Connecting Brownian and Poisson Random Bridges with Rectified Flows

Abstract

This paper connects rectified flows from generative modelling with a family of Markov random bridges acting as stochastic transports between pairs of distributions. More precisely, using the non-anticipative semimartingale representation of Lévy random bridges, we provide a unified framework with flow matching through the corresponding stochastic and ordinary differential equations. For a more detailed account, we focus on Brownian random bridges and Poisson random bridges, both of which we consider to be canonical constructs with purely continuous and purely discontinuous paths, respectively. Finally, we present numerical training and simulation algorithms followed by generative examples for demonstration.

Summary

This paper establishes a rigorous mathematical bridge between rectified flows — a deterministic ODE-based generative modelling framework — and Lévy random bridges, which are stochastic processes conditioned to transport between two distributions. The central insight is that for any Lévy driving process, the conditional expectation of the random bridge drift takes the identical functional form to the optimal rectified flow velocity field: E[Y | ξ_t] - ξ_t / (T - t). This holds universally across all Lévy processes with finite first moments.

The paper derives detailed results for two canonical cases. For Brownian random bridges (purely continuous paths), the bridge decomposes as rectified flow plus a standard Brownian bridge, and the probability flow ODE reveals a “half-speed” rectified flow tempered by source-dependent drift — the arithmetic mean of a source-push and target-pull velocity. Anderson’s reverse-time SDE is derived, and Nelson’s osmotic velocity identity emerges naturally. For Poisson random bridges (purely discontinuous paths), the bridge follows a product-binomial distribution, and the drift equals the intensity process of the counting process. A generalized Tweedie’s formula for the generalized exponential family is derived to handle the Poisson case.

The paper also provides practical training and simulation algorithms for the Poisson bridge, and sketches the connection to Iterative Proportional Fitting, showing that one IPF half-iteration recovers the random bridge drift.

Key Contributions

• Proves that the rectified flow velocity field and the random bridge conditional drift share the same functional form for any Lévy driving process
• Shows rectified flow is the zero-volatility limit of the Gaussian random bridge
• Derives the probability flow ODE for Gaussian random bridges as an average of source-push and target-pull velocities
• Establishes E[ξ_t^{x,y}] = R_t^{x,y} universally across all Lévy drivers
• Extends the framework to Poisson random bridges with product-binomial distributions
• Derives a generalized Tweedie’s formula for the generalized exponential family
• Connects Nelson’s osmotic velocity identity to the forward/backward bridge drift discrepancy
• Provides training and simulation algorithms for discrete (Poisson) bridges
• Sketches the connection between one IPF half-iteration and the random bridge drift

Methodology

The mathematical framework builds on the (Φ, Ψ)-bridge definition from Goria et al. (2025), using non-anticipative semimartingale representations of Lévy random bridges. The key equation is the drift representation: dξ_t^x = ∫₀ᵗ (E[Y | ξ_s^x] - ξ_s^x)/(T-s) ds + M_t, where M_t is a martingale. For the Brownian case, the anticipative representation ξ_t = (t/T)Y + σ(W_t - (t/T)W_T) + ((T-t)/T)X connects directly to stochastic interpolation. For the Poisson case, Doob’s h-transform gives transition probabilities via f_t(z-x)f_{T-t}(y-z)/f_T(y-x). Training uses MSE regression of a neural network on E[Y | ξ_t], with bridge states sampled from the known conditional distributions.

Key Findings

• The bridge probability flow ODE is dξ_t = ½[(ξ_t - x)/t + (E[Y|ξ_t] - ξ_t)/(T-t)]dt, exposing the dual source-push / target-pull structure
• Anderson’s reverse-time SDE for the Gaussian bridge simplifies to dξ̄_t = (ξ̄_t - x)/t dt̄ + σdW̄_t due to exact score cancellation
• The Poisson bridge intensity λ_{tT} = (E[Y|ξ_t] - ξ_t)/(T-t) equals the rectified flow velocity
• Temporal consistency: E[ξ_u | ξ_t] - ξ_t / (u-t) = E[Y|ξ_t] - ξ_t / (T-t) for any u ∈ (t,T]
• One half-iteration of IPF with zero-drift backward pass recovers the random bridge forward drift

Important References

1. Random-Bridges as Stochastic Transports for Generative Models — Foundation paper defining (Φ,Ψ)-bridges and their properties under Lévy drivers
2. Flow Straight and Fast — Rectified flow framework for learning ODE transports along straight paths
3. Lévy Random Bridges and the Modelling of Financial Information — Original Lévy random bridge theory by Hoyle, Hughston, Macrina

Atomic Notes

• Lévy random bridge
• Brownian random bridge
• Poisson random bridge
• non-anticipative semimartingale representation
• probability flow ODE
• Nelson’s osmotic velocity
• generalized Tweedie’s formula

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