form, we find $$\partial_{t}S(\rho) + \Div(S(\rho)u)+\Div u[\rho S'(\rho)-S(\rho)] +\nabla \Phi(\rho)\circ \dot{W}=0.$$ To convert from Stratonovich to Itô, introduce an additional flux $\Psi$ defined by $\Psi'(\rho)=b^{2}(\rho)S'(\rho)$. Inspecting reveals that the cross variation of $\nabla \Phi(\rho)$ with $W$ is $-\Delta \Psi(\rho)$. Hence, we find that $$\label{eq:EntropyBalance:Ito}
\partial_{t}S(\rho) + \Div(S(\rho)u)-\frac{1}{2}\Delta \Psi(\rho)+\Div u[\rho S'(\rho)-S(\rho)] +\nabla \Phi(\rho) \cdot \dot{W}=0.$$ In weak form, reads as follows: given a test function $\varphi$ and a time $t \in [0,T]$ it holds $$\begin{aligned}
 \label{eq:formalBd:weightedEntropy}
\int_{\R^{d}}S(\rho_{t})\varphi \dx=\int_{\R^{d}}S(\rho_{0})\varphi \dx&+\int_{0}^{t}\int_{\R^{d}}S(\rho)\left [\partial_{t}\varphi + u \cdot \nabla_{x}\varphi \right]\dx\ds \nonumber\\
&+\int_{0}^{t}\int_{\R^{d}} \left ( \frac{1}{2}\Delta \varphi \Psi(\rho)-\varphi [\rho S'(\rho)-S(\rho)]\Div u \right ) \dx \ds \nonumber \\
&+ \int_{0}^{t}\int_{\R^{d}}\Phi(\rho)\nabla \varphi \dx \cdot \dee W_{s}.\end{aligned}$$ In the present context, the $\varphi$ should be thought of as a convenient choice of weight to be selected. $$\text{Step 2: Choice of test function}$$ We choose a constant $M>0$ depending on the velocity field $u$ and a weight $\varphi$ such that on $[0,T] \times \R^{d}$ it holds $$\begin{aligned}
\label{eq:FormalBd:EllipticProblem} \partial_{t}\varphi + u \cdot \nabla \varphi + \mathcal{M}^{+}_{\frac{\lambda}{4},\Lambda}(D^{2}\varphi)+(\Div u)_{-}\varphi &\leq M\\
\label{eq:FormalBd:bounds}1 \leq \varphi &\leq M,\end{aligned}$$ where $\mathcal{M}^{+}_{\frac{\lambda}{4},\Lambda}$ is Pucci’s maximal operator, defined by $$\mathcal{M}^{+}_{\frac{\lambda}{4},\Lambda}(D^{2}\varphi)=\sup_{\frac{\lambda}{4} I \leq A \leq \Lambda I} A: D^{2}\varphi.$$ Recall that $\lambda$ and $\Lambda$ are defined in Hypothesis \[hyp:noiseFlux\]. 