Let $X_t$ be a Markov process in $\mathbb R$ with generator $L$. Let $\mu_t$ be the law of the Markov process starting with an initial distribution $\mu$, in other words $$\mu_t(A)=\int_{\mathbb R}P(X_t\in A|X_0=x)d\mu(x),$$ for every measurable $A\subseteq \mathbb R$. Call $L^*$ the adjoint operator of $L$. I found some notes in which they say that, by the Chapmann Kolmogorov equations, $$\frac{d\mu_t}{dt}=L^*\mu_t$$ I guess that the previous equation has to be thought in the weak sense, that is $$\frac{d}{dt}<\mu_t, f>=L^*<\mu_t, f>,$$ for every $f:\mathbb R\to \mathbb R$, where the notation $<\mu, f>:=\int_{\mathbb R}f(x)d\mu(x)$. Is this correct?
Someone refers to the Chapmann Kolmogorov equation as the forward Fokker Planck equation. I have confusion between the forward and the backward equations. Could someone give me an help or suggest me a good reference about this topic?
Thank you!