Memory Multi-Fractional Brownian Motion

• Memory multi-fractional Brownian motion is a Gaussian process with a time-varying Hurst exponent that retains long-memory effects by encoding the entire history of past increments.
• Various formulations—including continuous-memory, Volterra-kernel, switching, and multi-mixed models—offer distinct mechanisms to model non-stationarity while preserving dependence on historical data.
• These approaches have practical implications in fields like superconducting-qubit decoherence and cellular tracking, where adaptive kernels and regime-switching capture complex noise dynamics.

Memory multi-fractional Brownian motion (mmfBm) denotes a family of Gaussian, generally non-stationary extensions of fractional Brownian motion in which a time-dependent Hurst or memory exponent is combined with an explicit mechanism that preserves dependence on past increments. In the current literature the designation is not fully standardized: it is used for continuous-memory Volterra processes with exponent $\alpha(t)$, for kernel-based models of the form $\mathcal M(t)=\int_0^t K(t,s)\,dB^{H(s)}(s)$, for switching constructions with piecewise-constant $H(t)$ and diffusivity $D(t)$, and, under the same acronym, for “multi-mixed fractional Brownian motion,” a superposition of independent fBms with different Hurst indices (Wang et al., 2023, Haq, 27 Jul 2025, Balcerek et al., 2023, Almani et al., 2021). What unifies these constructions is the attempt to go beyond constant-Hurst fBm and beyond multifractional schemes in which changing the local exponent effectively resets the memory structure.

1. Terminological scope and canonical constructions

The literature uses closely related but non-identical definitions of mmfBm. The common theme is that the roughness or memory index varies with time while long-memory effects are retained rather than reinitialized.

Formulation in the literature	Representative definition	Distinguishing feature
Continuous-memory MMFBM	$$X(t)=\int_0^t \sqrt{\alpha(s)}\,(t-s)^{(\alpha(s)-1)/2}\,dB(s)$$	Entire history of $\alpha(s)$ enters the kernel
Volterra-kernel mmfBm	$\mathcal M(t)=\int_0^t K(t,s)\,dB^{H(s)}(s)$	Separate causal memory kernel and local Hurst field
Switching-FBM / mmfBm	$$X(t)=\int_0^t \sqrt{2D(s)H(s)}\,(t-s)^{H(s)-1/2}\,dB(s)$$	Piecewise-constant or stochastic regime changes
Multi-mixed fBm	$M_t=\sum_{k=1}^\infty \sigma_k B_t^{H_k}$	Superposition over a spectrum of Hurst indices

In the continuous-memory formulation of Pagnini, Sposini, and Metzler, the exponent is written as $\alpha(t)\in(0,2]$ and the process is defined by a Riemann–Liouville-type Volterra–Itô integral; the defining feature is that the kernel uses the past values $\mathcal M(t)=\int_0^t K(t,s)\,dB^{H(s)}(s)$0 rather than the current value $\mathcal M(t)=\int_0^t K(t,s)\,dB^{H(s)}(s)$1 alone (Wang et al., 2023). In the Volterra-kernel formulation used for superconducting-qubit charge noise, the process is written as

$\mathcal M(t)=\int_0^t K(t,s)\,dB^{H(s)}(s)$2

where $\mathcal M(t)=\int_0^t K(t,s)\,dB^{H(s)}(s)$3 is smooth and $\mathcal M(t)=\int_0^t K(t,s)\,dB^{H(s)}(s)$4 is a causal kernel of Volterra type (Haq, 27 Jul 2025). In the switching-FBM model, both the local Hurst exponent $\mathcal M(t)=\int_0^t K(t,s)\,dB^{H(s)}(s)$5 and the generalized diffusivity $\mathcal M(t)=\int_0^t K(t,s)\,dB^{H(s)}(s)$6 jump between states, but the kernel still reaches over the full past interval $\mathcal M(t)=\int_0^t K(t,s)\,dB^{H(s)}(s)$7, so the process does not forget its pre-switch history (Balcerek et al., 2023). By contrast, the “multi-mixed fractional Brownian motion” of Almani and Sottinen is not a time-varying-exponent Volterra process at all; it is

$\mathcal M(t)=\int_0^t K(t,s)\,dB^{H(s)}(s)$8

with distinct Hurst indices $\mathcal M(t)=\int_0^t K(t,s)\,dB^{H(s)}(s)$9 (Almani et al., 2021).

