Diffusions with Critical Divergence-Free Drifts: Varadhan Asymptotics and Path-Law Singularity
Abstract
We study the Euclidean Varadhan formula (EVF) and path laws for diffusions with critical divergence-free drifts in dimension d≥2. For drifts given by smooth bounded skew-symmetric stream matrices, we prove a quantitative lower bound under a sector condition with a lower-order term. Its constants depend only on the dimension and the sector condition data. By approximation, the locally uniform Euclidean lower bound holds for all divergence-free L∞tBMO−1x drifts. However, a time-dependent counterexample shows that EVF can fail even for a drift in L∞tLdx. A complementary upper bound is given under a structural decomposition condition. In particular, for every time-independent Ld,∞ drift (without any smallness assumption), and the decoupled diffusion associated with the two-dimensional vorticity equation, the EVF holds. Finally, we construct a time-independent Ld drift whose SDE has a unique strong solution from the origin and whose path law is mutually singular with the corresponding driftless Brownian law. The same diffusion satisfies EVF, showing that Brownian short-time endpoint costs can persist despite singularity of the path law.
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BibTeX
@misc{feng2026diffusions,
title = {{Diffusions with Critical Divergence-Free Drifts: Varadhan Asymptotics and Path-Law Singularity}},
author = {Jintao Feng and Guohuan Zhao},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.06341},
eprint = {2610.06341},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.06341},
}
RIS
TY - UNPB
TI - Diffusions with Critical Divergence-Free Drifts: Varadhan Asymptotics and Path-Law Singularity
AU - Jintao Feng
AU - Guohuan Zhao
PY - 2026
DA - 2026/10/05
PB - arXiv
DO - 10.48550/arxiv.2610.06341
UR - https://arxiv.org/abs/2610.06341
AB - We study the Euclidean Varadhan formula (EVF) and path laws for diffusions with critical divergence-free drifts in dimension d≥2. For drifts given by smooth bounded skew-symmetric stream matrices, we prove a quantitative lower bound under a sector condition with a lower-order term. Its constants depend only on the dimension and the sector condition data. By approximation, the locally uniform Euclidean lower bound holds for all divergence-free L^∞_tBMO^−1_x drifts. However, a time-dependent counterexample shows that EVF can fail even for a drift in L^∞_tL^d_x. A complementary upper bound is given under a structural decomposition condition. In particular, for every time-independent L^d,∞ drift (without any smallness assumption), and the decoupled diffusion associated with the two-dimensional vorticity equation, the EVF holds. Finally, we construct a time-independent L^d drift whose SDE has a unique strong solution from the origin and whose path law is mutually singular with the corresponding driftless Brownian law. The same diffusion satisfies EVF, showing that Brownian short-time endpoint costs can persist despite singularity of the path law.
ER -