Time-Interdependent Stochastic Processes

• Time-interdependent stochastic processes are families of random variables explicitly influenced by past histories, multiple time indices, and evolving temporal contexts.
• They employ advanced techniques such as empirical process CLT, path integrals, and scenario trees to model phenomena in epidemic control, reliability engineering, and multi-agent systems.
• Integrating algebraic structures and memory-dependent branching, these processes enhance our ability to analyze non-Markovian behaviors in statistical physics and complex systems.

Time-interdependent stochastic processes are families of random variables or state evolutions whose probabilistic laws exhibit explicit dependence on multiple time indices, historical filtrations, or temporal context, rather than conforming to memoryless or Markovian assumptions. Such processes arise naturally in fields ranging from empirical process theory, non-Markovian modelling, quantum probability, epidemic control, branching systems, and multi-agent decision frameworks. Through diverse constructions—time-change, functional data analytics, controlled stochastic optimization, or algebraic structure—these processes encapsulate phenomena where present and future randomness are inherently coupled with the evolution of past states.

1. Structural Aspects and Defining Criteria

Time-interdependence in stochastic processes manifests as nontrivial dependency on the temporal path, past histories, or non-Markovian filtrations. In empirical process theory, the uniform CLT for classes such as $$\{ I_{X_t \le x} - P(X_t \le x) : t \in E, x \in \mathbb{R} \}$$ (Kuelbs et al., 2010) is established through tight local control conditions on the distributional transforms over $(t,x)$, requiring oscillation bounds in a metric $\rho$ defined by an auxiliary Gaussian process: $$\rho(s, t) = \left( E[ (H(s)-H(t))^2 ] \right)^{1/2}$$ with convergence in $\ell^{\infty}(E \times \mathbb{R})$ holding under the L condition for time-indexed stochastic data. Processes exhibiting such functional time dependence—sample path continuity, distribution function smoothness, and oscillation control—are paradigmatic examples of time-interdependent structures.

In non-Markovian models, such as time-changed population dynamics (Orsingher et al., 2014), the process $X^f(t) = X(H^f(t))$ depends on a random, increasing time process $H^f(t)$ with Laplace exponent $f$, and all temporal transitions are subordinated, directly influencing interevent distributions and extinction or explosion criteria.

For multi-stage stochastic optimization, time-interdependence is represented by scenario trees with nested filtration structures (Timonina-Farkas, 25 Aug 2025), where each node encodes conditional distributions dependent on entire previous histories, and the scenario evolution cannot be described by simple stage-wise Markov transitions.

2. Key Theoretical Constructs and Analytic Tools

Multiple analytic methodologies underpin the paper of time-interdependent processes:

• Empirical Process CLT and Distributional Transforms: Donsker properties and weak convergence in $\ell^{\infty}(E \times \mathbb{R})$ are achieved not by classical VC theory or bracketing, but via direct control of the local behavior of transformed distribution functions—highlighting the necessity for tailored conditions adapted to time-indexed data (Kuelbs et al., 2010).
• Integral Equations and Convolution Functionals: In continuous-time mixed state branching processes (MSB) (Chen et al., 2021), dynamics and distributions are governed by solutions to coupled stochastic equations, such as

$$V_i(t, \lambda) = \lambda_i - \int_0^t \psi_i(V(s, \lambda)) ds$$

highlighting memory effects as the evolution at each $t$ depends on the entire trajectory up to $t$.

• Path Integral and Algebraic Expansion: Time series path integral expansions employ time-ordered evolution operators, coherent states, and reproducing kernels (e.g., Doi–Peliti formalisms and $\mathfrak{su}(1,1)$ algebra) (Greenman, 2021). Unlike Dyson series methods that perturb about a time-invariant operator, direct time-slicing enables analytic computation of temporal effect series for processes with nonstationary or quadratic rates.
• Set-theoretic Characterization of Independence Times: For Lévy processes, independence of post-$R$ increments requires events to split into past and future components, via representations such as (Vidmar, 2017)

$\{R=n\} = F_n \cap \{\Delta_n X \in \Gamma\}$

providing sufficient and partly necessary conditions for temporal disentanglement in continuous and discrete time.

