Continuous Horizon Conditioning

• Continuous horizon conditioning is a framework that models, samples, and controls stochastic processes under time-extensive constraints, extending classical endpoint methods.
• It employs a generalized Doob–h-transform and variational principles to enforce pathwise constraints using tilting functions that encode survival probabilities and functional targets.
• The approach supports applications in physical forecasting, rare-event simulation, and modern ML architectures by enabling continuous-time querying and flexible control strategies.

Continuous horizon conditioning is a theoretical and algorithmic framework for modeling, sampling, or controlling stochastic processes, dynamical systems, and time-dependent learning architectures subject to constraints or conditioning targets that are specified across an entire finite or infinite time interval—rather than only at isolated endpoints. Unlike traditional bridge constructions, which fix initial and terminal states, continuous horizon conditioning enables pathwise enforcement of survival probabilities, absorption laws, functional observables, and arbitrary future constraints, supporting applications ranging from physical forecasting to rare-event simulation and parameterized control.

1. Mathematical Foundations of Continuous Horizon Conditioning

At the core of continuous horizon conditioning is the construction of a new, pathwise probability measure on a stochastic process or dynamical system, such that specified statistical or path constraints are met throughout a time interval $[0,T]$ or potentially up to $T \to \infty$. This is achieved via a generalized Doob–$h$-transform, in which a non-negative “tilting” function $Q_T(x,t)$ is synthesized to encode the desired horizon-wide constraints.

Consider a diffusion process $X_t$ on $\mathbb{R}^d$ with generator $$\mathcal{F} = \mu(x) \cdot \nabla + D(x) \Delta - k(x)$$, potentially with killing or absorption, and let $P(x,t|x_0,0)$ denote its transition density. To impose a final-time constraint such as survival at $T$ ($P^*(y,T)$) as well as a continuous-time constraint (e.g., a killing-time law $T \to \infty$0), one constructs

$T \to \infty$1

and defines the conditioned process as

$T \to \infty$2

with drift and killing rates

$T \to \infty$3

This explicit construction is agnostic to the constraint detail: $T \to \infty$4 can encode boundary conditions, first-passage targets, time-additive observables, or path functionals (Mazzolo et al., 2022, Mazzolo et al., 2022, Mazzolo et al., 2022).

For infinite-horizon or quasi-stationary conditioning ($T \to \infty$5), $T \to \infty$6 converges to a stationary tilting function $T \to \infty$7 determined by the persistent survival or killing statistics, e.g.,

$T \to \infty$8

where $T \to \infty$9 is the unconditioned survival probability (Mazzolo et al., 2022).

2. Variational and Large Deviations Principles

Continuous-horizon conditioning can be interpreted as a solution to a dynamical large deviations problem at Level 2.5: among all possible pathwise evolutions $h$0 compatible with the process’s continuity equation (Fokker–Planck or Kolmogorov forward dynamics) and the imposed conditioning constraints, select the process law that minimizes the relative entropy (or rate functional)

$h$1

subject to the conservation law and specified marginals, absorption, or functional constraints (Mazzolo et al., 2022, Mazzolo et al., 2022, Mazzolo et al., 2022).

This connection to the Schrödinger bridge and optimal transport theory also provides the control interpretation: the optimal conditioned drift is the minimizer of the Kullback–Leibler cost over admissible steering policies, while the solution can be equivalently characterized by a coupled system of forward (Fokker–Planck) and backward (Hamilton–Jacobi–Bellman) PDEs (Achdou et al., 2019).

