# Conditional Flow Matching for Transport Between Markov Processes

arXiv: https://arxiv.org/abs/2610.07229 · HTML: https://mlsanity.com/paper/2610.07229/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Placed in Flow matching, time series and science, one of 14 directions in Diffusion models.
- Builds on 7 of its 53 references that are in the map.
- First posted on arXiv 5 Oct 2026.

## Placed in

- Diffusion models / [Flow matching, time series and science](https://mlsanity.com/diffusion/flow-matching-time-series-and-science/)

## Builds on (most cited first, 7 of 7)

- [Rectified Flow: A Marginal Preserving Approach to Optimal Transport](https://mlsanity.com/paper/2209.14577/) (2022)
- [Perturbation theory for Markov chains via Wasserstein distance](https://mlsanity.com/paper/1503.04123/) (2015)
- [Low-Dimensional Adaptation of Rectified Flow: A Diffusion and Stochastic Localization Perspective](https://mlsanity.com/paper/2601.15500/) (2026)
- [Is Flow Matching Just Trajectory Replay for Sequential Data?](https://mlsanity.com/paper/2602.08318/) (2026)
- [Generative Modeling with Continuous Flows: Sample Complexity of Flow Matching](https://mlsanity.com/paper/2512.01286/) (2025)
- [ENIGMA: EEG-to-Image in 15 Minutes Using Less Than 1% of the Parameters](https://mlsanity.com/paper/2602.10361/) (2026)
- [SCORE: Subject Coordinate Recovery for Label-Free Cross-Subject EEG-to-Image Retrieval](https://mlsanity.com/paper/2608.19134/) (2026)

## Abstract (arXiv)

Motivated by sequence-to-sequence transport in the context time-series domain adaptation, we study the problem of transportation between trajectories of Markov processes. Given a limited number of trajectories from source distribution and the target distribution, we formulate a flow matching based algorithm which learns a transport map from the source to target trajectory distribution, while preserving the Markov structure. We show that this is consistent in the population limit and derive finite-sample error bounds under mixing time assumptions, following the analysis of classical statistical problems including regression (Nagaraj et al., 2020), principal component analysis (Kumar and Sarkar, 2023), and matrix concentration (Neeman et al., 2024) in the Markov setting. We complement that with a lower-bound construction showing that a mixing-time dependent sample complexity is unavoidable even with regular Gaussian conditional transitions. We evaluate on synthetic and real-world data. For image retrieval from electroencephalography (EEG) on THINGS-EEG2 (Gifford et al., 2022), the task is to identify the viewed image from EEG signals captured from human subjects, which suffers from high inter subject variability. We augment the ENIGMA decoder (Kneeland et al., 2026) with a conditional flow before its subject-specific temporal map. This improves mean top-5 retrieval accuracy from 43.87% to 49.05%, an 11.82% relative improvement.