Let
- $(E,\mathcal E)$ be a measurable space
- $(\kappa_t)_{t\ge0}$ be a Markov semigroup on $(E,\mathcal E)$ and $$\kappa_tf:=\int\kappa_t(\;\cdot\;,{\rm d}y)f(y)$$ for $$f\in F_0:=\left\{f:E\to\mathbb R\mid f\text{ is }\mathcal E\text{-measurable and }\left(\kappa_t\left|f\right|\right)(x)<\infty\text{ for all }x\in E\text{ and }t\ge0\right\}$$
- $F\subseteq F_0$ be a $\mathbb R$-Banach space containing the elementary $\mathcal E$-measurable functions as a dense subset
Assume $$T(t)f:=\kappa_tf\;\;\;\text{for }f\in F\text{ and }t\ge0$$ is a $C^0$-semigroup on $F$. Let $$\operatorname{orb}f:[0,\infty)\to F\;,\;\;\;t\mapsto T(t)f$$ for $f\in F$ and $(\mathcal D(A),A)$ denote the infinitesimal generator of $(T(t))_{t\ge0}$. We know that $$\left(\operatorname{orb}f\right)'(t)=AT(t)f=T(t)Af\;\;\;\text{for all }f\in\mathcal D(A)\text{ and }t\ge0.\tag1$$
Now, assume that there is a $\mathcal E\otimes\mathcal E$-measurable $p_t:E\times E\to[0,\infty)$ with $$\kappa_t(x,B)=\int_Bp(x,y)\lambda\left({\rm d}y\right)\;\;\;\text{for all }(x,B)\in E\times\mathcal E\tag2$$ for some measure $\lambda$ on $(E,\mathcal E)$ for all $t\ge0$. It's easy to see that $$p_{s+t}(x,\;\cdot\;)=\int p_s(x,y)p_t(y,\;\cdot\;)\lambda\left({\rm d}y\right)\;\;\;\lambda\text{-almost everywhere for all }x\in E\tag3$$ for all $s,t\ge0$.
Does $(p_t)_{t\ge0}$ satisfy an identity like $(1)$?
I guess the answer is yes and the corresponding identity is called the Fokker-PLanck equation. However, I couldn't find this equation in this general setting. Clearly, we need regularity assumptions on $(p_t)_{t\ge0}$, but which assumptions do we really need to impose?
I could imagine that we're able to show something like $$\frac\partial{\partial t}p_t(x,y)=(Ap_t(x,\;\cdot\;)(y).$$