Mixed Fractional Brownian Noise

• Mixed fractional Brownian noise is defined as the increment process of a multi-component fBm with distinct Hurst parameters, generalizing classical white noise and fractional Gaussian noise.
• Its covariance and spectral properties exhibit power-law scaling and long-range dependence, where the smallest Hurst exponent governs local regularity and p-variation.
• The rough path framework and Malliavin–Skorohod techniques enable precise statistical inference and simulation in models with multi-scale, non-Markovian dynamics.

Mixed fractional Brownian noise refers to the stochastic noise generated by increments of mixed (or multi-component) fractional Brownian motion (MFBM)—a Gaussian process formed as a finite or infinite superposition of independent fractional Brownian motions (fBm) with distinct Hurst exponents and weights, and potentially including classical Brownian motion components. It generalizes both classical white noise and fractional Gaussian noise, producing increment processes with richer covariance structure, regularity phenomena, and long-memory characteristics. Mixed fractional Brownian noise is central in modern stochastic analysis, rough path theory, modeling with colored noise, and statistical inference of systems exhibiting multi-scale temporal correlations.

1. Formal Definition and Noise Structure

Given $N \geq 1$ components, Hurst parameters $H = (H_1,\dots,H_N) \in (0,1)^N$, and nonzero weights $a = (a_1,\dots,a_N) \in \mathbb{R}^N$, generalized mixed fractional Brownian motion (GMFBM) is defined as

$$M_t^H(a) = \sum_{k=1}^N a_k B_t^{H_k}, \quad t\geq 0,$$

where each $B^{H_k}$ is an independent fractional Brownian motion with Hurst index $H_k \in (0,1)$. The associated mixed fractional Brownian noise is the increment process,

$$\Delta M_t^H(a) = M_{t+\Delta t}^H(a) - M_t^H(a) = \sum_{k=1}^N a_k \left( B_{t+\Delta t}^{H_k} - B_t^{H_k} \right).$$

In the infinite-component (“multi-mixed”) case, one may specify a finite measure $\mu$ on $(0,1)$ and take

$$X_t = \int_{H\in(0,1)} B_t^H\,\mu(dH) = \sum_{k=1}^\infty \sigma_k B_t^{H_k}$$

for a sequence $(H_k, \sigma_k)_{k\ge 1}$, yielding “multi-mixed fractional Brownian noise” (Almani et al., 2021).

2. Covariance and Spectral Properties

The covariance structure is determined by the independence of fBm increments and their characteristic covariance: $$\begin{align*} \mathbb{E}[\Delta M_s^H(a) \Delta M_t^H(a)] &= \sum_{k=1}^N a_k^2\, \mathbb{E}[(B^{H_k}_{s+\Delta t} - B^{H_k}_s)(B^{H_k}_{t+\Delta t} - B^{H_k}_t)] \ &= \frac{1}{2}\sum_{k=1}^N a_k^2 \Big( |s+\Delta t - (t+\Delta t)|^{2H_k} + |s-t|^{2H_k} \ &\qquad - |s+\Delta t - t|^{2H_k} - |s - (t+\Delta t)|^{2H_k} \Big). \end{align*}$$ For single increments, the variance becomes

$$\operatorname{Var}[\Delta M_t^H(a)] = \sum_{k=1}^N a_k^2 |\Delta t|^{2H_k}.$$

For infinite mixtures, the spectral density of the increment noise is

$$S(\omega) = \sum_{k=1}^\infty \sigma_k^2\, S_{H_k}(\omega), \qquad S_{H_k}(\omega) \sim | \omega |^{-(2H_k + 1) } \text{ as } \omega \to 0,$$

generalizing white and colored Gaussian noise (Almani et al., 2021, Cottone et al., 2013).

3. Pathwise Regularity and Long-Range Dependence

The local regularity of mixed fractional Brownian noise is governed by the smallest Hurst exponent:

• Hölder regularity: $M_t^H(a)$ is locally $C^{\gamma}$ for all $\gamma < \min_k H_k$; thus, the noise increments exhibit $O(|t-s|^{\min_k H_k})$ fluctuations (Lechiheb, 24 Nov 2025, Almani et al., 2021).
• $p$-Variation: For $p > 1/\min_k H_k$, the canonical rough-path lift of $M_t^H(a)$ exists, and $p$-variation is finite (Lechiheb, 24 Nov 2025). Equidistant $p$-variation of infinite mixtures displays a phase transition at $p = 1/\inf_k H_k$ (Almani et al., 2021).
• Long-range dependence: The auto-covariance of increments exhibits slow, power-law decay: for any $k$ with $H_k > 1/2$, $$E[(\Delta M_t^H(a))(\Delta M_{t+n}^H(a))] \sim n^{2H_k - 2}$$. If at least one $H_k > 1/2$, the process has genuine long-range dependence (Mliki et al., 2021, Almani et al., 2021).

