Energy solutions to SDEs with supercritical distributional drift: An extension and weak convergence rates
Lukas Gräfner
TL;DR
This work tackles SDEs with supercritical distributional drift in $d\ge 2$, formulating energy solutions for $dX_t = b(t,X_t)dt + \sqrt{2}\,dB_t$ under a divergence-free condition. It extends prior super-critical well-posedness results by merging two regimes and introducing a time-dependent singular set $K$, using a stopping-time framework to prove global uniqueness and Markov properties whenever $X$ remains Hölder continuous with exponent $\alpha$ satisfying a dimension-regularity constraint on $K$. The paper also derives quantitative weak convergence rates for mollified, time-independent drifts $b^n = \rho^n * b$, with rates $n^{-\beta}$ for $\beta$ up to $1-2/p-\gamma$ when $b\in B^{-\gamma}_{p,\infty}$, and provides a semi-explicit treatment of local singularities. Overall, the results bridge PDE- and probabilistic-language in the supercritical regime and yield practical convergence estimates for approximations of the SDE. The techniques rely on a Krylov–Itô type control, Besov-space bounds, and a stopping-time argument around the singular set $K$ to achieve global well-posedness.
Abstract
In this work we consider the SDE \begin{equation} \text{d} X_t = b (t, X_t) \text{d} t + \sqrt{2} \text{d} B_t, \label{mainSDE} \end{equation} in dimension $d \geqslant 2$, where $B$ is a Brownian motion and $b : \mathbb{R}_+ \rightarrow \mathcal{S}' (\mathbb{R}^d , \mathbb{R}^d)$ is distributional, scaling super-critical and satisfies $\nabla \cdot b \equiv 0$. We partially extend the super-critical weak well-posedness result for energy solutions from [GP24] by allowing a mixture of the regularity regimes treated therein: Outside of neighbourhoods of a small (and compared to [GP24] ''time-dependent'') local singularity set $K \subset \mathbb{R}_+ \times \mathbb{R}^d$, $b$ is assumed to be in a certain supercritical $L^q_T H^{s, p}$-type class that allows a direct link between the PDE and the energy solution from a-priori estimates up to the stopping time of visiting $K$. To establish this correspondence, and thus uniqueness, globally in time we then show that $K$ is actually never visited which requires us to impose a relation between the dimension of $K$ and the Hölder regularity of $X$. In the second part of this work we derive weak convergence rates for approximations of the above equation in the case of time-independent drift, in particular with local singularities as above.