Pearson Diffusion Processes

• Pearson diffusion processes are one-dimensional SDEs defined by a linear drift and quadratic variance function, ensuring invariant Pearson distributions.
• They comprise distinct classes like Ornstein–Uhlenbeck, CIR, and Jacobi, each yielding classical invariant laws such as Gaussian, Gamma, and Beta.
• Recent fractional and non-local extensions capture memory and long-range dependence, enhancing applications in diverse fields from finance to population genetics.

A Pearson diffusion process is a class of one-dimensional stochastic differential equations (SDEs) characterized by a linear drift function and a quadratic variance (squared diffusion) function, constructed such that the invariant (steady-state) distribution lies within the Pearson system of distributions, which encompasses the normal, gamma, beta, Student, Fisher–Snedecor, and other classical probability laws. The mathematical tractability of the Pearson system, in combination with its ability to model skewness, kurtosis, and heavy tails, underlies the system’s foundational role in ergodic diffusions, statistical physics, mathematical finance, population genetics, and applied probability.

1. Mathematical Definition and Classes of Pearson Diffusion Processes

Pearson diffusions are defined as solutions to SDEs of the form: $dX_t = \mu(X_t) dt + \sigma(X_t) dW_t$ where the drift $\mu(x)$ is a first-degree polynomial and the variance $\sigma^2(x)$ is a polynomial of degree at most two. The canonical formulation (centered at the long-term mean $\mu$) is: $dX_t = -\theta(X_t - \mu) dt + \sigma(X_t) dW_t,$ where $\theta > 0$ is the mean-reversion speed and $W_t$ is standard Brownian motion (0711.1789).

Distinct choices of $\sigma(x)$ correspond to the different sub-families within the Pearson class:

• Ornstein–Uhlenbeck: $\sigma(x)\equiv$ constant; invariant law is Gaussian.
• Cox–Ingersoll–Ross (CIR)/Feller: $\sigma(x)\propto \sqrt{x}$; invariant law is Gamma.
• Jacobi/Wright–Fisher: $\sigma^2(x)\propto x(1-x)$; invariant law is Beta.
• Pearson IV/Type IV: $\sigma(x)=\sqrt{2\theta a(x+1)}$; invariant law is heavy-tailed, involving the arctangent function (0711.1789, Monthus, 2023).

The invariant density $f(x)$ can typically be written in closed form using the scale and speed measures: $$s(x) = \exp\left(-2\int_{x_0}^x \frac{b(y)}{\sigma^2(y)} dy\right), \qquad m(x) = \frac{1}{\sigma^2(x) s(x)}$$ and

$f(x) = \frac{m(x)}{G}, \quad G = \int m(x) dx$

with $b(y)$ the drift polynomial (0711.1789).

2. Invariant Measures and Information-Theoretic Quantities

Once the invariant density $f(x)$ is established, it is possible to compute explicit forms of various information-theoretic measures:

• Rényi entropy of order $\alpha>0$:

$$R_\alpha(f) = \frac{1}{1-\alpha} \log \int f(x)^\alpha dx$$

For the Gaussian case (Ornstein–Uhlenbeck), $$R_\alpha(f) = \frac{1}{2}\log(2\pi)-\frac{1}{2(1-\alpha)}$$.

• Shannon entropy ($\alpha \to 1$ limit):

$R_1(f) = -\int f(x)\log f(x) dx$

For the CIR (Gamma) case: $R_1(f) = \log \Gamma(\mu) - (\mu-1)\psi(\mu) + \mu$ with $\psi(\cdot)$ the digamma function.

• Song measure (curvature of the Rényi spectrum at $\alpha=1$):

$$S(f) = -2 \left.\frac{\partial}{\partial \alpha} R_\alpha(f)\right|_{\alpha=1}$$

$S(f)$ equals the variance of $\log f(X)$ and is a location- and scale-invariant summary of the distribution’s shape—sensitive to tail behavior and higher moments even when some (such as the fourth moment) do not exist (0711.1789).

For more complex Pearson laws (e.g., Type IV), closed forms are available for specific $\alpha$ using classical integral formulas (e.g., via cosine integrals over $( -\pi/2, \pi/2 )$), and limits computed using analytic techniques such as L’Hôpital’s Rule (0711.1789).

3. Spectral Structure and Polynomial Eigenfunctions

The generator $\mathcal{A}$ of a Pearson diffusion acts on sufficiently smooth test functions and, owing to the polynomial structure of drift and diffusion, admits a complete system of polynomial eigenfunctions: $\mathcal{A} Q_n(x) = -\lambda_n Q_n(x)$ where $Q_n$ are appropriately orthogonormal polynomials (e.g., Hermite for Ornstein–Uhlenbeck, Laguerre for CIR, Jacobi for Beta/Wright–Fisher), and $\lambda_n$ is the $n$th nontrivial eigenvalue.

This spectral decomposition yields explicit formulas for transition densities, moments, and relaxation times: $$p(t,x;y) = m(x)\sum_n e^{-\lambda_n t} Q_n(x) Q_n(y)$$ The relaxation rate (spectral gap $\lambda_1$) governs exponential decay to equilibrium (Monthus, 2023, Jafarizadeh, 30 Dec 2024). For Pearson diffusions with quadratic variance and linear drift, the eigenfunctions and eigenvalues are tractable and expressible via hypergeometric functions (Jafarizadeh, 30 Dec 2024).