A related neighboring object is the multifractional bifractional Brownian motion $H(t)$0, introduced by replacing the constant Hurst parameter in bifractional Brownian motion by a Hölder function $H(t)$1 while keeping $H(t)$2 fixed. It is not usually labeled mmfBm, but it belongs to the same broader program of time-varying Gaussian self-similar models (Ouahra et al., 2020).

2. Volterra kernels, covariance structure, and memory encoding

Most memory-preserving multifractional models are naturally expressed as Gaussian Volterra processes. In this representation the entire dependence structure is encoded in the kernel.

For the continuous-memory MMFBM,

$H(t)$3

so the response function is simply

$H(t)$4

The use of $H(t)$5 rather than $H(t)$6 is the mathematical origin of the “continuous correlations” emphasized in that work (Wang et al., 2023).

In the qubit-oriented Volterra model,

$H(t)$7

the covariance is given formally by

$H(t)$8

When the term $H(t)$9 is small, the integrand reduces to $D(t)$0, making the covariance resemble a locally weighted fBm covariance with an additional memory kernel (Haq, 27 Jul 2025).

A second qubit formulation uses the Riemann–Liouville representation

$D(t)$1

with adaptive memory kernel

$D(t)$2

Its exact two-point covariance is expressed in Gauss hypergeometric form: $D(t)$3 Under the adiabatic condition $D(t)$4, this is approximated by

$D(t)$5

which recovers a locally stationary fBm-like form with slowly drifting exponent (Haq, 18 May 2026).

For the multifractional bifractional process $D(t)$6, the covariance is

$D(t)$7

When $D(t)$8, one recovers standard mBm; when $D(t)$9, one recovers bifractional Brownian motion $$X(t)=\int_0^t \sqrt{\alpha(s)}\,(t-s)^{(\alpha(s)-1)/2}\,dB(s)$$0 (Ouahra et al., 2020).

3. Regularity, local asymptotics, and long-range dependence

The time-varying exponent controls local roughness, but the memory-preserving constructions also impose global correlation properties.

For $$X(t)=\int_0^t \sqrt{\alpha(s)}\,(t-s)^{(\alpha(s)-1)/2}\,dB(s)$$1, if $$X(t)=\int_0^t \sqrt{\alpha(s)}\,(t-s)^{(\alpha(s)-1)/2}\,dB(s)$$2 is $$X(t)=\int_0^t \sqrt{\alpha(s)}\,(t-s)^{(\alpha(s)-1)/2}\,dB(s)$$3-Hölder continuous, then the covariance kernel is positive-definite, so the centered Gaussian process exists. Under the additional condition $$X(t)=\int_0^t \sqrt{\alpha(s)}\,(t-s)^{(\alpha(s)-1)/2}\,dB(s)$$4, there is a two-sided increment estimate

$$X(t)=\int_0^t \sqrt{\alpha(s)}\,(t-s)^{(\alpha(s)-1)/2}\,dB(s)$$5

for all $$X(t)=\int_0^t \sqrt{\alpha(s)}\,(t-s)^{(\alpha(s)-1)/2}\,dB(s)$$6. Consequently, for every $$X(t)=\int_0^t \sqrt{\alpha(s)}\,(t-s)^{(\alpha(s)-1)/2}\,dB(s)$$7, the sample path is uniformly Hölder continuous of exponent $$X(t)=\int_0^t \sqrt{\alpha(s)}\,(t-s)^{(\alpha(s)-1)/2}\,dB(s)$$8 on $$X(t)=\int_0^t \sqrt{\alpha(s)}\,(t-s)^{(\alpha(s)-1)/2}\,dB(s)$$9, and the pointwise Hölder exponent at time $\alpha(s)$0 is almost surely $\alpha(s)$1 (Ouahra et al., 2020).

