Non-Stationary Anomalous Diffusion

• Non-stationary anomalous diffusion is characterized by time-dependent mean-squared displacement and evolving statistical properties such as ageing and weak ergodicity breaking.
• It encompasses models like CTRW, scaled Brownian motion, and random diffusivity processes, each revealing distinct mechanisms driving non-stationarity.
• Modern analysis leverages deep learning segmentation and advanced statistical diagnostics to reliably infer dynamic exponents and detect changepoints in trajectories.

Non-stationary anomalous diffusion refers to stochastic processes in which the mean-squared displacement (MSD) displays nonlinear, often time-dependent scaling in time, with the additional property that the statistical law governing increments changes over the observation period. Found in physical, biological, ecological, and complex engineered systems, non-stationary anomalous diffusion is characterized by breakdowns of time-translation invariance, evolving statistical properties, and signatures such as ageing, weak ergodicity breaking, and non-self-averaging. The mathematical and phenomenological frameworks capturing these behaviors span continuous-time random walks (CTRWs) with non-trivial waiting-time distributions, time-dependent or random diffusivity models, and random dynamical systems operating at the edge of stability. Recent advances have also focused on deep-learning segmentation of non-stationary anomalous trajectories, particularly in tracking single biomolecules and networked populations.

1. Defining Features and Taxonomy

Anomalous diffusion is defined through the scaling of the MSD:

$\langle |x(t)-x(0)|^2 \rangle \propto t^\alpha$

where $\alpha \neq 1$ signals deviation from classical Brownian motion. In non-stationary anomalous diffusion, the distribution of increments $\delta x_j \equiv x(j\Delta)-x((j-1)\Delta)$ is not invariant over time, i.e., the process is neither stationary nor homogeneous. In the terminology of the three-effect decomposition (Vilk et al., 2021), such processes manifest a "Moses effect," quantified by a nontrivial exponent $M$ extracted from time-averaged velocity scaling,

$\overline{|v|}(t) \propto t^{M-\frac12}$

and, more generally, by $\langle x^2(t)\rangle \propto t^{\alpha(t)}$. Non-stationarity (Moses effect) is to be distinguished from long-range temporal correlations (Joseph effect) and heavy-tailed increment laws (Noah effect), with anomalous transport obeying a sum rule $H=J+L+M-1$ ($H$ is the Hurst exponent).

2. Canonical Models and Mathematical Mechanisms

Several distinct mechanisms yield non-stationary anomalous diffusion:

• CTRWs with non-stationary waiting-time distributions: Processes with divergent mean trapping times or power-law-waiting times ($\psi(t)\sim t^{-1-\mu}$, $0<\mu<1$) generate subdiffusive, non-ergodic, and ageing dynamics. The number of steps grows sublinearly, and MSD scaling $\alpha \neq 1$0 (with model-dependent $\alpha \neq 1$1, $\alpha \neq 1$2) (Nigris et al., 2016).
• Time-dependent or stochastic diffusivity models: Langevin processes $\alpha \neq 1$3 with $\alpha \neq 1$4 following, e.g., $\alpha \neq 1$5 induce $\alpha \neq 1$6 (Cherstvy et al., 2016). Such Random Diffusivity Processes (RDPs) display weak ergodicity breaking and non-stationary time-averaged and ensemble-averaged MSDs.
• Scaled Brownian motion (SBM): This is a Gaussian process with time-varying diffusivity $\alpha \neq 1$7 producing $\alpha \neq 1$8 (Jeon et al., 2014). SBM is non-stationary due to the explicit time dependence of its covariance and fails to relax to a stationary state under confinement.
• Random dynamical systems at criticality: Stochastic alternation between expanding (diffusive) and contracting (localizing) maps, tuned such that the average Lyapunov exponent vanishes globally, induces subdiffusion characterized by $\alpha \neq 1$9, power-law waiting time statistics, explicit ageing, and non-self-averaging (Sato et al., 2018).
• Non-homogeneous, path-dependent Markov models: The generalized Pólya process (3p-BPM) specifies jump rates proportional to both the current state and inverse elapsed time, yielding MSD scaling $\delta x_j \equiv x(j\Delta)-x((j-1)\Delta)$0 and path-dependent autocorrelation (Barraza et al., 5 Mar 2025).

