Inprobability theory,Kolmogorov equations characterizecontinuous-time Markov processes. In particular, they describe how the probability of a continuous-time Markov process in a certain state changes over time. There are four distinct equations: the Kolmogorov forward equation for continuous processes, now understood to be identical to theFokker–Planck equation, theKolmogorov forward equation for jump processes, and twoKolmogorov backward equations for processes with and withoutdiscontinuous jumps.

Writing in 1931,Andrei Kolmogorov started from the theory of discrete time Markov processes, which are described by theChapman–Kolmogorov equation, 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",^[1] then you are led to what are calledjump processes.

The other case leads to processes such as those "represented bydiffusion and byBrownian 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".^[1]

For each of these two kinds of processes, Kolmogorov derived a forward and a backward system of equations (four in all).

The equations are named afterAndrei Kolmogorov since they were highlighted in his 1931 foundational work.^[2]

William Feller, in 1949, used the names "forward equation" and "backward equation" for his more general version of the Kolmogorov's pair,in both jump and diffusion processes.^[1] Much later, in 1956, he referred to the equations for the jump process as "Kolmogorov forward equations" and "Kolmogorov backward equations".^[3]

Other authors, such asMotoo Kimura,^[4] referred to thediffusion (Fokker–Planck) equation as Kolmogorov forward equation, a name that has persisted.

• In the context of acontinuous-time Markov process withjumps, seeKolmogorov equations (Markov jump process). In particular, innatural sciences the forward equation is also known asmaster equation.
• In the context of adiffusion process, for the backward Kolmogorov equations seeKolmogorov backward equations (diffusion). The forward Kolmogorov equation is also known asFokker–Planck equation.

The original derivation of the equations by Kolmogorov starts with theChapman–Kolmogorov equation (Kolmogorov called itfundamental equation) for time-continuous and differentiable Markov processes on a finite, discrete state space.^[2] In this formulation, it is assumed that the probabilities${\displaystyle P(x,s;y,t)}$ are continuous and differentiable functions of${\displaystyle t>s}$, where${\displaystyle x,y\in \mathrm{\Omega }}$ (the state space) and${\displaystyle t>s,t,s\in {\mathbb{R}}_{\ge 0}}$ are the final and initial times, respectively. Also, adequate limit properties for the derivatives are assumed. Feller derives the equations under slightly different conditions, starting with the concept ofpurely discontinuous Markov process and then formulating them for more general state spaces.^[5] Feller proves the existence of solutions of probabilistic character to theKolmogorov forward equations andKolmogorov backward equations under natural conditions.^[5]

For the case of acountable state space we put${\displaystyle i,j}$ in place of${\displaystyle x,y}$. TheKolmogorov forward equations read

${\displaystyle \frac{\mathrm{\partial }{P}_{ij}}{\mathrm{\partial }t}(s;t)=\sum _k{P}_{ik}(s;t)A_{kj}(t)}$,

where${\displaystyle A(t)}$ is thetransition rate matrix (also known as the generator matrix),

while theKolmogorov backward equations are

${\displaystyle \frac{\mathrm{\partial }{P}_{ij}}{\mathrm{\partial }s}(s;t)=-\sum _k{P}_{kj}(s;t)A_{ik}(s)}$

The functions${\displaystyle P_{ij}(s;t)}$ are continuous and differentiable in both time arguments. They represent theprobability that the system that was in state${\displaystyle i}$ at time${\displaystyle s}$ jumps to state${\displaystyle j}$ at some later time${\displaystyle t>s}$. The continuous quantities${\displaystyle A_{ij}(t)}$ satisfy

${\displaystyle A_{ij}(t)={\left[\frac{\mathrm{\partial }{P}_{ij}}{\mathrm{\partial }u}(t;u)\right]}_{u=t},A_{jk}(t)\ge 0,j\ne k,\sum _k{A}_{jk}(t)=0.}$

Still in the discrete state case, letting${\displaystyle s=0}$ and assuming that the system initially is found in state${\displaystyle i}$, theKolmogorov forward equations describe an initial-value problem for finding the probabilities of the process, given the quantities${\displaystyle A_{jk}(t)}$. We write${\displaystyle p_k(t)=P_{ik}(0;t)}$ where${\displaystyle \sum _k{p}_k(t)=1}$, then

${\displaystyle \frac{d{p}_k}{dt}(t)=\sum _j{A}_{jk}(t)p_j(t);p_k(0)={\delta }_{ik},k=0,1,\dots .}$

For the case of a pure death process with constant rates the only nonzero coefficients are${\displaystyle A_{j,j-1}={\mu }_j,j\ge 1}$. Letting

${\displaystyle \mathrm{\Psi }(x,t)=\sum _k{x}^k{p}_k(t),}$

the system of equations can in this case be recast as apartial differential equation for${\displaystyle \mathrm{\Psi }(x,t)}$ with initial condition${\displaystyle \mathrm{\Psi }(x,0)=x^i}$. After some manipulations, the system of equations reads,^[6]

${\displaystyle \frac{\mathrm{\partial }\mathrm{\Psi }}{\mathrm{\partial }t}(x,t)=\mu (1-x)\frac{\mathrm{\partial }\mathrm{\Psi }}{\mathrm{\partial }x}(x,t);\mathrm{\Psi }(x,0)=x^i,\mathrm{\Psi }(1,t)=1.}$

One example from biology is given below:^[7]

${\displaystyle p_n^{\prime }(t)=(n-1)\beta p_{n-1}(t)-n\beta p_n(t)}$

This equation is applied to modelpopulation growth withbirth. Where${\displaystyle n}$ is the population index, with reference the initial population,${\displaystyle \beta }$ is the birth rate, and finally${\displaystyle p_n(t)=Pr(N(t)=n)}$, i.e. theprobability of achieving a certainpopulation size.

The analytical solution is:^[7]

${\displaystyle p_n(t)=(n-1)\beta e^{-n\beta t}{\int }_0^t{p}_{n-1}(s)e^{n\beta s}\mathrm{d}s}$

This is a formula for the probability${\displaystyle p_n(t)}$ in terms of the preceding ones, i.e.${\displaystyle p_{n-1}(t)}$.

• Feynman-Kac formula
• Fokker-Planck equation
• Kolmogorov backward equation

• Markov processes
• Stochastic models
• Mathematical and theoretical biology
• Population models

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