Energy solutions to SDEs with supercritical distributional drift: An extension and weak convergence rates

TL;DR

This paper extends the theory of energy solutions for supercritical distributional SDEs with divergence-free drift, establishing weak convergence rates for approximations, and shows that singularities are almost surely not visited under certain conditions.

Contribution

It advances the understanding of supercritical SDEs by extending weak well-posedness results and providing convergence rates for approximations with singular drifts.

Findings

Energy solutions are uniquely determined outside small singularity sets.

Approximation schemes converge weakly with quantifiable rates.

Singularities are almost surely not visited by the solution.

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 H\"older 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.