Fractionally Integrated fBm Noise

• Fractionally integrated fractional Brownian noise is a Gaussian process created by applying a non-integer order integral operator to fBm, producing self-similarity and long-range memory effects.
• It features persistence exponents and duality relations that quantify the asymptotic persistence of the process, challenging earlier conjectures such as e(2,H) = H(1-H).
• Its analysis leverages spectral methods, path-integral formulations, and tools like the Lamperti transform and generalized Slepian's lemma to connect covariance structure and sample path regularity.

Fractionally integrated fractional Brownian noise refers to a class of real-valued Gaussian processes constructed by applying a fractional (possibly non-integer order) integral operator to fractional Brownian motion (fBm) or its increments (fractional Gaussian noise, fGn). These processes interlace the long-range dependence and self-similarity properties of fBm with further smoothing and memory effects stemming from the degree of integration. The main technical parameterization involves the integration order $a > 0$ and the Hurst parameter $H \in (0,1)$ of the underlying fBm. Such processes arise naturally in stochastic analysis, physical modeling of memory-driven systems, and the paper of persistence or crossing probabilities for non-Markovian dynamics.

1. Mathematical Definition and Properties

Fractionally integrated fractional Brownian noise is defined by

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

where $w_H$ denotes fractional Brownian motion (fBm) with Hurst parameter $H$, and $a>0$ is the (possibly non-integer) fractional integration order (Molchan, 12 Sep 2025). The process $I_{a,H}(t)$ generalizes multiple integrals of fBm to arbitrary order. The resulting process is Gaussian, exhibits self-similarity with index $K = a + H - 1$, and its sample paths are almost surely Hölder continuous of order $< \min\{H, a\}$.

Key characteristics:

• Self-similarity: $$I_{a,H}(\lambda t) \stackrel{d}{=} \lambda^K I_{a,H}(t)$$ for $\lambda > 0$, with $K = a + H - 1$.
• Long-range dependence: In most cases (notably for $a + H > 1$), the process exhibits strong persistence, governed by the interplay between tail index of the fBm and the smoothing effect of integration.
• Covariance structure: The covariance of $I_{a,H}$ tracks both the memory in fBm (via $H$) and additional smoothing from the integral kernel, with asymptotics scaling as $|t-s|^{2K}$ and explicit spectral density given for high frequencies.
• Increment process: For $a = 1$, $I_{1,H}(t)$ is the (classical) integral of fBm. For $a$ non-integer, it is a genuine fractional integration in the sense of Riemann–Liouville.

2. Persistence Exponents and Power-Law Crossing Probabilities

A major focus of current research is understanding the persistence probabilities:

$$\mathbb{P}(I_{a,H}(t) < c,\,\forall t \in [0,T]) \sim T^{-e(a,H)} \quad \text{as}~ T \to \infty$$

where $e(a,H)$ is the persistence exponent (Molchan, 12 Sep 2025). The value of $e(a,H)$ quantifies the asymptotic likelihood that the process remains below a fixed threshold for extended intervals. Notable results:

• For $a=1$, $e(1,H) = 1-H$.
• For $a=2$, $H=\frac{1}{2}$, $e(2,1/2) = 1/4$.
• The exponent is strictly decreasing in $a$ and depends nontrivially on both $a$ and $H$.
• There is a nontrivial duality: $e(a,H) = e(a+2H-1,1-H)$, refuting the prior conjecture $e(2,H) = H(1-H)$.

This duality arises from symmetry arguments involving the Lamperti transform and extended versions of Slepian's comparison lemma, with continuity properties established via Gaussian process theory.

