Kolmogorov equations
Encyclopedia
Kolmogorov equations, including Kolmogorov forward equations and Kolmogorov backward equations, characterize random dynamic processes.
, and sought to derive a theory of continuous time Markov processes by extending this equation. He found that there are two kinds of continuous time Markov Processes, depending on the assumed behavior over small intervals of time:
If you assume that "in a small time interval there is an overwhelming probability that the state will remain unchanged; however, if it changes, the change may be radical") then you are led to what are called jump process
es.
The other case leads to processes such as those "represented by diffusion
and by Brownian motion
; there it is certain that some change will occur in any time interval, however small; only, here it is certain that the changes during small time intervals will be also small").
For each of these two kinds of processes, Kolmogorov derived a forward and a backward system of equations (four in all).
William Feller
makes reference to the equations for the jump process
as Kolmogorov's equations.
He later gives the names forward equation and backward equation to his (more general) version
of the equations and uses the same names as nicknames for each member of Kolmogorov's pair, while he refers to the diffusion equations as "forward" and "backward" Fokker–Planck equation.
Much later, by 1957, Feller refers to the equations for the jump process as Kolmogorov forward equations and Kolmogorov backward equations.
Other authors, such as Motoo Kimura
will refer to the diffusion (Fokker–Planck) equation as Kolmogorov forward equation, a name that has persisted.
Diffusion Processes vs. Jump Processes
Writing in 1931, Andrei Kolmogorov started from the theory of discrete time Markov processes, which are described by the Chapman-Kolmogorov equationChapman-Kolmogorov equation
In mathematics, specifically in probability theory and in particular the theory of Markovian stochastic processes, the Chapman–Kolmogorov equation is an identity relating the joint probability distributions of different sets of coordinates on a stochastic process...
, and sought to derive a theory of continuous time Markov processes by extending this equation. He found that there are two kinds of continuous time Markov Processes, depending on the assumed behavior over small intervals of time:
If you assume that "in a small time interval there is an overwhelming probability that the state will remain unchanged; however, if it changes, the change may be radical") then you are led to what are called jump process
Jump process
A jump process is a type of stochastic process that has discrete movements, called jumps, rather than small continuous movements.In physics, jump processes result in diffusion...
es.
The other case leads to processes such as those "represented by diffusion
Ito diffusion
In mathematics — specifically, in stochastic analysis — an Itō diffusion is a solution to a specific type of stochastic differential equation. That equation is similar to the Langevin equation, used in Physics to describe the brownian motion of a particle subjected to a potential in a...
and by Brownian motion
Brownian motion
Brownian motion or pedesis is the presumably random drifting of particles suspended in a fluid or the mathematical model used to describe such random movements, which is often called a particle theory.The mathematical model of Brownian motion has several real-world applications...
; there it is certain that some change will occur in any time interval, however small; only, here it is certain that the changes during small time intervals will be also small").
For each of these two kinds of processes, Kolmogorov derived a forward and a backward system of equations (four in all).
Kolmogorov equations: the modern view
- In the context of a continuous-time Markov process, the term refers to the Kolmogorov forward equations (a special case is known in Natural sciences as master equationMaster equationIn physics and chemistry and related fields, master equations are used to describe the time-evolution of a system that can be modelled as being in exactly one of countable number of states at any given time, and where switching between states is treated probabilistically...
) and the Kolmogorov backward equations.
- In the context of diffusionDiffusionMolecular diffusion, often called simply diffusion, is the thermal motion of all particles at temperatures above absolute zero. The rate of this movement is a function of temperature, viscosity of the fluid and the size of the particles...
equations these names refer to Fokker–Planck equation (forward equation) and to the Kolmogorov backward equations (diffusion).
History
The equations are named after Andrei Kolmogorov's since they were highlighted in his 1931 foundational work.William Feller
William Feller
William Feller born Vilibald Srećko Feller , was a Croatian-American mathematician specializing in probability theory.-Early life and education:...
makes reference to the equations for the jump process
Jump process
A jump process is a type of stochastic process that has discrete movements, called jumps, rather than small continuous movements.In physics, jump processes result in diffusion...
as Kolmogorov's equations.
He later gives the names forward equation and backward equation to his (more general) version
of the equations and uses the same names as nicknames for each member of Kolmogorov's pair, while he refers to the diffusion equations as "forward" and "backward" Fokker–Planck equation.
Much later, by 1957, Feller refers to the equations for the jump process as Kolmogorov forward equations and Kolmogorov backward equations.
Other authors, such as Motoo Kimura
Motoo Kimura
was a Japanese biologist best known for introducing the neutral theory of molecular evolution in 1968. He became one of the most influential theoretical population geneticists. He is remembered in genetics for his innovative use of diffusion equations to calculate the probability of fixation of...
will refer to the diffusion (Fokker–Planck) equation as Kolmogorov forward equation, a name that has persisted.