Chapman–Kolmogorov equation
E254907
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.
All labels observed (2)
| Label | Occurrences |
|---|---|
| Chapman–Kolmogorov equation canonical | 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.
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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.
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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.
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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.
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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 processes
ⓘ
surface form:
Markov chains
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 |
Andrei Kolmogorov
ⓘ
surface form:
Andrey Kolmogorov
Sydney Chapman ⓘ |
| relatedTo |
Fokker–Planck equation
ⓘ
Kolmogorov backward equation ⓘ Fokker–Planck equation ⓘ
surface form:
Kolmogorov forward 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 processes
ⓘ
surface form:
Markov chain theory
derivation of Kolmogorov backward equation ⓘ derivation of Kolmogorov forward equation ⓘ financial mathematics ⓘ population dynamics ⓘ queueing theory ⓘ statistical physics ⓘ |
How these facts were elicited
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
Instruction
You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10. # Requirements - If you don't know the subject at all, return an empty list. - If the subject is not a named entity, return an empty list. - Include at least one triple where predicate is "instanceOf". - Do not get too wordy. - Separate several objects into multiple triples with one object.
Input
Subject: Chapman–Kolmogorov equation Description of subject: 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.
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.
subject surface form:
Markov process
this entity surface form:
Markov property of Brownian motion