Kolmogorov equations (continuous-time Markov chains)

Kolmogorov Forward Equations (Markov Jump Processes)

In the context of Continuous-time Markov process Kolmgorov forward equations and Kolmogorov backward equations refer to a pair of differential equations that describe the time-evolution of the probability $P(x,s;yt)$ where $x,y\in \Omega$ (the states space) and $t>s$ are the final and initial time respectively.

History

In his 1931 paper Andrei Kolmogorov highlights two pair of equations within the analysis of Markov processes. The first pair, equations (52) and (57), are named "first (second) system of differential equations". Later, William Feller names the equations as "Kolmogorov equations" in , while in he names them Kolmogorov Forward Equation and Kolmogorov Backward Equation. The equations are known under these names today within the community of Markov Jump Processes (even the wider community, including applications in Natural Sciences). The physics community refers to a special case of (52) as the "Master equation".

The equations

(escribir, inclusive assumptions, conditions)

Background

The origin of the equations is the Chapman-Kolmogorov equation for time-continuous and differentiable Markov processes.

References

• Andrei Kolmogorov, 1931, Über die analytischen Methoden in der Wahrseheinliehkeitsrechnung [4]
• Willy Feller, 1940. On Boundaries and Lateral Conditions for the Kolmogorov Differential Equations

[5]

• William Feller, 1957. On Boundaries and Lateral Conditions for the Kolmogorov Differential Equations [6]

External links

• Continuous-time Markov process
• Master equation
• Fokker-Planck Equation
• Kolmogorov bakward equation (difussion)