ContinuousMarkovProcess[i0, q]
represents a continuous-time finite-state Markov process with transition rate matrix q and initial state .

ContinuousMarkovProcess[p0, q]
represents a Markov process with initial state probability vector .

ContinuousMarkovProcess[..., m, ]
represents a Markov process with transition matrix m and transition rates .

ContinuousMarkovProcess[..., g]
represents a Markov process transition rate matrix from the graph g.


  • ContinuousMarkovProcess is also known as a continuous-time Markov chain.
  • ContinuousMarkovProcess is a continuous-time and discrete-state random process.
  • The states of ContinuousMarkovProcess are integers between 1 and , where is the length of transition rate matrix q.
  • For infinitesimal time dt, gives the probability that the process transitions from state i to state j over the next dt units of time qi, jdt=Probability[x[t+dt]=jConditionedx[t]=i].
  • The time the process stays in state i before transitioning follows ExponentialDistribution[-qii].
  • The transition matrix m specifies conditional transition probabilities mi, j=Probability[x[tk+1]=jConditionedx[tk]=i], where is the state at time , and the transition rate specifies that the time between events in state follows ExponentialDistribution[i].
  • The transition matrix in the case of a graph g is constructed to give equal probability of transitioning to each incident vertex with unit transition rates.
  • ContinuousMarkovProcess allows q to be an × matrix where and for with rows that sum to 0, can be an integer between 1 and , is a vector of length of non-negative elements that sum to 1, m is an × matrix with non-negative elements and rows that sum to 1, and is a vector of length with positive elements.
  • ContinuousMarkovProcess can be used with such functions as MarkovProcessProperties, PDF, Probability, and RandomFunction.
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