**[1]** On a technical level, right continuity of paths for $ (X_t) $ implies condition 2, as proved in Theorem 2.12 of [[Liggett, 2010](https://jstac.github.io/continuous_time_mcs/zreferences.html#id8)]. Right continuity of paths allows for jumps, but insists on only finitely many jumps in any bounded interval.\n",
"\n",
"

**[2]** In the definition of $ P_t $ in [(3.9)](#equation-poissemi), we use the convention that $ 0^0 = 1 $, which leads to $ P_0 = I $ and $ \\lim_{t \\to 0} P_t(j, k) = I(j,k) $ for all $ j,k $. These facts, along with the semigroup property, imply that $ (P_t) $ is a valid Markov semigroup."
]
}
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"title": "The Markov Property"
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