High-Dimensional Non-Stationary OU Processes

• High-dimensional non-stationary OU processes are continuous-time stochastic models that generalize classical OU dynamics to include time-varying drifts, multiple noise sources, and infinite-dimensional state spaces.
• They employ explicit solution representations, characteristic functions, and Laplace transforms to analyze moment evolution, memory effects, and long-range dependencies in non-equilibrium systems.
• They support advanced statistical estimation methods, such as sparsity-inducing Lasso estimators, to robustly infer parameters in applications spanning finance, physics, and data science.

High-dimensional non-stationary Ornstein–Uhlenbeck (OU) processes encompass a broad spectrum of continuous-time stochastic models in which the classical OU structure is generalized to allow for non-stationary dynamics, high-dimensional or infinite-dimensional state spaces, various sources of noise (Gaussian, Lévy, multifractal, colored), and, crucially, settings where explicit ergodicity or equilibrium may fail or be postponed by long memory or model complexity. OU processes and their extensions form a central class for high-dimensional inference, time series modeling, random field analysis, non-equilibrium systems, and numerous applications across probability, mathematical physics, finance, and data science.

1. Generalized High-Dimensional Ornstein–Uhlenbeck Models

The high-dimensional OU process is naturally defined as a vector-valued, matrix-valued, or Hilbert-space-valued solution to a (possibly time-inhomogeneous and/or non-autonomous) stochastic differential equation (SDE) of the Langevin type:

$dX_t = A(t) X_t dt + \Sigma(t) dW_t + dL_t,$

where $A(t)$ is a potentially non-constant drift matrix/operator, $\Sigma(t)$ models (possibly time- or state-dependent) volatility, $W_t$ is an appropriate Gaussian process (Brownian motion or cylindrical Wiener process), and $L_t$ denotes general Lévy or colored noise. In the infinite-dimensional case, $X_t$ may take values in a separable Hilbert space with dynamics mediated by a $C_0$-semigroup $S(t)$ and driven by an $H$-valued or cylindrical Lévy process (Applebaum, 2014).

Extensions further comprise:

• Volterra-type OU processes: spatially and temporally nonlocal generalizations driven by convolution with a drift measure and a noise kernel, leading to stochastic Volterra equations with space-time convolution and potentially long-range dependence (Pham et al., 2016).
• Markov-modulated and non-Markovian OU processes: models where parameters (drift, volatility) are modulated by exogenous, possibly high-dimensional, stochastic processes (e.g., finite-state Markov chains, random environments, or multifractal corrections) (Huang et al., 2014, Chevillard et al., 2020).
• Generalized OU processes with memory kernels or arbitrary Lévy driving noise, including $\alpha$-stable or compound Poisson processes, fractional Brownian or Hermite noise, and deeply non-stationary settings (Stein et al., 2021, Es-Sebaiy, 2021).

Such generalizations provide the basis for modeling non-stationary and high-dimensional dependencies, with the classical exponential kernel (Markov property) becoming just a special case.

2. Probabilistic Structure, Stationarity, and Memory

The probabilistic structure of high-dimensional OU processes is governed by the interaction between the drift, noise regularity, and the possible presence of memory kernels:

• Stationarity and Markov Property: The exponential memory kernel uniquely characterizes Markovianity and time-stationarity; departures via general kernels yield non-Markovian and/or non-stationary processes (Stein et al., 2021). In high-dimensional settings, processes of the form

$X_t = S(t)X_0 + \int_0^t S(t-s) dL(s)$

admit explicit formulas for the evolution of moments and covariance structures, with stationarity occurring only for specific driver (kernel) and initial condition choices (Applebaum, 2014, Voutilainen et al., 2019).

• Memory Effects and Long-Range Dependence: The decay properties of a memory kernel or the spectral form of the convolution kernel in Volterra-type or multifractal OU processes control the persistence of autocorrelation and can lead to long memory (slow decay) or multiscaling (Pham et al., 2016, Chevillard et al., 2020).
• Multivariate and Infinite-Dimensional Generalizations: The covariance (or Lyapunov/Sylvester) equations generalize, requiring solution for operator-valued equations, e.g.,

$B S + S B^\top = 2D,$

encoding the multivariate equilibrium or non-equilibrium covariance (with $B$ drift/friction and $D$ the diffusion/noise covariance) (Godrèche et al., 2018).

