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

TL;DR

This paper establishes existence and uniqueness of solutions for multidimensional SDEs with singular drifts using paracontrolled distributions, enabling the rigorous construction of the three-dimensional polymer measure with white noise potential.

Contribution

It introduces a novel approach to handle SDEs with distributional drifts and constructs the polymer measure in three dimensions with white noise potential.

Findings

Proves well-posedness of SDEs with distributional drift

Provides a universal construction for the polymer measure with white noise

Utilizes paracontrolled distributions to interpret martingale problems

Abstract

We study existence and uniqueness of solution for stochastic differential equations with distributional drift by giving a meaning to the Stroock-Varadhan martingale problem associated such equations. The approach we exploit is the one of paracontrolled distributions introduced in [13]. As a result we make sense of the three dimensional polymer measure with white noise potential.