Interacting particles in random irregular media

In this paper, we first develop novel tools of independent interest in the field of stochastic calculus related to SDEs with distributional drift, namely the formulation of a quasi-Stroock-Varadhan martingale problem in the sense that the test functions are of class $C^{1,2}$ instead of solutions of Kolmogorov type PDE. This has several applications, among which, the time restriction and the notion of conditional rough martingale problem and related solution.