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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 transitionkernel $\Gamma_t$ and prove upper and lower heat kernel bounds for $\Gamma_t$with explicit dependence on $t$ and the norm of $b$.

Quantitative heat kernel estimates for diffusions with distributional drift