The generator matrix and long run behaviour

Question 1

Kolmogorov’s Equations (KFDEs)

Answer 1

The Kolmogorov forward differential equations (KFDEs) describe how the transition matrix P(t) evolves over time.

Question 2

Kolmogorov’s Backward Equations (KBDEs)

Answer 2

The Kolmogorov backward differential equations (KBDEs) describe how the transition probabilities evolve when viewed backward in time.

Question 3

The Generator Matrix, Q

Answer 3

The generator matrix Q captures the rates at which transitions occur between states in a continuous-time Markov chain.

Question 4

Solving the Kolmogorov Equations

Answer 4

The solution to the Kolmogorov equations is given by P(t) = exp(tQ), which is the matrix exponential of tQ.

Question 5

Invariant Distributions

Answer 5

An invariant distribution π satisfies πQ = 0, meaning if the Markov chain starts in this distribution, it remains in this distribution over time.

Question 6

Equilibrium Distribution

Answer 6

An equilibrium distribution π satisfies lim(t→∞) p_ij(t) = π_j, and it represents the long-run probability distribution that does not depend on the initial state.

Question 7

Limiting Behavior of Continuous-Time Markov Chains

Answer 7

If the chain is irreducible and positive recurrent, an invariant distribution exists and is unique. If it is null recurrent or transient, no invariant distribution exists.