The prevalence of non-stationarity in the above models is summarized below:

Model Type	Key Non-stationary Mechanism	Ergodicity
CTRW with diverging WTD	Age-dependent trapping, non-TI increments	Strongly broken
SBM	Time-dependent diffusivity, covariance depends on $\delta x_j \equiv x(j\Delta)-x((j-1)\Delta)$1	Weakly broken
RDP	Ensemble of $\delta x_j \equiv x(j\Delta)-x((j-1)\Delta)$2, weak ergodicity breaking	Weakly broken
Random dynamical map	On-off intermittency, infinite invariant density	Strongly broken
Path-dependent Markov	State & time-dependent jump probabilities	Strongly broken

3. Ageing, Weak Ergodicity Breaking, and Non-self-averaging

Non-stationary anomalous diffusion generically exhibits ageing, where two-time statistics are explicit functions of both measurement duration and elapsed process age. For example, in random dynamical systems at the critical point $\delta x_j \equiv x(j\Delta)-x((j-1)\Delta)$3, the two-time MSD scales as

$\delta x_j \equiv x(j\Delta)-x((j-1)\Delta)$4

with $\delta x_j \equiv x(j\Delta)-x((j-1)\Delta)$5 (Sato et al., 2018). Weak ergodicity breaking is observed when ensemble and time-averaged MSDs differ even at long measurement times. In the scaled Brownian and RDP frameworks, the time-averaged MSD decays algebraically or linearly with path length or is sensitive to the observation interval (Cherstvy et al., 2016, Jeon et al., 2014). Furthermore, in the presence of "common noise" or system-wide disorder, self-averaging can break down such that sample-to-sample fluctuations of statistical observables persist indefinitely (Sato et al., 2018).

4. Detection, Inference, and Segmentation Methodologies

Traditional inference of anomalous exponents $\delta x_j \equiv x(j\Delta)-x((j-1)\Delta)$6 from MSD scaling is rendered unreliable in non-stationary or switching scenarios. Modern methods incorporate machine learning—especially recurrent neural networks (RNNs)—and semantic segmentation:

• RNN and LSTM architectures can process discretized increments or positions, estimate time-resolved exponents $\delta x_j \equiv x(j\Delta)-x((j-1)\Delta)$7, generalized diffusion coefficients $\delta x_j \equiv x(j\Delta)-x((j-1)\Delta)$8, and changepoint locations (Bo et al., 2019, Lanza et al., 12 Mar 2025). For multidimensional tracts, architectures process batched block increments with heads for $\delta x_j \equiv x(j\Delta)-x((j-1)\Delta)$9, $M$0, and changepoint probabilities, using loss combinations targeting both regression and classification tasks.
• U-AnDi (U-Net with DCC and GAU): Deep convolutional networks employing dilated causal convolutions and gated activation units segment trajectories by exponent or model, leveraging long-range memory to faithfully capture transitions and temporal correlations (Qu et al., 2023).
• Physical-statistic diagnostics: Three-effect decomposition (Joseph, Noah, Moses) operates by regressing scaling exponents from time-averaged velocity and squared-velocity statistics across lags and trajectory samples (Vilk et al., 2021).

Benchmarks consistently show superiority of deep learning segmentation (e.g., RNN, U-AnDi) for irregularly sampled, piecewise-diffusive, or short trajectories where conventional MSD analysis cannot reliably identify exponents or changepoints (Bo et al., 2019, Lanza et al., 12 Mar 2025, Qu et al., 2023).

