Let
- $(\Omega,\mathcal A,\operatorname P)$ be a probability space
- $(E,d)$ be a locally compact complete separable metric space
- $D([0,\infty),E):=\left\{x:[0,\infty)\to E\mid x\text{ is càdlàg}\right\}$ be equipped with the Skorohod topology
- $X:\Omega\to D([0,\infty),E)$ be $(\mathcal A,\mathcal B(D([0,\infty),E)))$-measurable
- $\mu$ be a probability measure on $\mathcal B(D([0,\infty),E))$
How can we formalize the condition that $\mu$ is "essentially the distribution of $X$ up to the dependence on the initial value"?
Since you probably don't know what I mean exactly, let me elaborate on that: Since $E$ is Polish, $D([0,\infty),E)$ is Polish too and hence there is a Markov kernel $\kappa$ with source $(E,\mathcal B(E))$ and target $(D([0,\infty),E),\mathcal B(D([0,\infty),E)))$ with $$\operatorname P\left[X\in B\mid X_0\right]=\kappa(X_0,B)\;\;\;\text{almost surely for all }B\in\mathcal B(D([0,\infty),E)).\tag1$$ Note that $$\operatorname P\left[X\in B\right]=\int\operatorname P\left[X_0\in{\rm d}x_0\right]\kappa(x_0,B)\;\;\;\text{for all }B\in\mathcal B(D([0,\infty),E)).\tag2$$ Now, let $\mu_0$ be a probability measure on $\mathcal B(E)$ and $\mu:=\mu_0\kappa$ be the composition of $\mu_0$ and $\kappa$. Then $$\mu\left(\left\{x\in D([0,\infty),E):x(0)\in B_0\right\}\right)=\mu_0(B_0)\;\;\;\text{for all }B_0\in\mathcal B(E)\tag3$$ and $$\mu(B)=\int\mu_0({\rm d}x_0)\kappa(x_0,B)\;\;\;\text{for all }B\in\mathcal B(D([0,\infty),E)).\tag4$$
If $\mu$ is of the form $(4)$, then we see from $(2)$ that $\mu$ is "essentially the distribution of $X$ up to the dependence on the initial value". How can we formalize this?
Intuitively, $\mu$ and $\operatorname P\left[X\in\;\cdot\;\right]$ should coincide on some kind of "trace" of $\mathcal B(D([0,\infty),E)$. In particular, on $$\left\{B\in\mathcal B([0,\infty),E):\mu_0(\pi_0(B))=\operatorname P\left[X_0\in\pi_0(B)\right]\right\},\tag5$$ where $$\pi_t:D([0,\infty),E)\to E\;\;\;x\mapsto x(t)\tag6$$ for $t\ge0$. But is $\pi_0(B)\in\mathcal B(E)$ for all $B\in\mathcal B([0,\infty),E)$. We clearly know that $$\mathcal B([0,\infty),E)\supseteq\sigma(\pi_t:t\ge0)\tag7$$ (and by separability of $E$ we even got equality).
EDIT: In light of the fact that the projection of a Borel subset of $\mathbb R^2$ onto one of the components is not a Borel subset of $\mathbb R$ (see the discussion here: https://mathoverflow.net/q/34142/91890), I might run into trouble with my considerations, since I obviously need that the projection of a Borel subset of $D([0,\infty),E)$ onto the $0$th component is a Borel subset of $E$. On the other hand, $\mathcal B([0,\infty),E)=\sigma(\pi_t:t\ge0)$ might come to the rescue.