Displacement Moments and Path–Terminal Separation in Brownian Interpolant Sampling

Zhong Han

Submitted by Zhong Han

Published November 8, 2026 · Version v1, November 8, 2026 · DOI 10.66977/xsci.2611.000y

Machine Learning

Abstract

We prove early-stopped, target-to-output KL guarantees for Brownian interpolant Euler sampling from the true source under arbitrary Borel couplings, even with nonintegrable endpoints. At fixed stopping time and noise level, a fourth displacement moment yields an $O(N^{-1})$ discretization term. A second moment suffices for fixed-law convergence with grid refinement and vanishing drift error. For one fixed coupling with no displacement moment above two, the original and comparison paths have KL of orders one and $1/\log N$ relative to the exact-drift Euler path. On the same geometric grid, terminal KL is at most $C/N^2$. The path decomposition identifies the information about the source discarded by Markov projection. When the original bridge is Markov, current-state Euler sampling has an $O((M_2+d)/N)$ discretization term. We prove expected stopped-output KL consistency for a learner trained on iid endpoint pairs with fresh true-source initialization. In a spatial image control, iterative sampling has worse fair energy scores than direct prediction with resampled residuals. Uniform terminal rates for general second-moment couplings remain open.

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