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Merge pull request #52 from jstac/ucms
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ctmc_lectures/uc_mc_semigroups.md

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@@ -35,7 +35,7 @@ The main aim is to give an exact one-to-one correspondence between
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Conservativeness is defined below and relates to "nonexplosiveness" of the
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associated Markov chain.
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We will also give a brief discussion of intensity matricies that do not have
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We will also give a brief discussion of intensity matrices that do not have
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this property, along with the processes they generate.
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Our definition of a conservative intensity matrix works for the theory above
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but can be hard to check in appliations and lacks probabilistic intuition.
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Fortunately, we have the following simple charcterization.
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Fortunately, we have the following simple characterization.
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```{proof:lemma}
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### Jump Chain Decomposition
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Given a intensity matrix $Q$, set
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Given an intensity matrix $Q$, set
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$$
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\lambda(x) := -Q(x, x)
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$(P_t)$ where $P_t = e^{tQ}$ for $Q$ satisfying {eq}`jcinmat`.
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(Although our argument assumed finite $S$, the proof goes through when
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$S$ is contably infinite and $Q$ is conservative with very minor changes.)
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$S$ is countably infinite and $Q$ is conservative with very minor changes.)
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In particular, $(X_t)$ is a continuous time Markov chain with intensity matrix
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$Q$.

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