LDP for Time-Averaged Suboptimality in Diffusions

• The paper establishes a rigorous variational framework that characterizes the exponential decay of probabilities for time-averaged suboptimality in coupled slow–fast diffusion systems.
• It employs techniques such as exponential tightness and martingale analysis to derive an explicit rate function linking optimal trajectories with empirical measures.
• The construction of smooth approximations and regularity conditions supports practical computation of decay exponents and quantifies suboptimal performance in multiscale stochastic models.

The large-deviation principle (LDP) for time-averaged suboptimality investigates the exponential decay rate of the probability that a time-averaged performance metric, such as the average suboptimality in stochastic optimization or the time-averaged value of an observable in a stochastic process, deviates substantially from its typical value. This principle establishes not only the asymptotic probability scaling for such rare events but also provides a variational characterization of the rate function, elucidating the connection between optimal trajectories, averaging principles, and the “cost” of suboptimal excursions. The following sections present a rigorous and comprehensive account of the theoretical and practical framework developed for coupled diffusions with time scale separation, focusing on the construction, identification, and regularization of the large deviation rate function for pairs of slow variables and the empirical measure of fast components.

1. Large Deviation Principle for Coupled Slow–Fast Diffusions

Consider a fully coupled system of two Itô diffusions evolving on distinct time scales, with all coefficients (drift and diffusion) dependent on both the slow variable $X$ and the fast variable, and possibly mutually correlated noise terms. The small noise in $X$ (the slow component) creates an asymptotic separation of scales.

The primary object of paper is the joint process $(X, p)$, with $X$ the slow variable and $p$ the empirical measure of the fast process. The large-deviation principle is established for the joint law of $(X,p)$, rigorously quantifying the probabilities of observing atypical joint trajectories or empirical measures. The LDP is stated as follows: for suitable sets $A$ in the path/empirical measure space,

$$\mathbb{P}\big( (X, p) \in A \big)\asymp\exp\left[ - \inf_{(X,p)\in A} I(X,p) \right],$$

where $I$ is a good rate function characterizing the exponential scale of such deviations. The LDP is established under extensive technical conditions (local Lipschitz continuity, boundedness, controllability, etc.; see Conditions 2.1–2.3, (2.4b), (2.14d)) using a combination of exponential tightness and the analysis of exponential martingales.

2. Variational Structure and Identification of the Rate Function

The main theoretical achievement is the explicit identification and approximation of the rate function $I(X,p)$. In the most general form,

$I(X,p) = I_0(X_0) + I'(X,p),$

where $I_0$ denotes the rate function for the choice of initial condition $X_0$, and $I'(X,p)$ is an action-integral term encoding the cost of deviations along the trajectory, defined via a variational formula over function spaces. For instance, via a duality argument and mollified approximation (see Section 6 and especially (6.19)–(6.23)), it is shown that

$$I(X,p) = \frac{1}{2} \int_0^T \left\{ \int_{\mathbb{R}^d} \frac{|D m_s(x)|^2}{m_s(x)} dx + \int_{\mathbb{R}^d} \left| X_s - A_s(X_s, \cdot) * m_s(\cdot) - G_s(X_s, \cdot)\left[ \frac{D m_s(x)}{2 m_s(x)} - P_{s, m_s}(\cdot, X_s) \right] \right|^2 dx \right\} ds,$$

under regularity conditions on the density $m_s$ corresponding to the empirical measure $p$.

The functional $I$ captures the “penalty” for deviating both in the dynamics of $X$ and the stationary/ergodic properties of the fast system, incorporating coupling terms via $A_s$, $G_s$ (the slow–fast drift and control) and projection operators $Q_{s,m_s}$.

The explicit regular representations (e.g., Theorem 6.1, eqns (6.21)–(6.23)) link the rate function to Sobolev regularity of densities, drift-mismatch, and empirical measure fluctuations, providing a comprehensive variational structure.