The same process is locally asymptotically self-similar. For fixed $\alpha(s)$2 and $\alpha(s)$3,

$\alpha(s)$4

in finite-dimensional distributions, where the limit is a standard fBm of Hurst index $\alpha(s)$5. At large lags the same model exhibits long-range dependence: the correlation obeys a power law with exponent $\alpha(s)$6, hence $\alpha(s)$7 (Ouahra et al., 2020).

For mBm in the White Noise framework, increments near a fixed time $\alpha(s)$8 behave like those of an fBm with Hurst parameter $\alpha(s)$9, so the local Hölder exponent is $\mathcal M(t)=\int_0^t K(t,s)\,dB^{H(s)}(s)$0 and the local memory index is $\mathcal M(t)=\int_0^t K(t,s)\,dB^{H(s)}(s)$1 (Lebovits, 2013). In the qubit mmfBm model this same exponent appears in the non-stationary noise spectrum: $\mathcal M(t)=\int_0^t K(t,s)\,dB^{H(s)}(s)$2 which gives a slowly drifting $\mathcal M(t)=\int_0^t K(t,s)\,dB^{H(s)}(s)$3 law (Haq, 18 May 2026).

The continuous-memory MMFBM of (Wang et al., 2023) exhibits a stronger history effect than memory-reset multifractional models. Its mean-squared displacement is

$\mathcal M(t)=\int_0^t K(t,s)\,dB^{H(s)}(s)$4

For a step protocol

$\mathcal M(t)=\int_0^t K(t,s)\,dB^{H(s)}(s)$5

one obtains

$\mathcal M(t)=\int_0^t K(t,s)\,dB^{H(s)}(s)$6

and for $\mathcal M(t)=\int_0^t K(t,s)\,dB^{H(s)}(s)$7,

$\mathcal M(t)=\int_0^t K(t,s)\,dB^{H(s)}(s)$8

Thus, if $\mathcal M(t)=\int_0^t K(t,s)\,dB^{H(s)}(s)$9, the long-time scaling can be governed by the pre-step exponent rather than by the post-step exponent. This is a defining memory effect, not a perturbative correction (Wang et al., 2023).

4. Switching and superposed formulations

A major branch of mmfBm research studies regime changes rather than smoothly varying exponents. In the switching-FBM model, $$X(t)=\int_0^t \sqrt{2D(s)H(s)}\,(t-s)^{H(s)-1/2}\,dB(s)$$0 and $$X(t)=\int_0^t \sqrt{2D(s)H(s)}\,(t-s)^{H(s)-1/2}\,dB(s)$$1 jump between finitely many states and the process is defined by

$$X(t)=\int_0^t \sqrt{2D(s)H(s)}\,(t-s)^{H(s)-1/2}\,dB(s)$$2

Two switching laws are analyzed. In the Markovian two-state case, the active state $$X(t)=\int_0^t \sqrt{2D(s)H(s)}\,(t-s)^{H(s)-1/2}\,dB(s)$$3 has exponential dwell times with means $$X(t)=\int_0^t \sqrt{2D(s)H(s)}\,(t-s)^{H(s)-1/2}\,dB(s)$$4. In the scale-free intermittent case, one state has heavy-tailed waiting-time density

$$X(t)=\int_0^t \sqrt{2D(s)H(s)}\,(t-s)^{H(s)-1/2}\,dB(s)$$5

while the other remains exponential; this produces ageing and ergodicity breaking (Balcerek et al., 2023).

