Quantitative heat kernel estimates for diffusions with distributional drift

We consider the stochastic differential equation on $\mathbb{R}^d$ given by $$\, \mathrm{d}X_t = b(t,X_t) \, \mathrm{d}t + \, \mathrm{d} B_t,$$ where $B$ is a Brownian motion and $b$ is considered to be a distribution of regularity $> -\frac12$. We show that the martingale solution of the SDE has a transition kernel $\Gamma_t$ and prove upper and lower heat kernel bounds for $\Gamma_t$ with explicit dependence on $t$ and the norm of $b$.