Fokker Plank equation and Markov processes with generator $L$

Question

Let $X_t$ a Markov process with generator $L$. Let $L^*$ be the adjoint of $L$.

Is it true that, by the Fokker Plank equations, I can conclude that, $u_t(x)$, probability density function of the process satisfies

$$\frac{\partial}{\partial t}u_t(x)=L^*u_t(x)$$