I will discuss convergence rates for Euler-type approximations of martingale problems with distributional drift. The starting point is the usual regularisation procedure: replace the singular drift by a heat-regularised one, apply the classical Euler scheme to the regularised equation, and then choose the regularisation scale as a function of the time step.

The improvement comes from the stability part of this argument. Instead of estimating the effect of regularisation in the original Besov norm, we use estimates for the backward Kolmogorov equation with negative-regularity data to obtain stability in weaker Besov norms. This gives a sharper comparison between the original martingale problem and the regularised dynamics.

The discretisation error for the regularised equation is then treated separately, using stochastic sewing estimates to exploit the averaging of the Brownian increments along the Euler scheme. Optimising the two contributions gives improved weak convergence rates in any dimension. In dimension one, the same stability input, combined with a Zvonkin transformation, also improves the strong L^1-rate; in particular, for almost bounded and measurable drifts one recovers the Brownian order (1/2), up to arbitrarily small losses.

I will also indicate how the argument adapts to a randomised-time version of the scheme, whose role is to weaken the time-regularity assumptions on the drift without changing the spatial balance behind the rate.