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 $$ \frac{\partial}{\partial t} u_t(x)=\frac12 \frac{\partial^2}{\partial x^2}u_t(x)+b(u_t(x))+\dot{W}_{t}(x), \quad t>0,\, x\in D\subset\mathbb{R}, $$ and multidimensional stochastic differential equation (SDE) driven by fractional Brownian motion with the Hurst index $H\in(0,1/2)$ $$ d X_t=b(X_t) dt +d B_t^H,\quad t>0. $$ In both cases $b$ is a generalized function in the Besov space $\mathcal{B}^\alpha_{\infty,\infty}$, $\alpha<0$. Well-known pathwise uniqueness results for these equations do not cover the entire range of the parameter $\alpha$, for which weak existence holds. What happens in the range where weak existence holds but pathwise uniqueness is unknown has been an open problem. We settle this problem and show that for SHE weak uniqueness holds for $\alpha>-3/2$, and for SDE it holds for $\alpha>1/2-1/(2H)$; thus, in both cases, it holds in the entire desired range of values of $\alpha$. 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.