Continuous-Time Markov Chains

Continuous-time Markov chains extend the concept of Markov chains to processes where transitions can occur at any moment in time. While traditional Markov chains operate in discrete steps, continuous-time versions allow the system to change states at any continuous time point, with exponentially distributed waiting times between transitions.

These processes are characterized by a generator matrix Q, where off-diagonal elements qᵢⱼ (i≠j) represent transition rates from state i to state j, and diagonal elements qᵢᵢ = -∑ⱼ≠ᵢ qᵢⱼ ensure proper probability accounting. The probability of transitioning from state i to j after time t is given by the corresponding element of the matrix exponential e^(Qt).

The memoryless property of the exponential distribution is crucial here—it ensures that the future evolution depends only on the current state, not how long the process has been in that state.

These models are valuable for queueing systems, epidemiological models, reliability analysis, and modeling state transitions in complex systems where events don't occur at regular intervals.

In machine learning applications, continuous-time Markov chains model customer behavior in marketing analytics, patient progression through disease states in healthcare, and the evolution of complex systems like social networks or financial markets. Recent neural ODE (Ordinary Differential Equation) models incorporate continuous-time dynamics inspired by these processes to model irregularly-sampled time series data.