Well-Posedness for SDEs with Logarithmical Critical Distributional Drifts
Abstract
We study the stochastic differential equation d Xt=b(t,Xt)d t+√2d Wt on ℝd, where b is a time-dependent, divergence-free distributional drift of critical Hölder--Besov regularity −1, strengthened by an iterated-logarithmic correction. For every initial probability law, we construct a weak solution by smooth approximation and realize the singular drift as an additive functional. The main analytic ingredient is the Schauder estimate with a logarithmic smallness factor. Combined with uniform logarithmic Krylov estimates and a stochastic substitution formula for distributional test functions, this estimate allows us to apply a Zvonkin transformation and prove uniqueness in law among weak solutions satisfying the corresponding Krylov bounds. For solutions starting from deterministic points, we further show that their time-marginal distributions admit densities satisfying two-sided Aronson-type Gaussian estimates.
Related
- Global well-posedness and scattering for mass-critical Hartree equation
- Global well-posedness for the energy-critical focusing NLS on 𝕋3
- Well-posedness of the two-dimensional unsteady Prandtl system in Sobolev space with degenerate critical points
- Impact Mechanism of a Point Interaction in the Cubic Half-Wave Equation: Well-Posedness, Constrained Energy Levels and Near-Critical Rigidity
- Global well-posedness and scattering of energy-critical defocusing nonlinear Schrödinger equation on high-dimensional hyperbolic spaces
BibTeX
@misc{chen2026wellposedness,
title = {{Well-Posedness for SDEs with Logarithmical Critical Distributional Drifts}},
author = {Zikai Chen and Zimo Hao and Xicheng Zhang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.02022},
eprint = {2609.02022},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.02022},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Well-Posedness for SDEs with Logarithmical Critical Distributional Drifts
AU - Zikai Chen
AU - Zimo Hao
AU - Xicheng Zhang
PY - 2026
DA - 2026/09/02
PB - arXiv
DO - 10.48550/arxiv.2609.02022
UR - https://arxiv.org/abs/2609.02022
AB - We study the stochastic differential equation d X_t=b(t,X_t)d t+√2d W_t on ℝ^d, where b is a time-dependent, divergence-free distributional drift of critical Hölder--Besov regularity −1, strengthened by an iterated-logarithmic correction. For every initial probability law, we construct a weak solution by smooth approximation and realize the singular drift as an additive functional. The main analytic ingredient is the Schauder estimate with a logarithmic smallness factor. Combined with uniform logarithmic Krylov estimates and a stochastic substitution formula for distributional test functions, this estimate allows us to apply a Zvonkin transformation and prove uniqueness in law among weak solutions satisfying the corresponding Krylov bounds. For solutions starting from deterministic points, we further show that their time-marginal distributions admit densities satisfying two-sided Aronson-type Gaussian estimates.
ER -