For Markovian switching, the long-time occupation fractions are

$$X(t)=\int_0^t \sqrt{2D(s)H(s)}\,(t-s)^{H(s)-1/2}\,dB(s)$$6

and the ensemble-averaged time-averaged MSD is

$$X(t)=\int_0^t \sqrt{2D(s)H(s)}\,(t-s)^{H(s)-1/2}\,dB(s)$$7

The single-trajectory PSD satisfies the analogous weighted-sum rule

$$X(t)=\int_0^t \sqrt{2D(s)H(s)}\,(t-s)^{H(s)-1/2}\,dB(s)$$8

At short lag $$X(t)=\int_0^t \sqrt{2D(s)H(s)}\,(t-s)^{H(s)-1/2}\,dB(s)$$9, the increment distribution is approximately a bimodal Gaussian mixture,

$M_t=\sum_{k=1}^\infty \sigma_k B_t^{H_k}$0

whereas for $M_t=\sum_{k=1}^\infty \sigma_k B_t^{H_k}$1 it converges to a single Gaussian by the Central Limit Theorem (Balcerek et al., 2023).

The multi-mixed fractional Brownian motion of (Almani et al., 2021) is structurally different. Here

$M_t=\sum_{k=1}^\infty \sigma_k B_t^{H_k}$2

and

$M_t=\sum_{k=1}^\infty \sigma_k B_t^{H_k}$3

Its increment autocovariance behaves as

$M_t=\sum_{k=1}^\infty \sigma_k B_t^{H_k}$4

Hence the increments are long-range dependent if $M_t=\sum_{k=1}^\infty \sigma_k B_t^{H_k}$5 and short-range dependent if $M_t=\sum_{k=1}^\infty \sigma_k B_t^{H_k}$6. The almost-sure Hölder exponent is $M_t=\sum_{k=1}^\infty \sigma_k B_t^{H_k}$7, and the $M_t=\sum_{k=1}^\infty \sigma_k B_t^{H_k}$8-variation index is $M_t=\sum_{k=1}^\infty \sigma_k B_t^{H_k}$9 (Almani et al., 2021).

5. Stochastic integration and driven dynamics

Because these processes are typically not semimartingales, stochastic integration requires dedicated constructions. A key route is the White Noise approach for mBm. If $\alpha(t)\in(0,2]$0 is $\alpha(t)\in(0,2]$1-Hölder continuous and $\alpha(t)\in(0,2]$2 is the associated piecewise-constant approximation on a partition of $\alpha(t)\in(0,2]$3, then the patchwork process $\alpha(t)\in(0,2]$4 converges in law to $\alpha(t)\in(0,2]$5 in $\alpha(t)\in(0,2]$6. This makes it possible to define stochastic integration with respect to mBm as a limit of stochastic integrals with respect to tangent fBms (Lebovits, 2013).

In Hida–White Noise calculus, if $\alpha(t)\in(0,2]$7 is Bochner-integrable, the Wick–Itô integral with respect to fBm is

$\alpha(t)\in(0,2]$8

For mBm, the Hida derivative is

$\alpha(t)\in(0,2]$9

and the limiting fractional Wick–Itô integral becomes

$\mathcal M(t)=\int_0^t K(t,s)\,dB^{H(s)}(s)$00

The additional term is the explicit contribution of the time variation of the Hurst function (Lebovits, 2013).

In the qubit mmfBm model of (Haq, 18 May 2026), the noise enters a Young-integral SDE,

$\mathcal M(t)=\int_0^t K(t,s)\,dB^{H(s)}(s)$01

and when $\mathcal M(t)=\int_0^t K(t,s)\,dB^{H(s)}(s)$02 is effectively constant,

$\mathcal M(t)=\int_0^t K(t,s)\,dB^{H(s)}(s)$03

A related qubit charge-offset model embeds the Volterra mmfBm directly into

$\mathcal M(t)=\int_0^t K(t,s)\,dB^{H(s)}(s)$04

with $\mathcal M(t)=\int_0^t K(t,s)\,dB^{H(s)}(s)$05, $\mathcal M(t)=\int_0^t K(t,s)\,dB^{H(s)}(s)$06, $\mathcal M(t)=\int_0^t K(t,s)\,dB^{H(s)}(s)$07, and the target Hurst function calibrated from charge-noise data (Haq, 27 Jul 2025).