3. Algorithms and Frameworks Utilizing Continuous Horizon Conditioning

Continuous horizon conditioning has led to several algorithmic and deep learning frameworks, especially in contexts demanding flexible, arbitrarily-timed queries or functional constraints:

a. Neural ODE Emulation for Mesh-based Forecasting

In COGENT, the forecast trajectory of a spatially structured system is modeled as a node-wise latent Neural ODE, driven jointly by interpolated future forcings, a learned history context, and the continuous relative time variable $h$2, allowing direct querying at arbitrary future times and breaking the constraint of fixed step sizes. The conditioning machinery is as follows (Liu et al., 9 Jun 2026):

• Graph-based history encoder generates context $h$3 from past states/forcings.
• Initial latent $h$4 encodes the context, last observed state, and static features.
• Non-autonomous latent ODE $h$5 evolves dynamics, where $h$6 is the explicit “lead time” since the last history, normalized.
• At each ODE evaluation, $h$7 is tiled per node and concatenated as a feature, enabling time-dependent vector fields.
• Rollout-horizon sampling mechanisms expose the model to a variety of effective forecast horizons.

This enables direct continuous-horizon queries and avoids temporal drift plaguing autoregressive rollouts.

b. Graph Forecasting with Ephemeris Conditioning

IonoDGNN uses “ephemeris conditioning” to forecast ionospheric irregularities on dynamically changing graphs corresponding to GNSS links at times $h$8. The model conditions predictions on the entire future graph sequence $h$9 and node-wise future ephemeris-derived features, not just scalars (Turkmen et al., 20 Apr 2026):

• History encoding uses observed and ephemeris-derived features ($Q_T(x,t)$0) for $Q_T(x,t)$1.
• For $Q_T(x,t)$2 (forecast times), $Q_T(x,t)$3 and $Q_T(x,t)$4 are fully specified from known satellite trajectories, even for new nodes (satellites entering view only in the future).
• The full future graph structure and node/edge covariates are injected into the prediction GNN, providing a continuous horizon conditioning effect across variable spatiotemporal domains.

c. Diffusion Models with Timestamp Conditioning

In video prediction, a diffusion model can be trained to condition not only on past context frames but also on a parameterized, continuous timestamp $Q_T(x,t)$5, representing the target forecast offset in continuous time (Khurana et al., 2024):

• Context: $Q_T(x,t)$6, with corresponding $Q_T(x,t)$7.
• Conditioning: Continuous $Q_T(x,t)$8 is mapped to a sin-cos embedding and cross-attended into the diffusion UNet architecture, so the denoising process at each reverse step explicitly incorporates the future target time.
• Direct (non-AR) sampling: At test time, reverse diffusion is run independently for each desired $Q_T(x,t)$9, enabling non-recursive, horizon-agnostic queries.

Explicit timestamp conditioning allows for direct sampling at arbitrary time horizons and enables flexible new strategies (e.g., blended direct/autoregressive sampling) that outperform standard AR/HC approaches for long-range predictions.

4. Stochastic Process Conditioning: Doob–$X_t$0-Transform and Extensions

The canonical mathematical apparatus for continuous-horizon conditioning is the Doob–$X_t$1-transform. For a general Markov process $X_t$2 and harmonic function $X_t$3 (often the conditional future likelihood), the transformed infinitesimal generator is

$X_t$4

where $X_t$5 is chosen (or approximated) to enforce terminal, functional, or pathwise constraints (Corstanje et al., 2021). When $X_t$6 is intractable, a surrogate $X_t$7 can be used to define a guided process, with explicit likelihood ratios connecting the guide and target laws, and both exact and unbiased approximate sampling possible under technical conditions.

Generalizations include:

• Diffusion bridges, finite/infinite-horizon absorption or survival laws, interface or first-encounter properties (e.g., for multiple interacting diffusions (Mazzolo et al., 2022))
• Non-diffusive contexts, e.g., discrete-state Markov jump processes, via extension of the generator approach.

In every case, pathwise conditioning is represented by a time-dependent (or time-independent, in the infinite-horizon limit) tilting function $X_t$8 or $X_t$9, determining the transformation of the drift and any killing or control term.

