Displacement Moments and Path-Terminal Separation in Brownian Interpolant Sampling
Abstract
Brownian stochastic interpolants connect paired endpoint samples, yet their sampling error can depend on displacement moments even when neither endpoint has a first moment. For arbitrary Borel couplings, we prove early-stopped, target-to-output KL guarantees for standard Euler sampling from the true source. A fourth displacement moment gives an explicit O(N−1) bound; a second moment suffices for fixed-law convergence as drift error vanishes. For one fixed coupling with no displacement moment above two, original bridge-path KL stays of order one, comparison-path KL is of order 1/log N, and terminal KL is at most C/N2 on the same grid. The path decomposition identifies the information about the source discarded by Markov projection. When the original bridge is Markov, current-state Euler sampling instead has a second-moment O((M2+d)/N) bound; finite-entropy Schroedinger bridges for the probability Brownian reference are included. We also construct a current-state learner with expected output KL consistency from iid endpoint pairs. Numerical diagnostics illustrate the distinct errors. Conditional diffusion identities specify a restoration interface, while image controls find no iterative advantage over cheap probabilistic predictors. Uniform terminal rates for general second-moment couplings remain open.
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BibTeX
@misc{han2026displacement,
title = {{Displacement Moments and Path-Terminal Separation in Brownian Interpolant Sampling}},
author = {Zhong Han},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.37413},
eprint = {2610.37413},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.37413},
}
RIS
TY - UNPB
TI - Displacement Moments and Path-Terminal Separation in Brownian Interpolant Sampling
AU - Zhong Han
PY - 2026
DA - 2026/10/27
PB - arXiv
DO - 10.48550/arxiv.2610.37413
UR - https://arxiv.org/abs/2610.37413
AB - Brownian stochastic interpolants connect paired endpoint samples, yet their sampling error can depend on displacement moments even when neither endpoint has a first moment. For arbitrary Borel couplings, we prove early-stopped, target-to-output KL guarantees for standard Euler sampling from the true source. A fourth displacement moment gives an explicit O(N^−1) bound; a second moment suffices for fixed-law convergence as drift error vanishes. For one fixed coupling with no displacement moment above two, original bridge-path KL stays of order one, comparison-path KL is of order 1/log N, and terminal KL is at most C/N^2 on the same grid. The path decomposition identifies the information about the source discarded by Markov projection. When the original bridge is Markov, current-state Euler sampling instead has a second-moment O((M_2+d)/N) bound; finite-entropy Schroedinger bridges for the probability Brownian reference are included. We also construct a current-state learner with expected output KL consistency from iid endpoint pairs. Numerical diagnostics illustrate the distinct errors. Conditional diffusion identities specify a restoration interface, while image controls find no iterative advantage over cheap probabilistic predictors. Uniform terminal rates for general second-moment couplings remain open.
ER -