Sample-Path MDPs in Stochastic Processes

• Sample-path MDPs are a framework for quantifying deviations of entire stochastic trajectories, not just scalar endpoints.
• They employ methods such as weak convergence, exponential martingales, and control theory to derive quadratic rate functions.
• This approach unifies central limit and large deviation theories, though challenges remain in handling model-specific boundaries and path-space topologies.

A sample-path moderate deviation principle (MDP) is a path-space large deviation principle for a centered and normalized stochastic process at an intermediate scale between the central limit theorem and the full large deviation regime. In the contemporary literature, such principles are formulated for trajectories in spaces such as $D([0,T],\mathcal S'(\mathbb R^d))$, $C([0,T];L^p([0,1]))$, $C([0,T];\mathbb R^n)$, $D([0,T];\ell^2)$, and $\mathcal D([0,T],\mathbb R)$, with speeds including $a_n^2/n^d$, $a_N^2/N$, $h(\epsilon)^2$, $b_n^2$, and $\varepsilon/\lambda(\varepsilon)^2$ depending on the model class (Zhao, 2024, Xue et al., 2023, Morse et al., 2016, Budhiraja et al., 2015, Wang et al., 5 Mar 2025, Bourguin et al., 2022). Across these settings, the rate function is typically quadratic and often admits a control or skeleton representation, making the sample-path MDP the canonical intermediate-scale asymptotic theory for process-valued fluctuations.

1. Scope of the notion

A genuine sample-path MDP is process-level rather than terminal-time or one-dimensional. In the weakly asymmetric simple exclusion process (WASEP), the principle is proved for the trajectory $C([0,T];L^p([0,1]))$0 on $C([0,T];L^p([0,1]))$1 (Zhao, 2024). In the one-dimensional symmetric simple exclusion process (SSEP), the current and tagged particle satisfy sample-path MDPs in $C([0,T];L^p([0,1]))$2 (Xue et al., 2023). For slow-fast diffusions, the centered slow motion satisfies a Laplace-principle formulation on $C([0,T];L^p([0,1]))$3 (Morse et al., 2016). For weakly interacting particle systems, the empirical measure process satisfies a path-space LDP in $C([0,T];L^p([0,1]))$4 for interacting diffusions and in $C([0,T];L^p([0,1]))$5 for pure jump models (Budhiraja et al., 2015).

This usage excludes several nearby but distinct results. The Maki–Thompson rumour model proves an MDP only for the final proportion of ignorants, and explicitly states that the result is “not a sample-path MDP” (Wang et al., 9 May 2026). The Dyck-path paper studies the maximum height of a random Dyck path and emphasizes that it is a moderate deviation principle for a path functional, not for the whole path (Kousha, 2010). Good’s coverage estimator yields a pointwise MDP for a scalar statistic, not a functional or sample-path principle (Gao, 2013). The hierarchical “linear response and moderate deviations” program often treats integrals of processes or fields against boxes or test functions; these results are functional in a broad sense, but not always full path-space LDPs with an explicit topology on the underlying process space (Tsirelson, 2016, Tsirelson, 2017, Tsirelson, 2018).

A persistent misconception is therefore terminological: not every moderate deviation result involving a stochastic process is sample-path. The decisive criterion is whether the LDP is formulated for trajectories in a path space, rather than for a single-time marginal, a terminal functional, or an integrated observable.

2. Scaling regimes, path spaces, and topologies

The moderate-deviation scaling is always intermediate. In WASEP, the fluctuation field is normalized by $C([0,T];L^p([0,1]))$6 under the constraints

$C([0,T];L^p([0,1]))$7

and the MDP speed is $C([0,T];L^p([0,1]))$8 on $C([0,T];L^p([0,1]))$9 (Zhao, 2024). In the SSEP current and tagged-particle problem, the regime is

$C([0,T];\mathbb R^n)$0

with speed $C([0,T];\mathbb R^n)$1 in $C([0,T];\mathbb R^n)$2 (Xue et al., 2023). For queues with waiting-time-dependent interarrival and service times, the centered workload process

$C([0,T];\mathbb R^n)$3

satisfies an MDP in the Skorokhod $C([0,T];\mathbb R^n)$4 topology on $C([0,T];\mathbb R^n)$5 with speed $C([0,T];\mathbb R^n)$6 (Feng et al., 31 Oct 2025). For JSQ$C([0,T];\mathbb R^n)$7, the occupancy process is studied in $C([0,T];\mathbb R^n)$8 with moderate scaling $C([0,T];\mathbb R^n)$9 and speed $D([0,T];\ell^2)$0 (Wang et al., 5 Mar 2025).

