Kolmogorov backward equation
E48986
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.
All labels observed (3)
| Label | Occurrences |
|---|---|
| Kolmogorov backward equation canonical | 5 |
| Kolmogorov backward equation for expectations | 1 |
| Kolmogorov equations | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T378947 — 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: Kolmogorov backward equation Context triple: [Fokker–Planck equation, relatedTo, Kolmogorov backward equation]
-
A.
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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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.
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.
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D.
Itô calculus
Itô calculus is a branch of stochastic analysis that extends classical calculus to functions of stochastic processes, particularly Brownian motion, enabling rigorous treatment of stochastic differential equations.
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E.
Girsanov theorem
Girsanov theorem is a fundamental result in stochastic calculus that describes how the dynamics of stochastic processes, particularly Brownian motion, change under an equivalent change of probability measure.
- 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: Kolmogorov backward equation Target entity description: 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.
-
A.
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.
-
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.
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.
-
D.
Itô calculus
Itô calculus is a branch of stochastic analysis that extends classical calculus to functions of stochastic processes, particularly Brownian motion, enabling rigorous treatment of stochastic differential equations.
-
E.
Girsanov theorem
Girsanov theorem is a fundamental result in stochastic calculus that describes how the dynamics of stochastic processes, particularly Brownian motion, change under an equivalent change of probability measure.
- F. None of above. chosen
Statements (49)
| Predicate | Object |
|---|---|
| instanceOf |
Kolmogorov equation
ⓘ
equation in stochastic processes ⓘ partial differential equation ⓘ |
| appliesTo |
Itô processes
ⓘ
surface form:
Itô diffusion
Markov processes ⓘ diffusion processes ⓘ |
| associatedWith |
Markov semigroup
ⓘ
transition function of a Markov process ⓘ |
| characterizes | evolution of conditional expectations ⓘ |
| complements |
Fokker–Planck equation
ⓘ
Fokker–Planck equation ⓘ
surface form:
Kolmogorov forward equation
|
| contrastedWith | forward equation for probability density ⓘ |
| describes | time evolution of expected values of functionals of Markov processes ⓘ |
| equivalentTo | backward Fokker–Planck equation ⓘ |
| field |
mathematical physics
ⓘ
probability theory ⓘ stochastic analysis ⓘ stochastic processes ⓘ |
| hasComponent |
first-order spatial derivative terms
ⓘ
second-order spatial derivative terms ⓘ time derivative term ⓘ |
| hasForm | ∂u/∂t + Lu = 0 ⓘ |
| historicalPeriod | 20th century ⓘ |
| involves |
boundary conditions
ⓘ
diffusion coefficient ⓘ drift coefficient of the diffusion ⓘ terminal condition ⓘ |
| mathematicalNature | linear partial differential equation ⓘ |
| namedAfter |
Andrei Kolmogorov
ⓘ
surface form:
Andrey Kolmogorov
|
| relatedTo |
Dynkin formula
ⓘ
Itô calculus ⓘ generator of a Markov process ⓘ infinitesimal generator of a diffusion ⓘ parabolic partial differential equation ⓘ semigroup of operators ⓘ stochastic differential equation ⓘ |
| solutionMethod | probabilistic representation via Feynman–Kac formula ⓘ |
| solutionType | value function of a stochastic process ⓘ |
| timeDirection | backward in time ⓘ |
| usedFor |
characterizing transition probabilities of Markov processes
ⓘ
computing conditional expectations of functionals of stochastic processes ⓘ optimal control of stochastic systems ⓘ pricing of derivative securities in mathematical finance ⓘ |
| usedIn |
chemical reaction kinetics
ⓘ
epidemiological modeling ⓘ neuroscience modeling of membrane potentials ⓘ population dynamics modeling ⓘ queueing theory ⓘ reliability theory ⓘ |
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: Kolmogorov backward equation Description of subject: 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.
Referenced by (7)
Full triples — surface form annotated when it differs from this entity's canonical label.
subject surface form:
Markov process
this entity surface form:
Kolmogorov equations
this entity surface form:
Kolmogorov backward equation for expectations