6. Physical applications

The most developed application domain in the supplied literature is superconducting-qubit decoherence. In the unified stochastic drift model of (Haq, 18 May 2026), the classical sector uses a time-dependent Hurst exponent $\mathcal M(t)=\int_0^t K(t,s)\,dB^{H(s)}(s)$08 and adaptive memory kernel $\mathcal M(t)=\int_0^t K(t,s)\,dB^{H(s)}(s)$09 to represent non-stationary $\mathcal M(t)=\int_0^t K(t,s)\,dB^{H(s)}(s)$10 noise, while the quantum extension is a time-dependent Caldeira–Leggett bath with spectral density

$\mathcal M(t)=\int_0^t K(t,s)\,dB^{H(s)}(s)$11

For a pure-dephasing qubit with

$\mathcal M(t)=\int_0^t K(t,s)\,dB^{H(s)}(s)$12

the reduced dynamics take the exact time-local Lindblad form

$\mathcal M(t)=\int_0^t K(t,s)\,dB^{H(s)}(s)$13

with

$\mathcal M(t)=\int_0^t K(t,s)\,dB^{H(s)}(s)$14

in the low-frequency limit. The corresponding coherence envelopes are stretched exponentials,

$\mathcal M(t)=\int_0^t K(t,s)\,dB^{H(s)}(s)$15

The paper reports four central results: relaxation and noise amplitudes act independently on energy decay; time-varying $\mathcal M(t)=\int_0^t K(t,s)\,dB^{H(s)}(s)$16 matches experimental $\mathcal M(t)=\int_0^t K(t,s)\,dB^{H(s)}(s)$17 spectra more accurately than any constant exponent; adaptive kernel dynamics preserve correlations without artificial damping; and simulations predict coherence times $\mathcal M(t)=\int_0^t K(t,s)\,dB^{H(s)}(s)$18 and $\mathcal M(t)=\int_0^t K(t,s)\,dB^{H(s)}(s)$19 when charge noise dominates. It also identifies a thermal crossover time

$\mathcal M(t)=\int_0^t K(t,s)\,dB^{H(s)}(s)$20

with temperature-independent dephasing below $\mathcal M(t)=\int_0^t K(t,s)\,dB^{H(s)}(s)$21 mK and $\mathcal M(t)=\int_0^t K(t,s)\,dB^{H(s)}(s)$22 above $\mathcal M(t)=\int_0^t K(t,s)\,dB^{H(s)}(s)$23 mK (Haq, 18 May 2026).

The charge-noise model of (Haq, 27 Jul 2025) calibrates the mmfBm parameters to qubit data by fitting the relaxation rate to known $\mathcal M(t)=\int_0^t K(t,s)\,dB^{H(s)}(s)$24 times $\mathcal M(t)=\int_0^t K(t,s)\,dB^{H(s)}(s)$25, setting the noise amplitude by the low-frequency charge-noise level $\mathcal M(t)=\int_0^t K(t,s)\,dB^{H(s)}(s)$26, using a spectral exponent such as $\mathcal M(t)=\int_0^t K(t,s)\,dB^{H(s)}(s)$27, and converting it to a target Hurst function

$\mathcal M(t)=\int_0^t K(t,s)\,dB^{H(s)}(s)$28

Simulation uses the causal kernel

$\mathcal M(t)=\int_0^t K(t,s)\,dB^{H(s)}(s)$29

with Euler–Maruyama integration and an ensemble of $\mathcal M(t)=\int_0^t K(t,s)\,dB^{H(s)}(s)$30 realizations. Coupled to a transmon Hamiltonian, the model yields fidelity $\mathcal M(t)=\int_0^t K(t,s)\,dB^{H(s)}(s)$31 with $\mathcal M(t)=\int_0^t K(t,s)\,dB^{H(s)}(s)$32, coherence envelopes with non-Markovian revivals, and excited-state population below $\mathcal M(t)=\int_0^t K(t,s)\,dB^{H(s)}(s)$33 (Haq, 27 Jul 2025).

