In the book of Villani titled Hypocoercivity, there is the following proposition.
Let $\sigma \in C^2(\mathbb{R}^n; \mathbb{R}^{n \times m})$ and $\xi \in C^1(\mathbb{R}^n; \mathbb{R}^n)$, and let $(X_t)_{t\ge 0}$ be a stochastic process solving the autonomous stochastic differential equation $$ dX_t = \sqrt{2}\sigma(X_t) dB_t + \xi(X_t)dt$$, where $(B_t)_{t \ge 0}$ is a standard Brownian motion in $\mathbb{R}^m$. Then the law $(\rho_t)_{t \ge 0}$ of $X_t$ satisfies the diffusion equation $$ \frac{\partial \rho}{\partial t} = \nabla \cdot (D\nabla \rho - \xi \rho), \quad D := \rho^* \rho$$
Actually, the proof of this proposition says it is a classical consequence of Ito's formula, but I don't understand how they are related. Could you help me understand this? Thanks in advance!