3. Applications in Stochastic Modelling

Time-interdependent stochastic processes serve as foundational models in several applied domains:

• Epidemic Control: Markovian epidemic processes generalized to admit time-varying infection and cure rates $\lambda(t), \mu(t)$ allow nonautonomous control via BeLLMan equations, with convergence of optimal control policies under mean-field scaling to deterministic ODEs whose coefficients retain time-dependence (Lu et al., 2017).
• Reliability Engineering: Extensions of the Poisson process—replace-after-fixed-time (RaFT) and replace-after-random-time (RaRT) processes—model component replacement where failure and scheduled interventions both contribute to the renewal count, leading to modified interarrival time distributions and explicit, time-dependent renewal structure (Marengo et al., 2018).
• Inventory Optimization: Robust inventory control with time-interdependent demand uses scenario tree quantization and projected gradient descent under linear non-anticipativity constraints, accommodating demand processes with embedded historical dependencies. By optimizing against nested distance bounds between conditional distributions, the method achieves accurate discrete approximations of complex stochastic processes (Timonina-Farkas, 25 Aug 2025).
• Multi-agent Systems and Epistemic Uncertainty: Meeting time analysis for interdependent agents incorporates imprecise Markov chain models to exactly compute tight bounds on expected meeting times given uncertainty in action selections and stochastic couplings, scalable to multiple agents with symmetry reductions on the joint state space (Sangalli et al., 10 Jul 2025).

4. Algebraic and Structural Generalizations

Time-interdependence extends to algebraic representations, for example in quadratic stochastic processes (QSPs) (Casas et al., 2017), where state transitions are not by square matrices but by cubic stochastic matrices, and the Kolmogorov–Chapman equation is adapted to involve cubic matrix multiplications—reflecting bilinear interactions dependent on prior states. Here, the evolutionary law is written as

$M^{[s, t]} = M^{[s, \tau]} *_p M^{[\tau, t]}$

with the specific multiplication rule $*_p$ encoding the temporal interdependencies.

Also, in the paper of continuous time mixed state branching with immigration, the inclusion of external flows and memory-integrated branching mechanisms enable the identification of stationary distributions and exponential ergodicity in Wasserstein metrics, essential for quantifying stability in non-Markovian dynamics (Chen et al., 2021).

5. Connections to Statistical Physics and Risk Theory

The probabilistic formalism of time-interdependent processes aligns with thermodynamics of trajectories and large deviations theory (Ryazanov, 2023). Moment generating functions and cumulant expansions (Lundberg equation, SCGF) describing first-passage times and dynamic activity translate directly to partition functions in statistical physics,

$Z_K(y) = \sum_{K} e^{-yK}\,P(K)$

with correspondence between roots of the Lundberg equation and nonequilibrium free energies. Deviations arise when trajectory observables admit both positive and negative excursions, exposing differences between risk-theoretic MGFs and the thermodynamic formalism, but nonetheless providing unified tools for analyzing fluctuations and rare-event statistics in non-equilibrium systems.

6. Open Challenges and Research Directions

The paper of time-interdependent stochastic processes continues to evolve:

• Scenario Tree Representations: Improving quantization procedures for high-dimensional, nested filtrations to mitigate the curse of dimensionality and achieve tighter approximations of non-Markovian interdependencies (Timonina-Farkas, 25 Aug 2025).
• Algorithmic Scalability: Addressing combinatorial explosion in multi-agent meeting time analysis through symmetry exploitation and quotient spaces (Sangalli et al., 10 Jul 2025).
• Integration Across Domains: Connecting set-theoretic independence time principles, renewal theory, and large deviations to statistical physics, quantum probability, and complex systems modelling, as proposed in thermodynamics of trajectories (Ryazanov, 2023).
• Statistical Inference and Robust Optimization: Developing estimation, hypothesis testing, and uncertainty quantification techniques tailored to time-dependent stochastic environments in reliability, finance, and control.
• Mathematical Foundations: Further refinement of local time-space integration, functional CLT conditions, and algebraic multiplications in stochastic matrices to generalize current results to broader classes of non-Markovian and functional stochastic processes.

Conclusion

Time-interdependent stochastic processes capture intricate dependencies across temporal dimensions, necessitating a blend of advanced analytic, algebraic, and computational methods. Their theoretical rigor and practical relevance extend across empirical process theory, population dynamics, multi-agent systems, optimal control, reliability engineering, and statistical physics. Continued research in quantization, ergodicity, independence times, and scenario-based optimization is essential for advancing the mathematical and applied understanding of these complex processes.