Chapman–Kolmogorov equation

E254907

mathematical equation probability theory concept stochastic process concept

The Chapman–Kolmogorov equation is a fundamental relation in the theory of stochastic processes that expresses how transition probabilities of a Markov process over longer time intervals can be obtained by integrating over intermediate states.

Try in SPARQL Jump to: Surface forms Disambiguation Statements Elicitation Referenced by

All labels observed (3)

Label	Occurrences
Chapman–Kolmogorov equation canonical	2
Chapman–Kolmogorov equation in stochastic processes	1
Markov property of Brownian motion	1

How this entity was disambiguated

This entity first appeared as the object of triple T2325347 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.

NED1 Entity disambiguation (via context triple) gpt-5-mini-2025-08-07

Target entity: Chapman–Kolmogorov equation
Context triple: [Markov process, relatedTo, Chapman–Kolmogorov equation]

A. Kolmogorov backward equation
The Kolmogorov backward equation is a fundamental partial differential equation in stochastic processes that characterizes the time evolution of expected values of functionals of Markov processes, complementary to the Fokker–Planck (forward) equation.
B. Fokker–Planck equation
The Fokker–Planck equation is a partial differential equation that describes the time evolution of the probability density function of a stochastic (random) process, such as Brownian motion.
C. Feynman–Kac formula
The Feynman–Kac formula is a fundamental result connecting solutions of certain partial differential equations with expectations over stochastic processes, forming a bridge between quantum mechanics, probability theory, and mathematical finance.
D. Itô’s lemma
Itô’s lemma is a fundamental result in stochastic calculus that generalizes the chain rule to functions of stochastic processes, especially Brownian motion.
E. Ornstein–Uhlenbeck process
The Ornstein–Uhlenbeck process is a continuous-time stochastic process that models mean-reverting random motion, widely used in physics and quantitative finance to describe systems fluctuating around a long-term equilibrium.
F. None of above. chosen
G. Unsure - the case is ambiguous/there is not enough information to decide.

NED2 Entity disambiguation (via description) gpt-5-mini-2025-08-07

Target entity: Chapman–Kolmogorov equation
Target entity description: The Chapman–Kolmogorov equation is a fundamental relation in the theory of stochastic processes that expresses how transition probabilities of a Markov process over longer time intervals can be obtained by integrating over intermediate states.

A. Kolmogorov backward equation
The Kolmogorov backward equation is a fundamental partial differential equation in stochastic processes that characterizes the time evolution of expected values of functionals of Markov processes, complementary to the Fokker–Planck (forward) equation.
B. Fokker–Planck equation
The Fokker–Planck equation is a partial differential equation that describes the time evolution of the probability density function of a stochastic (random) process, such as Brownian motion.
C. Feynman–Kac formula
The Feynman–Kac formula is a fundamental result connecting solutions of certain partial differential equations with expectations over stochastic processes, forming a bridge between quantum mechanics, probability theory, and mathematical finance.
D. Itô’s lemma
Itô’s lemma is a fundamental result in stochastic calculus that generalizes the chain rule to functions of stochastic processes, especially Brownian motion.
E. Ornstein–Uhlenbeck process
The Ornstein–Uhlenbeck process is a continuous-time stochastic process that models mean-reverting random motion, widely used in physics and quantitative finance to describe systems fluctuating around a long-term equilibrium.
F. None of above. chosen

Statements (48)

Predicate	Object
instanceOf	mathematical equation ⓘ probability theory concept ⓘ stochastic process concept ⓘ
appliesTo	Markov chains ⓘ linked to: Markov processes Markov diffusion processes ⓘ Markov processes ⓘ continuous-time Markov processes ⓘ discrete-time Markov processes ⓘ
assumes	Markov property of the process ⓘ time parameter can be ordered ⓘ
category	Markov process theory ⓘ
coreIdea	transition probabilities over long intervals can be obtained via intermediate states ⓘ
describes	relation between transition probabilities at different times ⓘ
discreteTimeForm	P_{ij}(m+n) = \sum_k P_{ik}(m) P_{kj}(n) ⓘ
ensures	compatibility of transition probabilities over overlapping time intervals ⓘ
equivalentTo	semigroup property P_{s,t} = P_{s,u} P_{u,t} ⓘ
expresses	composition law for transition probabilities ⓘ semigroup property of Markov transition operators ⓘ
field	Markov processes ⓘ probability theory ⓘ stochastic processes ⓘ
formalStatement	P(X_t \in B \mid X_s = x) = \int P(X_t \in B \mid X_u = y) \, P(X_u \in dy \mid X_s = x) for s < u < t ⓘ
generalizes	law of total probability for Markov processes ⓘ
implies	Markov property over multiple time steps ⓘ
involves	integration over intermediate states for continuous state spaces ⓘ summation over intermediate states for discrete state spaces ⓘ
mathematicalDomain	functional analysis ⓘ measure-theoretic probability ⓘ
namedAfter	Andrey Kolmogorov ⓘ linked to: Andrei Kolmogorov Sydney Chapman ⓘ
relatedTo	Fokker–Planck equation ⓘ Kolmogorov backward equation ⓘ Kolmogorov forward equation ⓘ linked to: Fokker–Planck equation Markov semigroup ⓘ master equation ⓘ transition probability kernel ⓘ
relates	short-time transition probabilities to long-time transition probabilities ⓘ
role	consistency condition for finite-dimensional distributions of Markov processes ⓘ fundamental relation for Markov transition functions ⓘ
usedFor	computing multi-step transition probabilities from one-step transitions ⓘ constructing Markov processes from transition kernels ⓘ
usedIn	Markov chain theory ⓘ linked to: Markov processes derivation of Kolmogorov backward equation ⓘ derivation of Kolmogorov forward equation ⓘ financial mathematics ⓘ population dynamics ⓘ queueing theory ⓘ statistical physics ⓘ

How these facts were elicited

Referenced by (4)

Full triples — surface form annotated when it differs from this entity's canonical label.

Markov process → relatedTo → Chapman–Kolmogorov equation ⓘ

subject linked to: Markov processes

Brownian filtration → relatedTo → Markov property of Brownian motion ⓘ

linked to: Chapman–Kolmogorov equation

Sydney Chapman → notableWork → Chapman–Kolmogorov equation ⓘ

Sydney Chapman → knownFor → Chapman–Kolmogorov equation in stochastic processes ⓘ