Let
- $(E,\mathcal E)$ be a measurable space;
- $\mathcal E_b:=\left\{f:E\to\mathbb R\mid f\text{ is bounded and }\mathcal E\text{-measurable}\right\}$ be equipped with the supremum norm;
- $(\kappa_t)_{t\ge0}$ be a Markov semigroup on $(E,\mathcal E)$;
- $(\tilde E,\tilde{\mathcal E})$ be a measurable space;
- $\varphi:E\to\tilde E$ be $\left(\mathcal E,\tilde{\mathcal E}\right)$-measurable with $\varphi(E)=\tilde E$ and $$\kappa_t\left(\tilde f\circ\varphi\right)=\tilde g\circ\varphi\tag1$$ for all $\tilde f\in\tilde{\mathcal E}_b$ and $t\ge0$.
Note that $\tilde g$ is uniquely determined and hence $$\tilde\kappa_tf:=\tilde g$$ is well-defined. We easily see that $\left(\tilde\kappa_t\right)_{t\ge0}$ is a Markov semigroup on $(\tilde E,\tilde{\mathcal E})$.
Now, we can treat $(\kappa_t)_{t\ge0}$ and $\left(\tilde\kappa_t\right)_{t\ge0}$ as contraction semigroups on $\mathcal E_b$- and $\tilde{\mathcal E}_b$-semigroups, respectively. Let $A$ and $\tilde A$ denote the corresponding generators.
Is there a relation between $A$ and $\tilde A$ similar to the relation between $\kappa_t$ and $\tilde\kappa_t$ in $(1)$?