Outside quantum hardware, the switching-FBM framework has been validated against single-particle tracking of quantum dots in the cytoplasm of live mammalian cells, where intermittent changes in transport parameters coexist with persistent correlations along trajectories (Balcerek et al., 2023).

7. Conceptual distinctions, misconceptions, and related multifractional models

A recurrent misconception is that any time-dependent Hurst exponent automatically produces a memory-preserving process. The literature explicitly distinguishes between memory-reset multifractional Brownian motion and continuous-memory MMFBM. In the memory-reset model,

$\mathcal M(t)=\int_0^t K(t,s)\,dB^{H(s)}(s)$34

the current value $\mathcal M(t)=\int_0^t K(t,s)\,dB^{H(s)}(s)$35 reweights the entire past kernel. As a result, $\mathcal M(t)=\int_0^t K(t,s)\,dB^{H(s)}(s)$36, a step in $\mathcal M(t)=\int_0^t K(t,s)\,dB^{H(s)}(s)$37 produces an abrupt change in the MSD slope, the linear response to a short perturbation is identically zero for $\mathcal M(t)=\int_0^t K(t,s)\,dB^{H(s)}(s)$38, and the autocovariance immediately switches to the new scaling. In MMFBM, by contrast, the kernel keeps the whole history of $\mathcal M(t)=\int_0^t K(t,s)\,dB^{H(s)}(s)$39, so pre-step exponents remain visible in the MSD, autocovariance, and response function (Wang et al., 2023).

A second source of confusion is terminological. The acronym “mmfBm” is used both for memory multi-fractional Brownian motion and for multi-mixed fractional Brownian motion. The former varies the exponent in time and encodes memory through a Volterra kernel or related construction; the latter mixes independent fBms with fixed but different Hurst indices through a superposition measure $\mathcal M(t)=\int_0^t K(t,s)\,dB^{H(s)}(s)$40 (Almani et al., 2021).

A third distinction concerns nearby but non-equivalent generalizations. Multifractional bifractional Brownian motion $\mathcal M(t)=\int_0^t K(t,s)\,dB^{H(s)}(s)$41 preserves the bifractional parameter $\mathcal M(t)=\int_0^t K(t,s)\,dB^{H(s)}(s)$42 and reduces to mBm when $\mathcal M(t)=\int_0^t K(t,s)\,dB^{H(s)}(s)$43; its local Hölder exponent is $\mathcal M(t)=\int_0^t K(t,s)\,dB^{H(s)}(s)$44, not $\mathcal M(t)=\int_0^t K(t,s)\,dB^{H(s)}(s)$45, and its long-range dependence follows from the asymptotic power-law behavior of its correlation function (Ouahra et al., 2020). Related multifractional, rather than explicitly memory-kernel, models also appear in finance: a multifractional Black–Scholes calibration on SPX ATM calls reported mean squared errors $\mathcal M(t)=\int_0^t K(t,s)\,dB^{H(s)}(s)$46 for the multifractional model, $\mathcal M(t)=\int_0^t K(t,s)\,dB^{H(s)}(s)$47 for the fractional model with fixed $\mathcal M(t)=\int_0^t K(t,s)\,dB^{H(s)}(s)$48, and $\mathcal M(t)=\int_0^t K(t,s)\,dB^{H(s)}(s)$49 for classical Black–Scholes, illustrating the empirical value of time-varying roughness even outside the mmfBm terminology (Araneda, 2023).

Taken together, the arXiv literature presents mmfBm not as a single canonical process but as a research program: Gaussian models with time-varying roughness, explicit history dependence, and analytically tractable covariance structure. The central technical question across these variants is how to allow $\mathcal M(t)=\int_0^t K(t,s)\,dB^{H(s)}(s)$50 or $\mathcal M(t)=\int_0^t K(t,s)\,dB^{H(s)}(s)$51 to change without destroying long-memory effects already accumulated in the past.