3. Integral Representations and Path Integral Formulation

Fractionally integrated fBm is naturally represented using Riemann–Liouville fractional integrals, which underpin both probabilistic representations and rigorous functional integration formulations. For subdiffusive ($H<1/2$) and superdiffusive ($H>1/2$) regimes, path-integral representations unify apparently different fractional processes (Benichou et al., 2023):

• All fBm variants (Lévy, Mandelbrot–van Ness one-sided, two-sided) may be written as fractional integrals of white noise, differing only by the domain of integration.
• Path integral actions for fBm and fractionally integrated noise can be expressed succinctly via nonlocal quadratic forms involving the same fractional kernel of order determined by $H$, with only the integration domain changing between variants.
• Covariance functions and actions (in the path-integral sense) are rooted in the same operator framework, clarifying that these processes—though superficially distinct—belong to one mathematical family.

For example, the action functional for Lévy fBm (finite interval) is

$$S[x] = \frac{1}{2} \int_0^T \left\{ \frac{d}{dT}\left[ I_t^{1/2-H} x(T) \right] \right\}^2 dT,$$

and analogous representations hold for MvN constructions, with explicit Riemann–Liouville kernels.

4. Spectral Properties, Memory, and Covariance Analysis

Spectral analysis reveals that fractionally integrated fBm processes possess a spectral density

$$f_{a,H}(\lambda) \sim C_{a,H} |\lambda| (1 + o(1)) \quad \text{as}~ |\lambda| \to \infty$$

with explicit constants and decay rates depending on $a$ and $H$ (Molchan, 12 Sep 2025). This structure determines sample path regularity and the strength of temporal correlations. Covariance analysis demonstrates:

• For $K=a+H-1 > 0$, sample paths are in Hölder space of order $K$ almost surely.
• The covariance structure for the process's increments conforms to that of long-memory Gaussian sequences, with explicit asymptotics available.
• Persistence and upper/lower bounds for exponents are established using spectral and covariance-theoretic methods.

5. Theoretical Methodology and Analytical Tools

Analysis of fractionally integrated fractional Brownian noise leans on a suite of Gaussian process techniques:

• Continuity lemma for persistence: Guarantees that persistence exponents remain stable under smooth changes in the parameters $a$ and $H$.
• Generalized Slepian’s lemma: Extended to parameterized families of Gaussian processes, allowing comparison of persistence probabilities and establishing monotonicity or duality relations.
• Lamperti transform: Connects self-similar processes to stationary ones, uncovering hidden symmetries in persistence behavior.
• Spectral theory and RKHS: Enables detailed understanding of sample path properties and supports functional limit theorems.

6. Applications, Controversies, and Open Problems

Fractionally integrated fractional Brownian noise provides a tractable model in fields requiring quantification of persistence, roughness, or long-range dependence, including hydrology, turbulence, mathematical finance, and stochastic signal analysis. Its precise persistence exponents are pivotal in characterizing rare events, metastable periods, and threshold crossings. Notably, the identity $e(a, H) = e(a + 2H - 1, 1 - H)$ refutes longstanding conjectures (e.g., $e(2,H) = H(1-H)$), reshaping theoretical understanding of such processes and influencing their interpretation in applications (Molchan, 12 Sep 2025).

Significant open problems include deriving explicit formulas for $e(a,H)$ for general $(a,H)$, connecting persistence to multifractal properties of sample paths, and developing simulation schemes that exploit the operator-theoretic and path-integral representations for scalable modeling.

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Summary Table: Persistence Exponents for Fractionally Integrated fBM

Parameter Case	Known Exponent	Duality/Identity
$a=1$, $H$ arbitrary	$e(1,H) = 1-H$	$e(1,H) = e(1+2H-1,1-H)$
$a=2$, $H=1/2$	$e(2,1/2) = 1/4$	$e(2,1/2) = e(2,1/2)$
$a=2$, $H$ arbitrary	NOT $H(1-H)$	$e(2,H) = e(2 + 2H - 1, 1-H)$
General $(a,H)$, $a+H>1$	Strictly decreasing	$e(a,H) = e(a + 2H - 1, 1-H)$

The structure and properties of these processes, as revealed by recent research, underpin the modern theory and applications of fractional Gaussian models with integration and memory effects.