behavior of $\sigma$: large jumps of $\sigma$ correspond to flat intervals (“trapping”) in $L$.

Inverse subordinators have the property that their densities and their derivatives in both $x$ (operational time) and $t$ (physical time) are smooth functions under explicit moment and analyticity conditions on $\phi$ (Ascione et al., 2023).

2. Governing Equations and Fractional PDE Connections

Inverse subordinators are a canonical source of non-local-in-time operators in the governing equations of stochastic processes:

• Let $X(t)$ be a Markov process with generator $L$. The process $Z(t) = X(L(t))$—that is, $X$ time-changed by the inverse subordinator—has transition densities $u(x,t) = \mathbb{E}[f(Z(t))]$ that solve generalized Cauchy problems: $D_t^{(f)} u(t,x) = L u(t,x),\quad u(0,x) = f(x),$ where $D_t^{(f)}$ is a convolution-type time derivative determined by the Bernstein function $\phi$ of the subordinator (Toaldo, 2013, Zhao et al., 2019, Hirsch et al., 2013).

In the important special case when the subordinator is $\alpha$-stable ($\phi(\lambda) = \lambda^\alpha$), the time-derivative operator is the Caputo fractional derivative $D_t^\alpha$, yielding

$$\frac{\partial^\alpha u(t,x)}{\partial t^\alpha} = L u(t,x),\quad u(0,x) = f(x).$$

This “fractional Cauchy problem” is foundational in anomalous diffusion (0705.0168). For even more general Bernstein functions, the time-derivative is replaced by a convolution-type operator whose Laplace symbol is $\phi$.

Explicit Laplace transform formulas for the inverse subordinator densities,

$$\mathcal{L}_t [f_\phi(x, t)](\lambda) = [\phi(\lambda)/\lambda] e^{-x \phi(\lambda)},$$

provide powerful methods for analysis and inversion (Colantoni et al., 2021, Ascione et al., 2023).

3. Regularity, Asymptotics, and Series Representations

Recent work employs Laplace inversion and refined saddle-point analyses to characterize the smoothness and asymptotics of densities of both subordinators and their inverses (Ascione et al., 2023). Under mild regularity conditions on $\phi$, $f_\phi(x,t)$ and all its mixed derivatives are $C^\infty$ in both arguments.

Uniform asymptotic expansions are obtained in regions of the $(x,t)$-plane where $t/x$ is in a regular regime, with explicit error bounds and coefficients written in terms of derivatives of the Laplace exponent: $$f_\phi(x,t) \sim \frac{1}{\sqrt{2\pi x (-\phi''(c))}}\, \frac{\phi^{\dagger}(c)}{c} e^{-c t + x \phi(c)},$$ where $c = (\phi')^{-1}(t/x)$ and $\phi^{\dagger}$ is related to the adjusted Laplace exponent (Ascione et al., 2023). These results extend to all derivatives $\partial_x^k \partial_t^{\ell} f_\phi(x,t)$.

Additionally, under analyticity assumptions, power series expansions in $x$ are available for $f_\phi(x,t)$ and their derivatives, with explicit coefficients depending on convolutions and derivatives of the Lévy measure’s tail.

4. Key Examples and Special Cases

Stable and Tempered Stable Subordinators:

• For stable subordinators ($\phi(\lambda) = \lambda^\alpha$, $0 < \alpha < 1$), the inverse subordinator is central to the fractional calculus of order $\alpha$. Explicit series and integral representations exist for the inverse density (Kumar et al., 2014).
• For tempered stable subordinators ($$\phi(\lambda) = (\lambda+\theta)^\alpha - \theta^\alpha$$), the inverse density is more regular (“tempered aging”) at large times, and explicit formulas incorporate the tempering parameter (Kumar et al., 2014, Tang et al., 2022).

Gamma Subordinator:

• For gamma subordinators ($\phi(\lambda) = a \ln(1 + \lambda/b)$), the inverse density’s moments and Laplace transforms are expressed in terms of higher transcendental Volterra-type functions and the exponential integral (Colantoni et al., 2021).

Inverse Gaussian Subordinator:

• For the inverse Gaussian subordinator ($$\phi(\lambda) = \delta(\sqrt{2\lambda + \gamma^2} - \gamma)),$$ the right-continuous inverse (first-exit time) has explicit Laplace representations and satisfies fractional pseudo-differential equations (Vellaisamy et al., 2011).

Multivariate Inverse Subordinators:

• For componentwise inverses of multivariate subordinators, the governing equations become multidimensional with generalized fractional derivatives acting in multiple time variables, enabling modeling of anisotropic subdiffusions (Beghin et al., 2019).

