used by Beck/Flandoli/Gubinelli/Maurelli [@beck2014stochastic] and Gess/Maurelli [@gess2017well]. In particular, a key idea of Gess/Maurelli [@gess2017well] is that regularization by noise results are available in some non-linear settings if one combines the duality method of [@beck2014stochastic] with some classical tricks developed in [@MR2064166] for the kinetic approach to scalar conservation laws.\
Let us begin by reviewing the duality approach in the linear setting. Namely, we aim to sketch the proof that solutions to are prevented from concentrating mass. For simplicity, we focus on non-negative solutions. Formally, given a solution $\rho$ to the linear stochastic continuity equation , applying the Itô formula to $\rho^{q}$ gives $$\partial_{t}(\rho^{q})+\Div(\rho^{q}u)+(q-1)\rho^{q}\Div u+\nabla(\rho^{q}) \circ \dot{W}=0.$$ Taking the expectation on both sides, we find the following closed parabolic equation for $\E[\rho^{q}]$ $$\partial_{t}\E[\rho^{q}]+\Div(\E[\rho^{q}]u)+(q-1)\E[\rho^{q}]\Div u-\frac{1}{2}\Delta \E[\rho^{q}]=0.\label{eq:paraRenorm1}$$ Given a test function $\varphi$, we find that $$\int_{\R^{d}}\varphi(t)\E[\rho^{q}(t)]\dx = \int_{\R^{d}}\varphi(0)\rho_{0}^{q}\dx+\int_{0}^{t}\int_{\R^{d}}\E[\rho^{q}]\left (\partial_{t}\varphi+u \cdot \nabla_{x}\varphi+\frac{1}{2}\Delta_{x}\varphi - (q-1)\varphi \Div u \right)\dx\ds.$$ Hence, if we may find a sub-solution $\varphi$ which is bounded from above and below and satisfies the $a.e.$ inequality $$\label{eq:linearParabolicDual}
\partial_{t}\varphi+u \cdot \nabla_{x}\varphi+\frac{1}{2}\Delta_{x}\varphi - (q-1)\varphi \Div u \leq M,$$ then by Gronwall’s inequality it follows that $\rho(t)$ belongs to $L^{q}(\R^{d})$ with probability one. In the non-linear setting of , the direct link between the stochastic 