Multidimensional SDEs with singular drift and universal construction of the polymer measure with white noise potential

We study the existence and uniqueness of solution for stochastic differential equations with distributional drift by giving a meaning to the Stroock.Varadhan martingale problem associated to such equations. The approach we exploit is the one of paracontrolled distributions introduced in (Forum Math. Pi 3 (2015) e6). As a result, we make sense of the three-dimensional polymer measure with white noise potential.