Curvature-Dimension Conditions Along Discrete Refinement Sequences
Abstract
We introduce the sequence curvature--dimension condition SCD(K,N) for complete refinement systems of finite metric probability spaces. The condition couples a transport action bound with entropy comparisons along a single probability section through the prescribed refinement layers. Within a bounded comparison range, we prove that midpoint comparisons on three consecutive layers characterize the condition on every fixed finite segment. If, in addition, the refinement sequence converges in the measured sense in its intrinsic bridge realization, then SCD(K,N) is equivalent both to the vanishing of single-layer midpoint defects and to the entropic curvature--dimension condition CDe(K,N) of the limit. In the reverse direction, a CDe(K,N) bound on the limit yields quantitative recovery estimates that identify shrinking transport scales on which finite layers retain its second-order curvature comparison. These estimates are uniform over endpoint measures and include Dirac masses. Examples illustrate the effects of the refinement rule, the layer metrics and the reference measures.
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BibTeX
@misc{tao2026curvaturedimension,
title = {{Curvature-Dimension Conditions Along Discrete Refinement Sequences}},
author = {Chen Tao},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.38538},
eprint = {2610.38538},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.38538},
}
RIS
TY - UNPB
TI - Curvature-Dimension Conditions Along Discrete Refinement Sequences
AU - Chen Tao
PY - 2026
DA - 2026/10/28
PB - arXiv
DO - 10.48550/arxiv.2610.38538
UR - https://arxiv.org/abs/2610.38538
AB - We introduce the sequence curvature--dimension condition SCD(K,N) for complete refinement systems of finite metric probability spaces. The condition couples a transport action bound with entropy comparisons along a single probability section through the prescribed refinement layers. Within a bounded comparison range, we prove that midpoint comparisons on three consecutive layers characterize the condition on every fixed finite segment. If, in addition, the refinement sequence converges in the measured sense in its intrinsic bridge realization, then SCD(K,N) is equivalent both to the vanishing of single-layer midpoint defects and to the entropic curvature--dimension condition CD^e(K,N) of the limit. In the reverse direction, a CD^e(K,N) bound on the limit yields quantitative recovery estimates that identify shrinking transport scales on which finite layers retain its second-order curvature comparison. These estimates are uniform over endpoint measures and include Dirac masses. Examples illustrate the effects of the refinement rule, the layer metrics and the reference measures.
ER -