Weak uniqueness for singular stochastic equations
Oleg Butkovsky, Leonid Mytnik
TL;DR
The paper advances weak well-posedness for stochastic equations with singular, distributional drifts driven by non-Markovian or infinite-dimensional noises by marrying stochastic sewing with generalized couplings. The authors establish weak uniqueness for a 1D stochastic heat equation with drift in $\mathcal{B}^\alpha_{\infty,\infty}$ when $\alpha>-3/2$ and prove weak uniqueness for SDEs driven by fractional Brownian motion with $H\in(0,1/2]$ when $\alpha>1/2-1/(2H)$, aligning with the known weak existence ranges. The key methodological novelty is the Control-and-Reimburse strategy that integrates ergodic-theoretic generalized couplings with stochastic sewing, enabling weak well-posedness beyond the regime of strong uniqueness. In addition, the paper provides strong existence and uniqueness results in dimension $d=1$ under refined drift conditions, thereby extending Catellier-Gubinelli and Gyöngy-Pardoux results to the weak setting and offering a robust framework potentially applicable to broader rough-noise SPDE/SDE problems.
Abstract
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 $\mathcal{B}^α_{\infty,\infty}$, $α<0$. Well-known pathwise uniqueness results for these equations do not cover the entire range of the parameter $α$, 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 $α>-3/2$, and for SDE it holds for $α>1/2-1/(2H)$; thus, in both cases, it holds in the entire desired range of values of $α$. This extends seminal results of Catellier and Gubinelli (2016) and Gyöngy 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 Lê.