Persistence Exponent Analysis

• Persistence exponent is a critical parameter that defines the asymptotic decay probability for a process remaining below a threshold, highlighting key universality in stochastic dynamics.
• Selective averaging, Lamperti transform, and spectral analysis are central methodologies used to rigorously compute persistence exponents in diffusion, Lévy, and Gaussian processes.
• Applications include analyzing first-passage problems, zero-crossing statistics, and complex system behavior, challenging classical hypotheses with refined spectral and process-dependent insights.

The persistence exponent is a critical parameter characterizing the long-time asymptotics of the probability that a stochastic process or field maintains a fixed sign or remains below a threshold, often reflecting deep universality and nontrivial dynamical correlations in both physical and mathematical models. Its definition, calculation, universality, and implications span a broad class of Gaussian and non-Gaussian processes, random walks, Lévy processes, self-interacting systems, and stochastic PDEs. The salient features and progress in this field are outlined in the following sections, integrating rigorous results and principal methodologies.

1. Definition, Universality, and Mathematical Formulation

The persistence exponent, typically denoted θ (alternatively e or θ₀ in specific contexts), is formally defined via the asymptotic relation

$$\mathbb{P} \big( X_t < c, \ \forall t \in [0, T] \big) \sim T^{-\theta}$$

as $T \to \infty$, where $X_t$ is a real-valued stochastic process or field, and $c$ is a fixed threshold (usually taken as zero or one). For discrete settings, an analogous definition holds over lattice times or indices. In higher dimensions, for random fields X(x, t), the exponent may characterize the decay of the probability that the field at a fixed point or over a region does not cross zero up to time t.

The persistence exponent is highly universal for broad classes of processes: for example, symmetric random walks and their integrals, Lévy processes, and their fractional and higher-order integrated analogues. For classical random walks, θ = 1/2; for the integrated process $A_n = S_1 + \cdots + S_n$, θ = 1/4 (Sinai’s result) (Aurzada et al., 2012). For Gaussian stationary processes, θ can depend on fine spectral properties, but universality still arises under summable correlation decay (Aurzada et al., 2020).

In random fields, the scaling forms generalize as

$$-\ln \mathbb{P}\left(X(t) < 1, t \in [0, T]^d\right) \sim e \cdot (\ln T)^d$$

with the persistence exponent e depending on the dimension d, self-similarity indices, and, for fields connected via Lamperti transform, the limit for the associated stationary process (Molchan, 2021).

2. Exact Results for Canonical Processes

Diffusion Equation: For the simple diffusion equation with random Gaussian initial condition, the exact persistence exponent is derived by selective averaging: $$\theta_0 = \frac{d}{4} \qquad \text{for}\ d \leq 4 \qquad \theta_0 = 1\ \text{otherwise}$$ where d is the spatial dimension. This result stands in contrast to previously accepted values from the independent interval approximation (IIA) and implies, e.g., θ₀ = 0.25 (1D), 0.5 (2D), 0.75 (3D), saturating for d > 4 (Sanyal, 2010).

Integrated Lévy Processes: For the integral $X_t = \int_0^t L_s ds$ of a strictly stable Lévy process L (with index α and positivity parameter ρ), rigorous analysis yields

$\theta = \frac{\rho}{1 + \alpha(1-\rho)}$

recovering, for integrated Brownian motion (α=2, ρ=1/2), θ = 1/4 (Profeta et al., 2014).

Fractional and Higher Integration: For fractionally integrated FBM,

$I_{a,H}(t) = \int_0^t (t-x)^{a-1}\ dW_H(x)$

where W_H is FBM with Hurst parameter H, key results include:

• For (a=1, H), $e(1,H) = 1-H$
• For (a=2, H=1/2), $e(2,1/2) = 1/4$
• For general (a, H) in $a + H > 1$, the persistence exponent $e(a, H)$ is strictly decreasing in a; most importantly, the widely conjectured formula $e(2, H) = H(1-H)$ is rigorously refuted. Instead, $e(a, H) = e(a+2H-1, 1-H)$ (Molchan, 12 Sep 2025), showing a nontrivial spectral symmetry and parameter dependence.

Brownian Scenery and Non-Gaussian Fields: For processes of the form $A_t = \int_{\mathbb{R}} L_t(x)\ dW(x)$, with L_t the local time of a self-similar Y, the persistence exponent is robust and universal: $$\mathbb{P} \left[ \sup_{0 \leq t \leq T} A_t \leq 1 \right] \sim T^{-1/2}(\log T)^{c}$$ for some explicit c, for a broad class of Y (Castell et al., 2014).