3. Explicit Solutions and Analytical Tools

A central achievement in recent work is the availability of nearly explicit solution formulas and characterization of distributions for this broad class:

• Explicit Solution Representations: For time-inhomogeneous Gaussian and Lévy-driven cases, one can express the solution and its mean/variance paths explicitly via integrating factors and stochastic convolution, e.g.,

$$X_t = e^{-\int_0^t \beta(u) du} \left( X_0 + \int_0^t e^{\int_0^s \beta(u) du} \alpha(s) ds + \int_0^t e^{\int_0^s \beta(u) du} \sigma(s) dW_s \right)$$

(Vrins, 2016). The solution of the associated integral or SDE for generalized kernels or memory effects often involves resolvent or convolutional relationships (Pham et al., 2016, Stein et al., 2021).

• Characteristic Functions and Laplace Transforms: For both Gaussian and Lévy-driven OU models (with or without time inhomogeneity), closed-form expressions for the Laplace or characteristic function of functionals (e.g., integrals) can be obtained, enabling precise analysis of marginals and increments. For example, the cumulant generating function of an integrated OU process with general Lévy noise reduces to a two-dimensional integral, which can often be solved exactly, e.g., for gamma-distributed jumps or compound Poisson drivers (Vrins, 2016).
• Multiple Regimes and Functional Central Limit Theorems: Markov-modulated and regime-switching models admit explicit moment recursion relations, functional central limit theorems describing time-averaged dynamics as the modulating process is accelerated, and highlighting how the limiting behavior depends critically on scaling exponents (Huang et al., 2014).

4. Statistical Estimation and Sparse Inference in High Dimensions

The high-dimensional, non-stationary OU framework requires methods that efficiently estimate model parameters given limited sample information, often under sparsity constraints:

• Sparse Estimation via Regularized Maximum Likelihood: For a non-stationary $d$-dimensional OU process observed over $N$ independent paths,

$dx_i(t) = A x_i(t) dt + dw_i(t),$

the drift matrix $A$ can be estimated via sparsity-inducing penalties: - Lasso Estimator: Minimizing empirical risk plus an $\ell_1$ penalty:

$$\hat{A}_L \in \arg\min_{A \in \mathbb{R}^{d \times d}} \{ L_N(A) + \lambda_L \|A\|_1 \}$$

• Slope Estimator: Using an ordered weighted $\ell_1$ norm with explicit weights.

Both estimators are proven to attain the minimax optimal error rate $$\| \hat{A} - A_0 \|_2 = O_P(\sqrt{s \log(ed^2 / s)/N})$$ where $s$ is the number of nonzero entries (Nakakita, 24 Oct 2025). Numerical experiments confirm that these methods dramatically outperform classical MLE in high dimensions and are able to recover targeted sparsity patterns even for moderate dimensions or sample size.

• Covariance and Riccati-Based Estimation: In the vector-valued generalized OU context, the drift parameter (matrix $\Theta$) can be consistently estimated by empirical plug-in solutions to continuous-time algebraic Riccati equations derived from empirical cross-covariances (Voutilainen et al., 2019).
• Direct Estimation Approaches: For partially observed, non-Markovian, or colored-noise driven models, estimation can rely on analytic expansions of conditional moments (via stochastic Taylor expansion) and regression against a finite basis of functions of the time increment, bypassing the need for velocity or full state observations (Lehle et al., 2017).

