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The exponential contraction in $L^1$-Wasserstein distance and exponentialconvergence in $L^q$-Wasserstein distance ($q\geq 1$) are considered forstochastic differential equations with irregular drift. When the irregulardrift drift is locally bounded, the exponential convergence are derived byusing the reflection coupling with a new auxiliary function for stochasticdifferential equations driven by multiplicative noise. Explicit convergencerate is obtained. When the irregular drift is not locally bounded, a newZvonkin's transformation is given by an ultracontractive reference diffusion,and the exponential convergence is derived by combining the Zvonkin'stransformation and related results for stochastic differential equations withlocally bounded irregular drift.

Exponential convergence in Wasserstein Distance for Diffusion Semigroups with Irregular Drifts