4. Fractional and Non-Local Extensions

Time-fractional Pearson diffusions generalize standard (Markovian) Pearson processes by replacing the integer-order time derivative, in the Kolmogorov forward or backward equations, with a non-local (fractional or distributed order) derivative: $\partial_t^\Phi u(t,x) = \mathcal{G} u(t,x)$ where $\partial_t^\Phi$ denotes a Caputo or convolution derivative associated with a Bernstein function $\Phi$ (Ascione et al., 2020, Ascione et al., 2020, Mijena et al., 2014). Solutions admit spectral expansions with temporal weights given by generalized Mittag–Leffler or Kilbas–Saigo functions (Beghin et al., 11 May 2025).

Time-change representations are central: letting $E_\Phi(t)$ be the inverse subordinator associated with $\Phi$, solutions can be represented as $X(E_\Phi(t))$ where $X$ is a standard Pearson diffusion (Mijena et al., 2014, Leonenko et al., 2017). In stretched non-local Pearson diffusions, the evolution is governed by a “stretched” Caputo derivative, and the time-dependent part is described by the Kilbas–Saigo function, whose Mellin–Barnes representation and asymptotics were newly derived (Beghin et al., 11 May 2025).

Key consequences include the emergence of long-range dependence: while classical Pearson diffusions have exponentially decaying correlations, the time-changed (fractional) versions exhibit power-law temporal decay, with the smallest order of the fractional derivative governing the tail index of the autocorrelation function (Mijena et al., 2014).

5. Large Deviations, Optimality, and Statistical Implications

Pearson diffusions admit explicit large deviation analyses for both time-averaged observables and the empirical (occupation) density:

• The scaled cumulant generating function for an observable $w(x)$ is the dominant eigenvalue $E_0(p)$ of the generator deformed by $-p w(x)$.
• The Level–2 large deviation rate function for the empirical density is

$$I_2[p(\cdot)] = \int \frac{[p(x)F(x) - D(x) p'(x)]^2}{4 D(x) p(x)} dx$$

which enables explicit computation of inference rates for parameters, giving quantifiable uncertainty for parameter recovery from long-run data (Monthus, 2023).

From an optimization perspective, Pearson processes of the Hypergeometric type maximize convergence rate (minimize relaxation time) for a given stationary law and fixed average variance. The optimal drift is always linear, the optimal relaxation rate $\lambda_1$ is the average variance divided by twice the stationary variance, and this is uniquely achieved with Pearson structure (Jafarizadeh, 30 Dec 2024).

6. Applications Across Mathematical and Applied Sciences

Pearson diffusions are prominent in:

• Financial mathematics: CIR (for interest rate or volatility), heavy-tailed Pearson Type IV (for returns), providing flexible models for option pricing that accurately reproduce skewness and kurtosis, with empirical evidence for superior pricing accuracy over Black–Scholes and Heston models (Kar et al., 20 Aug 2025).
• Population genetics: the Wright–Fisher (Beta) and related processes describe the stochastic evolution of gene frequencies, dual to coalescent trees, and possess spectral representations in Jacobi polynomials (Griffiths et al., 2010).
• Anomalous transport: fractional Pearson diffusions capture memory and long-range dependence, modeling subdiffusive and “bursty” phenomena in hydrology and physics (Mijena et al., 2014, Leonenko et al., 2017, Beghin et al., 11 May 2025).
• Statistical inference: the tractable explicit formulas for distances (e.g., power divergences, Hellinger integrals) between diffusions underpin hypothesis testing, Bayesian risk bounds, and parameter distinguishability (Kammerer et al., 2010).

7. Extensions, Open Problems, and Technical Innovations

Innovations include the use of stretched fractional derivatives (yielding Kilbas–Saigo temporal weights and generalized hyperbolic telegraph equations) (Beghin et al., 11 May 2025), explicit representation of strong solutions for non-local (memory-laden) Kolmogorov equations (Ascione et al., 2020, Ascione et al., 2020), detailed spectral classification for heavy-tailed cases (fractional Fisher–Snedecor and reciprocal gamma with split discrete/continuous spectra) (Leonenko et al., 2017), and operator-theoretic approaches to analyticity and regularity (Ascione et al., 2020).

Novel mathematical representations—Mellin–Barnes type for special functions, asymptotics for complex arguments—advance the toolkit for analyzing fractional diffusions. Empirical and analytical work demonstrates that despite non-Markovian evolution, long-time limiting distributions and invariant measures are preserved under non-local time deformation.

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Summary Table of Key Pearson Diffusion Cases:

Process	Squared Diffusion $\sigma^2(x)$	Invariant Density	Eigenbasis
Ornstein–Uhlenbeck	Constant	Normal	Hermite polynomials
Cox–Ingersoll–Ross (CIR)	$x$	Gamma	Laguerre polynomials
Jacobi/WF	$x(1-x)$	Beta	Jacobi polynomials
Pearson IV	$a(x+1)$	Pearson Type IV (HT)	Generalized polynomials/special fn.

HT: Heavy-tailed

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Pearson diffusion processes exhibit a spectrum of analytical tractability, spectral decomposition, and applicability across mathematical fields, underpinned by closed-form formulas for invariant measures, entropy and information, correlation structure, optimal convergence, and explicit representations for both local and non-local (fractional, stretched) dynamics. Their robustness under time changes and generalizations, as demonstrated by the explicit connection to fractional calculus and stochastic time-changes, renders them a foundational model for both probabilistic theory and real-world stochastic modeling.