Based on joint works with Leonid Mytnik (Technion - Israel Institute of Technology) and Konstantinos Dareiotis (University of Leeds). We consider the stochastic differential equation
$$
dX_t = b(X_t) dt + dB_t^H,
$$
where the drift $b$ is a Schwartz distribution in the space $\mathcal{C}^\alpha$, $\alpha < 0$, and $B^H$ is a fractional Brownian motion of Hurst index $H \in (0, 1/2]$. If $H = 1/2$, both weak and strong uniqueness theories for this SDE have been developed. However, the situation is much more complicated if $H < 1/2$, as the main tool, the Zvonkin transformation, becomes unavailable in this setting. The breakthroughs by Catellier and Gubinelli, and later by Le, established strong well-posedness of this SDE via sewing/stochastic sewing arguments. However, weak uniqueness for this SDE remained a challenge for quite some time, since a direct application of stochastic sewing alone does not seem very fruitful. I will explain how a combination of stochastic sewing with certain arguments from ergodic theory allows to show weak uniqueness in the whole regime where weak existence is known, that is, $\alpha > 1/2 - 1/(2H)$. If time permits, we will discuss weak uniqueness for rough SDEs
$$
d X_t = \sigma(X_t) d B_t^H,
$$
where $\sigma$ is a Hölder continuous (but not necessarily Lipschitz!) function.
[1] O. Butkovsky, L. Mytnik (2024). Weak uniqueness for singular stochastic equations. arXiv preprint arXiv:2405.13780.
