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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.

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