Conditional Flow Matching for Transport Between Markov Processes

Motivated by sequence-to-sequence transport tasks such as time-series domain adaptation, prompt and query rewriting, and machine translation, we study transport between trajectories of Markov processes. Given one source trajectory and one target trajectory, we develop a conditional flow matching algorithm that learns to transform a fresh source trajectory into a trajectory following the target dynamics. The flow conditions on the current source and generated target states, preserving the target Markov structure under recursive generation. Under suitable regularity and initialization assumptions, the population flow reproduces the target trajectory distribution. For finite-sample learning over a finite realizable velocity class, we establish an error bound that incurs a factor proportional to the mixing time compared with independent transitions, as in classical statistical problems including regression (Nagaraj et al., 2020), principal component analysis (Kumar and Sarkar, 2023), and matrix concentration (Neeman et al., 2024). We complement this result with a lower-bound construction showing that dependence on the mixing time is unavoidable, even for regular Gaussian conditional transitions. We evaluate the method on synthetic data and supervised image retrieval from electroencephalography (EEG) recordings in THINGS-EEG2 (Gifford et al., 2022). For each subject, we use 4,096 calibration images while keeping the ENIGMA encoder and projection network pretrained on other subjects frozen (Kneeland et al., 2026; Alljoined, 2026). Compared with adapting only the subject-specific temporal map, jointly training this map with a preceding conditional flow using 32 previous EEG samples raises mean top-5 retrieval accuracy across ten subjects from 43.87% to 49.05%, an 11.82% relative improvement.