Stochastic Differential Equations with Local Growth Singular Drifts

TL;DR

This paper investigates the properties of solutions to stochastic differential equations with singular drifts and diffusion coefficients, establishing weak differentiability, the strong Feller property, and stochastic flow characteristics under specific growth conditions.

Contribution

It introduces new conditions on singular drifts and diffusion gradients ensuring key regularity and flow properties of solutions to SDEs, extending previous results to more singular cases.

Findings

Proved weak differentiability of solutions under logarithmic growth conditions.

Established the strong Feller property for the associated diffusion semigroups.

Demonstrated the global stochastic flow property for SDEs with singular coefficients.

Abstract

In this paper, we study the weak differentiability of global strong solution of stochastic differential equations, the strong Feller property of the associated diffusion semigroups and the global stochastic flow property in which the singular drift $b$ and the weak gradient of Sobolev diffusion $\sigma$ are supposed to satisfy $||b(x){1}_{|x|\le R}||_{p_1}\le O((\log R)^{{(p_1-d)^2}/{2p^2_1}})$ and $||\nabla \sigma(x)1_{|x|\le R} ||_{p_1} \le O((\log ({R}/{3}))^{{(p_1-d)^2}/{2p^2_1}})$ respectively. The main tools for these results are the decomposition of global two-point motions, Krylov's estimate, Khasminskii's estimate, Zvonkin's transformation and the characterization for Sobolev differentiability of random fields.