5. Scaling Limits, Path Properties, and Long-Range Dependence

Aggregating many independent inverse subordinators—with subordinators having heavy-tailed (regularly varying) Lévy measures—yields scaling limits fundamental to network traffic, finance, physics, and renewal theory:

• Fractional Brownian motion arises as scaling limits in the fast-connection regime, with self-similarity and long-range dependence directly determined by the tail index of the Lévy measure (Kaj et al., 2012).
• In intermediate regimes, the normalized sum converges to a non-Gaussian, non-stable limit process (bridging fBm and $\alpha$-stable motion). This allows unified modeling of systems with both frequent small and rare large fluctuations (Kaj et al., 2012).

Sample paths of inverse subordinators and their functionals exhibit detailed regularity. For instance:

• Local Hölder exponents, iterated logarithm laws, and explicit limsup and liminf results are established for various convolved and integrated functionals, such as fractionally integrated inverse stable subordinators (FIISS) arising as limits of heavy-tailed renewal shot noise (Iksanov et al., 2016).
• Processes time-changed by inverse subordinators display sublinear law of large numbers scaling, explicit laws of the iterated logarithm, ergodic and mixing increment properties, and generalized martingale structures (Magdziarz et al., 2013).

6. Infinite Divisibility and Distributional Properties

A prominent and sometimes counterintuitive fact: while classic subordinators’ distributions are infinitely divisible (ID), their inverses—except for degenerate or trivial cases—almost never are (Kumar et al., 2018, Vellaisamy et al., 2011). This is rigorously established using asymptotic bounds on the tails. For example, the tails of inverse stable, tempered stable, and inverse Gaussian subordinators decay fast enough that —log-tail probabilities outpace $x\log x$, violating Steutel’s necessary condition for ID (Kumar et al., 2018): $$-\log \mathbb{P}(E(t) > x) \sim d x^{1/(1-\alpha)} \to \infty \text{ faster than } x\log x, \; x \to \infty.$$ Consequently, time-changed processes such as the fractional Poisson process (Poisson process subordinated by an inverse stable subordinator) are also not infinitely divisible (Meerschaert et al., 2010, Kumar et al., 2018).

7. Simulation, Numerical Methods, and Applications

Sampling and Monte Carlo:

Explicit "practically exact" simulation algorithms for inverse subordinators and their associated age/remaining lifetime processes are constructed based on their Markovian extension in an augmented state space (Biočić et al., 20 Dec 2024). The probability transition structures are deduced from the Lévy measure, enabling recursive sampling at finite collections of time points with finite expected computational cost per time step. For time-changed diffusions (e.g., SDEs),

• Monte Carlo estimators for functionals of $Y_t = M_{L_t}$ (with $M$ a Feller or Itô process) converge as $N \to \infty$ with a central limit theorem and Berry-Esseen estimates. For diffusion approximations (Euler–Maruyama), explicit strong error bounds as a function of discretization step $h$ are available:

$$\mathbb{E}\left[\sup_{s\in[0,t]} |Y_s - Y^h_s|^2\right] \leq \mathcal{C}_1 e^{\mathcal{C}_2 t} h^{\min\{2\gamma, 1\}}.$$

(Biočić et al., 20 Dec 2024)

Numerical Methods for Inverse Subordinator-Driven PDEs:

The nonlocal-in-time operators in fractional and generalized Fokker-Planck equations are efficiently discretized using schemes based on convolution quadrature (particularly backward Euler convolution weights derived from the complete Bernstein function of the subordinator), yielding maximal $L^p$ regularity and optimal-order convergence (Tang et al., 2022).

Application Domains:

• Anomalous diffusion and fractional kinetics: inverse subordinators generate waiting times capturing trapping and memory effects, with governing equations in the form of fractional PDEs (0705.0168, Meerschaert et al., 2010, Magdziarz et al., 2013, Iksanov et al., 2016).
• Semi-Markov processes, finance (option pricing with subordination), network traffic models, hydrology, and survival analysis (Meerschaert et al., 2010, Kaj et al., 2012, Beghin et al., 2019, Ascione et al., 2023).
• Stochastic integration, perpetuities, remainder/random time variable functionals, and scaling limit theorems (Hirsch et al., 2013).

Conclusion

Inverse subordinators fundamentally link the theory of Lévy processes, time-fractional and generalized nonlocal evolution equations, and nuanced probabilistic models displaying subdiffusion, heavy-tailed waiting, and memory effects. They possess rich regularity, fine-grained asymptotic, and sample path properties. The critical role of the Bernstein function and its analytic structure enables a unified theory of convolution-type fractional operators, governing equations, explicit simulation, scaling limits, and numerical methods. Applications pervade physics, stochastic modeling, finance, hydrology, and beyond, where anomalous temporal randomization is essential for accurate modeling of empirically observed dynamics.