5. Path Regularity, Long Memory, and Functional Properties

The regularity and memory of high-dimensional OU processes are shaped by both the noise driving term and memory kernels:

• Hölder and Càdlàg Paths: Sufficient smoothness of the noise kernel or the drift/convolution kernel ensures, respectively, locally Hölder-continuous or càdlàg paths (and even stronger notions, e.g., “triangular càdlàg” for one-dimensional space) in Volterra-type models (Pham et al., 2016).
• Long-Range Dependence and Multifractality: Fractional and multifractal generalizations display increment moments and autocovariances decaying as $|t-s|^{2H-2}$ (with $H$ the Hurst parameter), leading to non-ergodic, self-similar, or multifractal behaviors in the increments, essential for modeling turbulence and financial time series (Chevillard et al., 2020, Es-Sebaiy, 2021).
• Variance and Covariance Formulas: The solutions retain explicit representations for mean and covariance. For instance, the covariance structure for a process driven by stationary increments noise is given by:

$$\mathbb{E}[X_t X_s] = e^{-\theta (t-s)}\, \mathbb{E}[X_s^2] + e^{-\theta (t+s)} \int_0^s \int_0^t e^{\theta(u+v)} R_G(u, v) du dv$$

(Es-Sebaiy, 2021).

6. Non-equilibrium, Nonequilibrium Stationary States, and Oscillation Analysis

Nonequilibrium stationary states and properties such as entropy production and irreversibility can be characterized completely in terms of the antisymmetric part of the Onsager (kinetic) matrix $L = B S$:

• The entropy production rate in stationary high-dimensional OU processes is given by

$$\Phi = \operatorname{tr}( D^{1/2} S^{-1} D^{1/2} H^2 )$$

where $H$ is the matrix encoding irreversibility (Godrèche et al., 2018). The fluctuation–dissipation ratio (FDR) matrix $X$ further quantifies the deviation from equilibrium, with $X = (I + D^{-1} Q)^{-1}$.

Oscillation and variation properties, especially in non-stationary regimes or in the semigroup approach, are governed by sharp variation inequalities. For the Ornstein–Uhlenbeck semigroup $\mathcal{H}_t$ acting in $\mathbb{R}^n$,

• The $\varrho$-th order variation seminorm yields an operator of weak type $(1,1)$ (for $\varrho > 2$) with respect to the invariant measure, but fails for $\varrho \leq 2$ (Casarino et al., 1 May 2024). Fine bounds hinge on vector-valued Calderón–Zygmund methods and precise kernel estimates.

7. Applications and Model Selection

High-dimensional non-stationary OU processes are encountered in:

• Stochastic PDEs and Physics: As mild solutions to linear SPDEs with Lévy noise, or as building blocks for continuous-state branching systems, generalized Mehler semigroups, and bridges (e.g., Gaussian Bernstein processes) (Applebaum, 2014, Vuillermot et al., 2015).
• Finance and Statistical Modeling: Time-inhomogeneous and jump-driven models underpin short-rate modeling (Hull–White model), credit risk, and regime-switching models for multiple assets (Markov-modulated OU) (Vrins, 2016, Huang et al., 2014).
• Complex Systems and Statistical Mechanics: Modeling of nonequilibrium dynamics, oscillatory properties, entropy production, and fluctuations in spin chains, networks, and pattern theory (Godrèche et al., 2018, Vuillermot et al., 2015).
• Practical Estimation and Model Selection: Data-driven parameter selection via maximum likelihood, Bayesian, and goodness-of-fit tests (KS, AD, McCulloch), with flexibility to model either heavy tails (stable laws) or Gaussianity depending on application (e.g., high-frequency finance or epidemiological data) (Stein et al., 2021).

A plausible implication is that the convergence and stability properties observed in nonlinear and infinite-dimensional random walks—where a continuum of stationary measures and nontrivial conserved quantities coexist—can inform understanding and control in high-dimensional, non-ergodic, or metastable stochastic systems (Muzychka et al., 2011). These findings also have potential for extension to nonlinear, interacting, or mean-field settings, where phase coexistence and multiple equilibria may fundamentally alter long-time behavior.

In summary, high-dimensional non-stationary Ornstein–Uhlenbeck processes and their generalizations provide a unified, analytically tractable, and statistically rich class for modeling, inference, and analysis across a diverse range of high-dimensional stochastic systems, with rigorous mathematical frameworks supporting both probabilistic and statistical inference as well as pathwise, functional, and ergodic properties.