SDEs with Supercritical Distributional Drifts

Let d⩾2. In this paper, we investigate the following stochastic differential equation (SDE) in Rd driven by Brownian motion (Formula presented.) where b belongs to the space LTqHpα with α∈[-1,0] and p,q∈[2,∞], which is a distribution-valued and divergence-free vector field. In the subcritical case dp+2q<1+α, we establish the existence and uniqueness of a weak solution to the integral equation: (Formula presented.) Here, bn:=b∗ϕn represents the mollifying approximation, and the limit is taken in the L2-sense. In the critical and supercritical case 1+α⩽dp+2q<2+α, assuming the initial distribution has an L2-density, we show the existence of weak solutions and associated Markov processes. Moreover, under the additional assumption that b=b1+b2+diva, where b1∈LT∞B∞,2-1, b2∈LT2L2, and a is a bounded antisymmetric matrix-valued function, we establish the convergence of mollifying approximation solutions without the need to subtract a subsequence. To illustrate our results, we provide examples of Gaussian random fields and singular interacting particle systems, including the two-dimensional vortex models.