Is $I(\omega_0) = J(\omega_0)$ for this homogeneous continuous-time Markov chain?

Question

Assume $(X_t)_{t\ge 0}$ is a homogeneous continuous-time Markov chain on the probability space $(\Omega, \mathcal{G}, \mathbb{P})$. Moreover, $X_1(\omega_0) = X_2(\omega_0)$ for some $\omega_0 \in \Omega$. Consider two random variables:

• $I = \inf \left\{t \geq 1 \mid X_t \neq X_1\right\}$

• $J = \inf \left\{t \geq 2 \mid X_t \neq X_2\right\}$

I would like to ask if we can conclude $I(\omega_0) = J(\omega_0)$. Thank you so much for your help!

Answer 1

I convert @lan's comment as answer to close this question.

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As random variables? Definitely not. Nor are they necessarily the same for this particular $\omega_0$. Nor do they even necessarily have the same distribution, unless you also assume that $X_3$ is the same (because effectively you have a kind of "Brownian bridge" between time $1$ and $2$ that you don't have after time $2$).