We study questions of existence and uniqueness of weak and strong solutions for a one-sided Tanaka equation with constant drift lambda. We observe a dichotomy in terms of the values of the drift parameter: for $\lambda\leq 0$, there exists a strong solution which is pathwise unique, thus also unique in distribution; whereas for $\lambda > 0$, the equation has a unique in distribution weak solution, but no strong solution (and not even a weak solution that spends zero time at the origin). We also show that strength and pathwise uniqueness are restored to the equation via suitable ``Brownian perturbations".
Ioannis Karatzas. Albert Shiryaev. Mykhaylo Shkolnikov. "On the one-sided Tanaka equation with drift." Electron. Commun. Probab. 16 664 - 677, 2011. https://doi.org/10.1214/ECP.v16-1665