Assume I have a Feller process $(X_s)_{s \geq 0}$ with values in $\mathbb{R}^d$, and I consider its càdlàg modification. Recall that this also implies that the process is strong Markov.
Let $C$ be a compact set in $\mathbb{R}^d$ and let $x \in Int(C)$, the interior of $C$. Let $\tilde{\tau} = \inf\{t > 0, X_t \not \in C\}$.
Is there a way to show that $\mathbb{E}_x[\frac{1}{\tilde{\tau}}] < \infty$ without further assumptions ?
I believe this is true because for every fixed $\omega$, setting $\epsilon = dist(x, \partial C)$, I can just upper bound $\tilde{\tau}$ by the continuity modulus associated with $\epsilon/2$ (for instance) of the right-continuous function $s \to X_s(\omega)$. However I do not know how to translate this property in something useful for the control of the expectation.