Given a Markov process, are we able to construct another Markov process with the same transition semigroup but different inital law?

Let

• $$E$$ be a locally compact separable metric space
• $$(T(t))_{t\ge0}$$ be a strongly continuous contraction semigroups on $$C_0(E)$$ with generator $$(\mathcal D(A),A)$$
• $$(\Omega,\mathcal A,\operatorname P)$$ be a probability space
• $$(X_t)_{t\ge0}$$ be an $$E$$-valued càdlàg processes on $$(\Omega,\mathcal A,\operatorname P)$$ with $$\operatorname E\left[f(X_t)\mid(X_r)_{r\le s}\right]=(T(t-s)f)(X_s)\;\;\;\text{almost surely for all }f\in C_0(E)\text{ and }t\ge s\ge0\tag1$$

Now let $$\mu$$ be a probability measure on $$\mathcal B(E)$$. Are we able to construct a probability space $$(\tilde\Omega,\tilde{\mathcal A},\tilde{\operatorname P})$$ and an $$E$$-valued càdlàg process $$(\tilde X_t)_{t\ge0}$$ on $$(\tilde\Omega,\tilde{\mathcal A},\tilde{\operatorname P})$$ such that $$\tilde{\operatorname P}\circ\tilde X_0^{-1}=\mu$$ and $$(1)$$ still holds?

EDIT: Maybe we are even to show that we can leave $$(\Omega,\mathcal A)$$ and $$X$$ as they are and just construct a new probability measure $$\operatorname P_\mu$$ with the desired properties.

In general, we cannot find a Markov process corresponding to a given transition semigroup. However, since it's assumed here that there is a Markov process $$X$$ corresponding to $$(T(t))_{t\ge0}$$, we might be successful. On the other hand, the càdlàg path property might be problematic.