Weak uniqueness for singular stochastic equations

We put forward a new method for proving weak uniqueness of stochastic equations with singular drifts driven by a non-Markov or infinite-dimensional noise. We apply our method to study stochastic heat equation (SHE) driven by Gaussian space-time white noise t uₜ (x) =12 ² x²uₜ (x) +b (uₜ (x) ) +Ẇₓ (x), t>0, \, x D, and multidimensional stochastic differential equation (SDE) driven by fractional Brownian motion with the Hurst index H (0, 1/2) d Xₜ=b (Xₜ) dt +d BₜH, t>0. In both cases b is a generalized function in the Besov space B^, , -3/2, and for SDE it holds for >1/2-1/ (2H) ; thus, in both cases, it holds in the entire desired range of values of. This extends seminal results of Catellier and Gubinelli (2016) and Gy\"ongy and Pardoux (1993) to the weak well-posedness setting. To establish these results, we develop a new strategy, combining ideas from ergodic theory (generalized couplings of Hairer-Mattingly-Kulik-Scheutzow) with stochastic sewing of L\ᵉ.