5. Physical Realizations and Regime Transitions

Empirical systems demonstrating non-stationary anomalous diffusion span nanoscopic to macroscopic scales:

• Molecular and cell biology: Rhodamine transport on water nanofilms, cytoplasmic quantum dots, and transmembrane protein tracking display regime transitions and explicit Moses effects driven by environmental change (e.g., humidity, phase separation, desorption kinetics) (Vilk et al., 2021, Qu et al., 2023).
• Movement ecology: For animal tracking data (e.g., storks, vultures), mode switches correspond to behavioral or environmental transitions (foraging, commuting, rest-activity cycles) (Vilk et al., 2021).
• Epidemiology and network science: Early-stage COVID-19 propagation and queue-length fluctuations in telecommunication routers exemplify hyperballistic non-stationary regimes with power-law tail statistics (Barraza et al., 5 Mar 2025).

In many models, transitions between normal, sub-, super-, ballistic, and hyperballistic regimes are governed by the interplay of time-dependent process parameters (e.g., drift–damping ratio in 3p-BPM, scaling of diffusivity, Lyapunov exponents). The regime boundaries have sharp mathematical correlates (e.g., $M$1 in 3p-BPM, Lyapunov exponent sign in random dynamical systems).

6. Theoretical Issues, Limitations, and Outlook

Several conceptual and practical issues remain under active investigation:

• Model non-uniqueness: Scaling exponents $M$2, or their counterparts, often constrain but do not uniquely determine the model class due to coupling of effects and finite sample limitations (Vilk et al., 2021).
• Robustness of learning-based segmentation: Deep networks (e.g., sequandi, U-AnDi) require massive synthetic training data across multiple models and changepoint densities; reliability at extremes of parameter ranges or with exotic noise statistics may require further theoretical grounding (Lanza et al., 12 Mar 2025, Qu et al., 2023).
• Thermodynamic consistency: Certain classes (SBM under harmonic confinement) are unphysical in equilibrium, never reach stationarity, and thus must be interpreted as phenomenological or applicable only to non-equilibrium or driven systems (Jeon et al., 2014).
• Infinite invariant densities and weak statistical stability: The presence of non-normalizable stationary densities (e.g., $M$3) complicates the interpretation of longstanding steady-state statistics and further implies weak clustering phenomena (Sato et al., 2018).
• Open theoretical directions: Extension to mixed or crossover scaling, rigorous estimation of stochastic changepoints beyond strict power-law segments, and integration with black-box machine-learning estimators are active fields (Vilk et al., 2021, Lanza et al., 12 Mar 2025).

7. Comparative Table: Representative Models of Non-Stationary Anomalous Diffusion

Model	Mechanism (Key Non-Stationarity)	MSD Scaling	Notable Properties/Limitations	arXiv Reference
CTRW (with heavy-tailed WTD)	Diverging mean waiting times, ageing	$M$4, $M$5	Strong ageing, weak ergodicity breaking	(Nigris et al., 2016)
SBM	Power-law time-dependent diffusivity	$M$6, $M$7	Weakly non-ergodic, unphysical under confinement	(Jeon et al., 2014)
RDP	Time-fluctuating $M$8, deterministic $M$9	$\overline{|v|}(t) \propto t^{M-\frac12}$0	Strong non-stationarity, weak erg. breaking	(Cherstvy et al., 2016)
Random dynamical systems	Alternating expansion/contraction, Lyapunov tuning	$\overline{|v|}(t) \propto t^{M-\frac12}$1 at criticality	Infinite invariant density, strong non-self-averaging	(Sato et al., 2018)
3p-BPM generalized Pólya	State/time-dependent rates $\overline{|v|}(t) \propto t^{M-\frac12}$2	$\overline{|v|}(t) \propto t^{M-\frac12}$3	Hyperballistic/Brownian regimes, path dependence	(Barraza et al., 5 Mar 2025)

Non-stationary anomalous diffusion displays rich interconnections between time-inhomogeneous microscopic laws, emergent scaling structures, fundamentally non-ergodic behavior, and a diversity of real-world realizations. Identification and quantitative characterization increasingly rely on sophisticated machine learning, combined with the deep analytic and phenomenological understanding provided by contemporary stochastic process theory.