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