4. Geometric Rough Path Structure

For $\min_k H_k > 1/4$, the theory of geometric rough paths applies to $M_t^H(a)$. The canonical rough path $\mathbb{M} = (1, M, \mathbb{M}^{(2)})$ over $M^H(a)$ is constructed as the almost sure $p$-variation limit of piecewise-linear dyadic approximations: $$M^m_t = M^H_{t_{\ell-1}^m} + 2^m\,(t-t_{\ell-1}^m)(M_{t_\ell^m}^H - M_{t_{\ell-1}^m}^H), \;\; t \in [t_{\ell-1}^m, t_\ell^m],$$ and the Lévy area

$$\mathbb{M}^{(2)}_{s,t} = \int_s^t (M_u^m - M_s^m) \otimes dM_u^m.$$

For any $p > 1/\min_k H_k$, convergence holds:

• $\sup_m \|M^m\|_{p\text{-var}} < \infty$,
• $$\sup_m \|\mathbb{M}^{m,(2)}\|_{p/2\text{-var}} < \infty$$, leading to a canonical geometric $p$-rough path (Lechiheb, 24 Nov 2025).

5. Malliavin–Skorohod Representation and Lévy Area

The second-level iterated integrals (Lévy area) of the rough path above GMFBM admit an explicit Skorohod integral representation: $$\begin{aligned} \mathbb{M}_{s,t}^{(2),i,j} = &\;\tfrac{1}{2}M_{s,t}^{(1),i} M_{s,t}^{(1),j} + \tfrac{1}{2}\sum_{k=1}^N a_k^2\,I_{2}^{H_k}(\mathbf{1}_{[s,t]}^{\otimes 2})\,\delta_{ij} \ &+ \sum_{1\le k\ne \ell\le N} a_k a_\ell \int_s^t (B_u^{H_k,i} - B_s^{H_k,i})\,\delta B_u^{H_\ell,j}, \end{aligned}$$ where $I_{2}^{H_k}$ is the second Wiener-Itô integral in the $k$th chaos and the cross-term is a Skorohod integral, well-defined whenever $H_k + H_\ell > 1/2$ (Lechiheb, 24 Nov 2025).

6. Algebraic Signature and Universal Properties

The full rough path signature of $M^H(a)$ is the tensor series

$$S(\mathbb{M})_{s,t} = \sum_{n=0}^\infty \int_{s<u_1<\cdots<u_n<t} d\mathbb{M}_{u_1} \otimes \cdots \otimes d\mathbb{M}_{u_n}.$$

For $p > 1/\min_k H_k$, the signature converges, satisfies Chen's identity ($S_{s,t} = S_{s,u} \otimes S_{u,t}$), and uniquely characterizes the path up to tree-like equivalence. Linear functionals on truncated signatures are universal approximators for continuous functionals of the underlying path, connecting to machine learning and SDE statistics (Lechiheb, 24 Nov 2025).

7. Applications and Statistical Methods

Mixed fractional Brownian noise underpins modern models of complex stochastic dynamics and observations corrupted by multi-scale, non-Markovian noise:

• Stochastic Differential Equations: Analysis and numerical methods for SDEs and SPDEs with mixed noise use the rough path lift and Malliavin calculus detailed above (Lechiheb, 24 Nov 2025, Inahama et al., 2023).
• Averaging and Large Deviations: In slow–fast SDEs with mixed noise, the averaging principle and large deviation theory rely crucially on the combined Itô/fractional integral calculus. Weak convergence approaches and variational representations (Boué–Dupuis) exploit the underlying Cameron–Martin structure, which is stricter for memory components $H > 1/2$ (Inahama et al., 2023).
• Statistical Testing: Inference for mixed fBm and noise models (e.g., fBm plus white noise) is often based on sample autocovariance statistics, with test statistics deriving generalized chi-squared laws, allowing noise-robust and explicit model validation (Balcerek et al., 2019).
• Simulation: Arbitrary stationary colored Gaussian noise may be simulated as finite superpositions of fBm with coefficients inferred from the fractional moments of a target spectral density, providing an explicit noise synthesis framework (Cottone et al., 2013).
• Non-Markovianity and Semimartingale Property: With more than one component or $H \ne 1/2$, mixed fBm is non-Markovian, and is not a semimartingale except for specific index values (e.g., $H = 1/2$ or $H > 3/4$ for specific mixtures) (Chigansky et al., 27 Nov 2025, Mliki et al., 2021).

8. Variants and Extensions

• Mixed Generalized fBm: Encompasses mixtures of standard Brownian motion, “forward” and “backward” fBm, covering a larger family of non-self-similar and non-stationary increment processes (Mliki et al., 2021).
• Completely Correlated Mixtures: Using Molchan–Golosov representations driven by a single Brownian motion, “completely correlated” mixed fBm exhibits short-time Brownian regularity and long-time fBm correlation structure, supporting transfer principles, Girsanov transformations, and optimal prediction (Dufitinema et al., 2021).
• Infinite Mixtures (“multi-mixed”): Infinite superpositions enable modeling of noise with arbitrary regularity and memory characteristics; path properties, covariance structure, long-range dependence, $p$-variation, and full support are inherited from the extremal Hurst exponents of the mixture (Almani et al., 2021).

---

References: Core technical developments, rigorous pathwise theory, and statistical methods as detailed in (Lechiheb, 24 Nov 2025, Chigansky et al., 27 Nov 2025, Inahama et al., 2023, Mliki et al., 2021, Almani et al., 2021, Dufitinema et al., 2021, Balcerek et al., 2019), and (Cottone et al., 2013).