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For the transition probability of a Markov jump process from state $x$ to $y$ in a small time interval $\Delta t$, what is the meaning of $o(\Delta t)$? $\alpha$ is a constant $\in[0,1)$.

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That is expressing the fact that the probability to have two or more jumps in the time $\Delta T$ is negligible with respect to $\alpha \Delta t$, and this fact is expressed in the little o notation.

It could also be expressed in "Big O notation" as $$ \alpha \,\Delta t + O\left( {\Delta t^2 } \right) $$

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