What does $\mathbb{E}^x((X_t)_{t \in T})$ for some stochastic process $(X_t)_{t \in T}$ mean? I heard that it is some kind of conditional expectation given that $X(0)=x$, but I can't find a precise definiton. Is it the stochastic process $t\mapsto\mathbb{E}(X_t|\{X(0)=x\})$, where $0$ is the first element in $T$? If yes, is there a simpler way to rewrite this?
2026-03-28 15:56:05.1774713365
What is the meaning of $\mathbb{E}^x$?
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