3. Approximation Schemes and Regularization

A fundamental challenge is the potential irregularity of the pair $(X,p)$, particularly when $p$ lacks smooth density or when $X$ may not be sufficiently regular. To resolve this, the authors construct sequences of approximate pairs $(X^{(j)}, p^{(j)})$ with enhanced regularity (see Sections 7–8). These approximations:

• Satisfy improved Sobolev or Hölder regularity and positivity properties.
• Are built by mollifying both the controls/drifts and “test functions” over which the variational supremum in the rate function is taken.
• Are shown to converge in the relevant topology (e.g., uniform convergence on compacts, $L^2$ convergence of densities).
• Satisfy the crucial “upper-approximation” property:

$I^*(X^{(j)},p^{(j)}) \nearrow I^*(X,p)$

as $j\to\infty$.

This procedure, using Dini-type lemmas for inverse system topologies and properties of lower semicontinuity, ensures that the variational formula is valid “almost everywhere” and that the rate function is well defined, even for irregular data.

4. Explicit Representation Under Regularity Conditions

Under further assumptions—most importantly, if all invariant densities $m_s(x)$ associated with $p$ are in a Sobolev space and are bounded away from zero locally—one obtains even more explicit representations. In this case, Theorem 6.1 proves that

$$I(X,p) = \frac{1}{2} \int_0^T \left\{ \int_{\mathbb{R}^d} \frac{|D m_s(x)|^2}{m_s(x)} dx + \int_{\mathbb{R}^d} \left| X_s - J[A_s(X_s, \cdot)m_s(\cdot)] - \text{correction terms} \right|^2 dx \right\} ds,$$

with $X_s = J[A_s(X_s, \cdot)m_s(\cdot)]$ almost everywhere when $I(X,p)<\infty$. The “matching” relation between the slow variable and the invariant measure ensures the “averaging principle,” i.e., that large deviations away from the law-of-large-numbers limit (averaged behavior) incur a strictly positive rate function.

5. Lemmas for Uniform Approximation and Convergence

Multiple technical lemmas support the approximation strategy:

• Lemmas 7.1–7.4 establish uniform smoothing properties for candidate controls and approximating functions.
• Bounds derived under local Lipschitz and boundedness (Conditions 2.1, 2.2) ensure that errors in the variational representation can be made arbitrarily small.
• Uniform convergence of “cost” functionals is quantified over compact sets (see, e.g., (7.8)–(7.10)).

The diagonal selection and mollified approximations, together with upper semicontinuity results, complete the existence and identification argument for the minimizers in the variational representation.

6. Completion of the Large Deviation Argument

By combining these ingredients and conditioning on the initial state (e.g., $X_0=\hat u$), and using the known LDP for the initial condition (if it holds), Theorem 2.1 is established: the precise large deviation rate function for $(X,p)$ is given by the variational action integral above, and the LDP holds in full generality.

In summary, any irregular pair $(X,p)$ can be approximated by regular sequences for which the variational formula is valid, and thus the rate function for the coupled slow–fast diffusion is identified rigorously as the infimum (over admissible approximations) of the action-functional plus the initial cost.

7. Implications for Time-Averaged Suboptimality

The established framework quantifies the probability that the slow variable and the empirical measure of the fast component jointly realize atypical time-averaged behavior. The rate function $I(X,p)$ serves as the “cost” for observing a path-level deviation from the averaged optimality regime. Specifically:

• The “zero” cost trajectory is characterized by the averaged dynamics, where the slow variable is slaved to the fast component’s empirical measure through the invariant density relation.
• Suboptimal long-time averages (in the sense of sustained mismatch between $X_s$ and $J[A_s(X_s, \cdot)m_s(\cdot)]$) correspond to a strictly positive rate function and are exponentially unlikely in the large-deviations regime.
• The explicit representation of the rate function allows, under suitable regularity, practical computation or estimation of the decay exponent for rare events of time-averaged suboptimality.

This principle underlies the ability to establish exponential error quantifications for time-averaged costs, control performance degradation, and rare-event probabilities in high-dimensional multiscale stochastic systems.

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Overall, the large-deviation principle for time-averaged suboptimality in fully coupled slow–fast diffusions is rigorously constructed via an explicit, variationally defined rate function. Robust approximation and regularity arguments ensure that the action-functional fully characterizes the exponential tail probabilities of sustained departures from time-averaged optimality, providing a mathematically sound foundation for theoretical and applied analyses of rare-event phenomena in multiscale stochastic dynamics.