3. Methodologies: Selective Averaging, Spectral and Continuity Arguments

Numerous methods are employed in rigorous determination of persistence exponents:

• Selective Averaging: For the diffusion equation, fixing the field value at a point and averaging over the rest isolates initial-condition induced stochasticity (Sanyal, 2010).
• Lamperti Transform: Converts a self-similar process to stationary Gaussian, enabling the paper of persistence exponents as the exponent (or capacity) for the stationary process over expanding intervals or domains (Aurzada et al., 2012, Molchan, 2021).
• Spectral Methods and Comparison Lemmas: Explicit spectral representations for covariance kernels, with Gamma- and hypergeometric functions, allow precise control and monotonicity analysis. Generalizations of Slepian’s lemma enable rigorous monotonicity statements for exponents as parameters (e.g., a, H) vary (Molchan, 12 Sep 2025).
• Operator-Theoretic Approaches: For Markov processes and processes with tractable structure (such as MA or AR processes), the exponent is shown to be the spectral radius of a concrete integral operator, allowing for perturbative and eigenvalue-based expansions (Aurzada et al., 2017, Aurzada et al., 9 Jul 2024).

4. Refutation of Standard Hypotheses and Universality Classes

A major outcome in the paper of fractionally integrated FBM is the disproof of the widely cited hypothesis $e(2, H) = H(1-H)$ for the double-integrated process. Instead, the spectral symmetry $e(a, H) = e(a+2H-1, 1-H)$ and strict monotonicity in integration order have been established, using continuity and comparison principles for parameter-dependent Gaussian processes (Molchan, 12 Sep 2025). This marks a departure from simple multiplicative conjectures and highlights the structural complexity of persistence exponents in long-range correlated systems.

Similar universality breakdowns occur in the presence of disorder or memory: for instance, persistence exponents become continuously varying (even complex, leading to log-periodic corrections) in disordered contact processes (Bhoyar et al., 2020), or depend on initial bias in the Ising chain, revealing nonuniversality of the exponent under initial condition variation (Shukla, 2019).

5. Summarized Examples of Persistence Exponents

Model	Persistence Exponent θ / e	Reference
Diffusion eq. in d dimensions	θ = d/4 (d≤4), θ=1 (d>4)	(Sanyal, 2010)
Integrated random walk (Sinaï)	θ = 1/4	(Aurzada et al., 2012)
Integrated α-stable Lévy process	θ = ρ/(1 + α(1−ρ))	(Profeta et al., 2014)
Fractionally integrated FBM, a=1	e(1,H) = 1–H	(Molchan, 12 Sep 2025)
Double-integrated BM, a=2, H=1/2	e(2,1/2) = 1/4	(Molchan, 12 Sep 2025)
Fractionally integrated FBM, a>0	e(a,H) strictly decreasing in a	(Molchan, 12 Sep 2025)
Brownian scenery	θ = 1/2 (log-corrections)	(Castell et al., 2014)
Random walk records (superior walks)	β ≈ 0.382258	(Ben-Naim et al., 2014)
SIRW: self-attracting (SATW₍φ₎)	θ = φ/2	(Brémont et al., 24 Oct 2024)
SIRW: polynomially self-repelling (PSRW₍γ₎)	θ = 1/4	(Brémont et al., 24 Oct 2024)

6. Structure, Spectral, and Process Dependence

Persistence exponents can depend sensitively on the structure of correlations:

• For positive, absolutely continuous spectral measures, θ > 0 if and only if the spectral density at zero is finite (Feldheim et al., 2021).
• For self-similar or stationary Gaussian processes, the Lamperti transform not only allows reduction to stationary problems but also yields explicit connections between the exponents for original and transformed fields (Molchan, 2021).
• For weighted sums or non-summable correlations, θ encodes both the decay of correlations (parameter H) and the polynomial growth rate of the weights (p) (Aurzada et al., 2020).

Continuity of θ in process parameters and definition under both continuous and discrete sampling is rigorously established, with monotonicity claims proved via parameter-differentiable Slepian-like comparison (Feldheim et al., 2021, Molchan, 12 Sep 2025).

7. Applications and Implications

Persistence exponents are pivotal for:

• Determining first-passage behavior, zero-crossing statistics, and rare-event survival in disordered systems, interface fluctuations, reaction-diffusion fronts, and random polynomials (Aurzada et al., 2012, Poplavskyi et al., 2018).
• Quantifying fractal or Hausdorff dimensions in hydrodynamic or turbulent flows (e.g., regular Lagrangian points in Burgers turbulence) (Molchan, 12 Sep 2025).
• Characterizing universal decay rates in Markov chains and ARMA processes, with perturbation theory and operator-theoretic analysis enabling explicit eigenvalue representations for the exponents (Aurzada et al., 2017, Aurzada et al., 9 Jul 2024).
• Analyzing the effect of memory and learning rules in non-Markovian self-interacting systems (SIRWs), with exact solution for the exponent in core universality classes (Brémont et al., 24 Oct 2024).

In summary, precise determination of persistence exponents necessitates an overview of spectral, probabilistic, and operator-theoretic techniques, revealing both unexpected universality and sensitive dependence on process structure, integration order, initial conditions, and underlying disorder. The persistence exponent remains a central nontrivial invariant encapsulating the interplay of memory, correlation, and fluctuation in complex stochastic systems.