Small-noise diffusions and SPDEs use analogous intermediate scalings. For the stochastic generalized Burgers–Huxley equation, the rescaled process

$D([0,T];\ell^2)$1

is studied in $D([0,T];\ell^2)$2, with $D([0,T];\ell^2)$3 and $D([0,T];\ell^2)$4 (Kumar et al., 2024). For slow-fast diffusions, the centered slow motion

$D([0,T];\ell^2)$5

satisfies a sample-path MDP on $D([0,T];\ell^2)$6 when $D([0,T];\ell^2)$7 and $D([0,T];\ell^2)$8 (Morse et al., 2016). For multiscale McKean–Vlasov SDEs, the deviation process

$D([0,T];\ell^2)$9

is treated in $\mathcal D([0,T],\mathbb R)$0 with speed $\mathcal D([0,T],\mathbb R)$1 (Hong et al., 2023). For fBm-driven multiscale systems, the analogous deviation process is

$\mathcal D([0,T],\mathbb R)$2

in $\mathcal D([0,T],\mathbb R)$3 (Bourguin et al., 2022).

The topology is model-dependent but never incidental. Diffusive hydrodynamic fields naturally live in distribution spaces (Zhao, 2024), empirical queue-length profiles in $\mathcal D([0,T],\mathbb R)$4-valued Skorokhod space (Wang et al., 5 Mar 2025), additive functionals of DDSDEs in $\mathcal D([0,T],\mathbb R)$5 rather than a path space (Ren et al., 2021), and SPDE solutions in the same function space as the underlying well-posedness theory (Kumar et al., 2024).

3. Rate functions and canonical structures

The characteristic rate function is quadratic, but its concrete realization varies.

For WASEP, the good rate function splits into dynamic and initial components,

$\mathcal D([0,T],\mathbb R)$6

with $\mathcal D([0,T],\mathbb R)$7 expressed by a variational supremum over $\mathcal D([0,T],\mathbb R)$8 and $\mathcal D([0,T],\mathbb R)$9 by a supremum over $a_n^2/n^d$0 (Zhao, 2024). When finite, the path solves a linearized heat equation with an $a_n^2/n^d$1-type forcing term. For the SGBH SPDE, the rate is

$a_n^2/n^d$2

which is the Budhiraja–Dupuis control cost associated with the linearized skeleton (Kumar et al., 2024).

In slow-fast diffusions, the rate function takes the explicit absolutely continuous form

$a_n^2/n^d$3

for $a_n^2/n^d$4 absolutely continuous, and $a_n^2/n^d$5 otherwise (Morse et al., 2016). The multi-scale McKean–Vlasov setting has two distinct regimes. In Regime 1, the rate depends only on the slow diffusion $a_n^2/n^d$6, whereas in Regime 2 it includes the additional covariance term

$a_n^2/n^d$7

through the matrix $a_n^2/n^d$8 (Hong et al., 2023). For multiscale systems driven by fBm, the rate function involves the nonlocal operator $a_n^2/n^d$9, and the paper emphasizes that this action functional is generally discontinuous in $a_N^2/N$0 at $a_N^2/N$1 (Bourguin et al., 2022).

Some rate functions are not absolutely continuous in the classical sense. For the SSEP current and tagged particle, the path-rate is

$a_N^2/N$2

where $a_N^2/N$3 is the reproducing-kernel space of fractional Brownian motion with Hurst index $a_N^2/N$4 (Xue et al., 2023). The occupation-time MDP for one-dimensional WASEP similarly identifies the path rate with the Cameron–Martin-type action of fractional Brownian motion with Hurst index $a_N^2/N$5 (Zhao, 2024).

Queueing models display another structural distinction. When the fluid equilibrium is positive, the rate function is an explicit quadratic action without reflection. When the equilibrium is zero, the rate function is expressed through the linearly generalized Skorokhod map $a_N^2/N$6, and reflection remains part of the pathwise cost (Feng et al., 31 Oct 2025).

4. Proof architecture

Two proof paradigms dominate.

The first is the weak-convergence and stochastic-control method of Budhiraja–Dupuis. It underlies the SGBH SPDE (Kumar et al., 2024), slow-fast diffusions (Morse et al., 2016), multiscale McKean–Vlasov SDEs (Hong et al., 2023), fBm-driven multiscale systems (Bourguin et al., 2022), and weakly interacting particle systems (Budhiraja et al., 2015). In this approach, one introduces controlled versions of the original dynamics, proves tightness of controlled trajectories and occupation measures, identifies the limiting skeleton equation, and derives the Laplace upper and lower bounds. Viable pairs, Poisson equations, and explicit local variational problems are recurrent technical components in multiscale models (Morse et al., 2016, Hong et al., 2023, Bourguin et al., 2022).

The second paradigm uses exponential martingales and hydrodynamic replacement theory. In WASEP, the field-level MDP is obtained from an exponential martingale; the symmetric part yields the Laplacian and the quadratic term, while the asymmetric part contributes a correction $a_N^2/N$7 that is shown to be superexponentially negligible via replacement lemmas and block estimates (Zhao, 2024). The lower bound is then handled by a tilted generator and a Girsanov change of measure.

