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

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 $d⩾2$, where $B$ is a Brownian motion and $b:{\mathbb{R}}_{+}\to {\mathcal{S}}^{\prime }({\mathbb{R}}^d,{\mathbb{R}}^d)$ 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 $L_T^q{H}^{s,p}$-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.

20 pages