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  "body": "\n## Definition\n\nLet $S$ be a countable set.\nA *time-homogeneous continuous-time Markov chain* on $S$ is a [[stochastic process|0QH5ENKZM1]] $(X_t)_{t \\ge 0}$ with values in $S$ and right-continuous paths that has the following property.\nThere are [[transition matrices|0F462493EA]] $P(t) = (p_{ij}(t))_{i, j \\in S}$, $t \\ge 0$, such that, for all $n \\ge 0$, all times $0 \\le t_0 \u003c t_1 \u003c \\cdots \u003c t_{n+1}$ and all states $i_0, \\ldots, i_{n+1} \\in S$ with $P(X_{t_0} = i_0, \\ldots, X_{t_n} = i_n) \u003e 0$,\n$$P(X_{t_{n+1}} = i_{n+1} \\mid X_{t_0} = i_0, \\ldots, X_{t_n} = i_n) = p_{i_n i_{n+1}}(t_{n+1} - t_n).$$\n\n## Remarks\n\nThe condition is the [[Markov property|0B77KZCQ51]] in continuous time, and it implies the [[Chapman–Kolmogorov equation|01APZ8GDQH]] $P(s + t) = P(s) P(t)$.\nFor a finite $S$ there is a rate matrix $Q = P'(0)$, the [[generator|VPA41PWRR1]], with $q_{ij} \\ge 0$ for $i \\ne j$ and zero row sums, and $P(t) = e^{tQ}$ is a [[matrix exponential|020M04FN1H]].\nThe chain stays in state $i$ for an [[exponential|0E99F51GVB]] time with rate $q_i = -q_{ii}$, then jumps to $j \\ne i$ with probability $q_{ij}/q_i$, when $q_i \u003e 0$.\nExample: the Poisson process of rate $\\lambda \u003e 0$ is a chain on $\\N$ with $q_{i, i+1} = \\lambda$ and $q_{ii} = -\\lambda$, and $p_{ij}(t) = e^{-\\lambda t} (\\lambda t)^{j-i} / (j-i)!$ for $j \\ge i$.\nNon-example: a process on $\\Z$ that moves up by $1$ at the times $1, 2, 3, \\ldots$ is not time-homogeneous.\nIndeed $P(X_{t+1/2} = X_t + 1)$ is $1$ for $t = 3/4$ and $0$ for $t = 1/4$, so no single $P(1/2)$ works.\n",
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