Poisson-random-measure variational representations form a third major subroutine. They are central for JSQ$a_N^2/N$8, where the occupancy process is represented as a PRM-driven jump equation and the rate function becomes a quadratic cost over controlled intensities $a_N^2/N$9 (Wang et al., 5 Mar 2025). The same PRM machinery appears in the jump-model part of weakly interacting particle systems (Budhiraja et al., 2015).

Exponential tightness is indispensable in all path-level results. For the SSEP current and tagged particle, it is the bridge from finite-dimensional MDPs to the sample-path MDP in $h(\epsilon)^2$0 (Xue et al., 2023). For WASEP, tightness combines martingale moment bounds with control of short-time increments (Zhao, 2024). For queues, it is used together with martingale and supermartingale arguments to show that the error terms are exponentially equivalent to zero (Feng et al., 31 Oct 2025).

5. Representative model classes

The range of sample-path MDPs already established is broad.

Model class	Path space	Distinctive feature
WASEP fluctuation field (Zhao, 2024)	$h(\epsilon)^2$1	dynamic-plus-initial quadratic rate
SSEP current and tagged particle (Xue et al., 2023)	$h(\epsilon)^2$2	fractional Brownian motion $h(\epsilon)^2$3 action
SGBH equation (Kumar et al., 2024)	$h(\epsilon)^2$4	multiplicative Gaussian noise, skeleton control cost
Slow-fast diffusions (Morse et al., 2016)	$h(\epsilon)^2$5	unified averaging and homogenization
Weakly interacting particle systems (Budhiraja et al., 2015)	$h(\epsilon)^2$6, $h(\epsilon)^2$7	empirical-measure path MDP
JSQ$h(\epsilon)^2$8 occupancy process (Wang et al., 5 Mar 2025)	$h(\epsilon)^2$9	PRM-driven infinite-dimensional queueing dynamics
Waiting-time-dependent queue (Feng et al., 31 Oct 2025)	$b_n^2$0	generalized Skorokhod reflection at zero

In interacting-particle systems, the sample-path MDP often tracks empirical measures or fluctuation fields rather than finite-dimensional coordinates (Budhiraja et al., 2015, Zhao, 2024). In exclusion processes, observables such as current, occupation time, and tagged-particle displacement produce non-Markovian path-rate functions linked to fractional Brownian motion (Zhao, 2024, Xue et al., 2023). In SPDEs and small-noise multiscale systems, the rate function is usually encoded by a deterministic skeleton equation and a quadratic control energy (Kumar et al., 2024, Morse et al., 2016, Bourguin et al., 2022).

A plausible implication is that the sample-path MDP has become the natural mesoscopic analogue of both the FCLT and the path-space LDP: it preserves the pathwise geometry of the underlying model while retaining the quadratic structure characteristic of Gaussian fluctuation theory.

6. Limitations, gaps, and boundary cases

The present theory is not uniform across all models, and several papers make the limitations explicit.

First, some results stop short of a full two-sided path-space principle. Part II of the hierarchical approach for uniformly splittable random fields proves an upper moderate-deviation estimate for box integrals, not a full MDP with a matching lower bound, and explicitly records a logarithmic gap in the scale $b_n^2$1 (Tsirelson, 2017). Part I for splittable stationary processes proves a scalar MDP for long-time integrals only in the regime $b_n^2$2, and notes that the usual region $b_n^2$3 is not covered (Tsirelson, 2016).

Second, several results are functional but not fully sample-path in the strongest sense. Part IV of the hierarchical series proves moderate deviations for random fields integrated against compactly supported continuous test functions; this is close to a functional MDP for the associated random signed measure, but it is not formulated as a full LDP on a measure space (Tsirelson, 2018). The same distinction reappears in box-integral results for splittable CMS random fields (Tsirelson, 2017).

Third, model-specific boundary phenomena alter the structure of the rate. In the waiting-time-dependent queue, the positive-equilibrium and zero-equilibrium cases have different pathwise rate functions precisely because reflection disappears in the first case and persists through the generalized Skorokhod map in the second (Feng et al., 31 Oct 2025). In multiscale McKean–Vlasov systems, Regime 1 and Regime 2 have different effective covariances because the fast noise survives only in the latter (Hong et al., 2023). In the fBm-driven multiscale system, the action functional is generally discontinuous at $b_n^2$4, so the Brownian and fractional cases are not linked by a smooth parameter limit (Bourguin et al., 2022).

Finally, the boundary between sample-path and non-sample-path results remains methodologically important. Exact-distribution and combinatorial methods can prove sharp moderate-deviation asymptotics for scalar observables, as in the Maki–Thompson rumour model or Dyck-path maxima, without yielding a process-level theory (Wang et al., 9 May 2026, Kousha, 2010). Conversely, path-space MDPs typically require exponential tightness, control representations, and continuity properties of the solution map that are absent from terminal-time analyses.

In this sense, the sample-path MDP is both broader and stricter than a scalar MDP: broader because it encodes full trajectory deviations, stricter because it demands a path-space topology, a good rate function on trajectories, and a proof architecture capable of controlling the entire evolution rather than a single endpoint.