What does the generator tell you about a Markov chain?

It gives the instantaneous rates of transition between states; from it the whole time evolution of transition probabilities follows, on finite state spaces as the matrix exponential of the generator.

How do the forward and backward equations differ?

The backward equation differentiates with respect to the starting state and is useful for hitting and expectation problems, while the forward equation differentiates with respect to the current state and describes the evolving probability distribution.

Kolmogorov Equations and Generators

The infinitesimal generator encodes the instantaneous transition rates of a continuous-time Markov chain, and the Kolmogorov forward and backward equations describe how its transition probabilities evolve in time.

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Definition

The infinitesimal generator of a continuous-time Markov chain is the matrix of transition rates giving the instantaneous rate of change of transition probabilities, and the Kolmogorov forward and backward equations are the differential equations the transition probability matrix satisfies as a function of time.

Scope

This topic covers the definition of the generator as the time derivative of the transition semigroup at zero, the forward (Fokker-Planck type) and backward Kolmogorov equations, the transition matrix as the matrix exponential of the generator, semigroup properties, and conditions for uniqueness, conservativeness, and the absence of explosion.

Core questions

How is the generator obtained as the derivative of the transition semigroup?
What is the difference between the forward and backward Kolmogorov equations?
When is the transition matrix the matrix exponential of the generator?
What conditions guarantee a unique, non-exploding solution?

Key theories

Backward and forward Kolmogorov equations: The transition probability matrix satisfies two coupled systems of linear differential equations driven by the generator, the backward equation differentiating in the initial state and the forward equation in the final state, and for finite state spaces both have the matrix exponential as their common solution.
Generator and semigroup correspondence: The family of transition operators forms a strongly continuous semigroup whose infinitesimal generator determines the process; this correspondence connects Markov chains to the analytic theory of operator semigroups and underlies convergence and approximation results.

Clinical relevance

The forward equation is the master equation of chemical kinetics and statistical physics, governing the probability distribution of molecular counts over time, while the generator formalism provides the computational basis for transient analysis of reliability, queueing, and epidemic models.

History

Kolmogorov's 1931 paper introduced the differential equations for transition probabilities, Feller resolved questions of existence, uniqueness, and explosion in the 1930s and 1940s, and the semigroup and generator viewpoint was systematised through the later work of Hille, Yosida, and Dynkin on Markov processes.

Key figures

Andrey Kolmogorov
William Feller
Thomas Kurtz

Seminal works

norris1997

Frequently asked questions

What does the generator tell you about a Markov chain?: It gives the instantaneous rates of transition between states; from it the whole time evolution of transition probabilities follows, on finite state spaces as the matrix exponential of the generator.
How do the forward and backward equations differ?: The backward equation differentiates with respect to the starting state and is useful for hitting and expectation problems, while the forward equation differentiates with respect to the current state and describes the evolving probability distribution.