5. Applications Across Scientific and Engineering Domains

Continuous-horizon conditioning is applied in various fields, driven by its ability to model, forecast, or simulate under dense, temporally extended constraints:

• Physical simulation emulation: Emulating mesh-based physical systems (e.g., ice-sheet flow) with Neural ODEs using continuous-horizon conditioning on lead time, future forcings, and context (Liu et al., 9 Jun 2026).
• Dynamic graph forecasting: Prediction on dynamic spatial graphs where future sampling topology (satellite orbits, sensor layout) is known, by conditioning directly on the full forecast-window topology and node features (Turkmen et al., 20 Apr 2026).
• Video and trajectory generation: Diffusion-based video and trajectory models that can output at arbitrary future times, leveraging timestamp conditioning for sampling flexibility (Khurana et al., 2024).
• Stochastic control and rare-event simulation: Optimal path sampling or feedback control for survival, first-passage, and absorption constraints over finite/infinite horizons (Mazzolo et al., 2022, Monthus et al., 2022, Achdou et al., 2019).

An illustrative set of method-application correspondences is given in the following table:

Method/Framework	Conditioning Mechanism	Primary Domain
Neural ODEs (COGENT)	Lead-time $\mathbb{R}^d$0 as continuous feature	Geospatial mesh forecasting
IonoDGNN	Ephemeris-conditioned future graphs	Ionospheric GNSS prediction
Diff. Video Models	Timestamp sincos embedding	Video trajectory prediction
Doob–$\mathbb{R}^d$1-transform	$\mathbb{R}^d$2/$\mathbb{R}^d$3 tilting	SDEs, Markov processes

6. Infinite-Horizon Limits and Quasi-Stationary Conditioning

A distinctive aspect of continuous horizon approaches is the systematic treatment of the infinite-horizon ($\mathbb{R}^d$4) limit, crucial for quasi-stationary, ergodic, or rare-event regimes. For diffusions or jump processes, the large-$\mathbb{R}^d$5 form of $\mathbb{R}^d$6 yields a stationary tilting, driving the process conditioned on e.g. non-absorption, fixed long-time empirical observables, or prescribed rare events (Mazzolo et al., 2022, Mazzolo et al., 2022). In this regime:

• The h-transform kernel $\mathbb{R}^d$7 is time-invariant.
• Stationary conditioned dynamics correspond to quasi-stationary distributions (Q-processes).
• The rate functional reduces to the large-deviation “cost” of sustaining the target constraint over infinite time, and the principal eigenfunction of a deformed generator often governs path statistics.

Explicit formulas clarify limiting drift structures: e.g. for taboo or survival conditioning, the drift near an absorbing boundary universalizes to forms like $\mathbb{R}^d$8, as seen in generalizations from Brownian and tanh-drift processes (François-Élie et al., 27 Mar 2026).

7. Structural and Identifiability Aspects: Conditioning in Parametric and Learning Models

In regularized regression and system identification, continuous-horizon conditioning is used to orthogonalize basis functions over variable time (or maturity) horizons, improving numerical stability and parameter identifiability. In the Nelson–Siegel–Svensson (NSS) curve model, the continuous-horizon Gramian is constructed over $\mathbb{R}^d$9, yielding orthogonal bases, exact diagonal Fisher matrices conditional on decay parameters, and explicit diagnostics for model reduction as a function of horizon and parameter collinearity (Flassig et al., 21 Apr 2026).

This principle extends to time-dependent control and filtering, where block-Toeplitz preconditioners for predictive control exploit the horizon-independence of spectral symbols to achieve uniform, horizon-independent conditioning and accelerate optimization algorithms regardless of rollout length (McInerney et al., 2020).

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In summary, continuous horizon conditioning unifies a suite of theoretical and practical tools for steering, sampling, or modeling time-evolving systems under temporally extended or functional probability constraints. It generalizes classical bridge and endpoint conditioning to arbitrary time windows, supports modern machine learning architectures for spatiotemporal prediction, clarifies the structure and limits of quasi-stationary processes, and provides rigorously grounded algorithms for stochastic control, simulation, and inference across stochastic, dynamical, and data-driven contexts (Liu et al., 9 Jun 2026, Turkmen et al., 20 Apr 2026, Mazzolo et al., 2022, Mazzolo et al., 2022, Khurana et al., 2024).