Markov semigroup
E262082
A Markov semigroup is a family of linear operators describing the time evolution of probability distributions in a Markov process, forming a semigroup under composition and preserving positivity and total mass.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Markov semigroup canonical | 2 |
How this entity was disambiguated
This entity first appeared as the object of triple T2393134 — 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: Markov semigroup Context triple: [Kolmogorov backward equation, associatedWith, Markov semigroup]
-
A.
Markov processes
Markov processes are stochastic processes in which the future evolution depends only on the present state and not on the past history.
-
B.
Chapman–Kolmogorov equation
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.
-
C.
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.
-
D.
Itô processes
Itô processes are a class of stochastic processes, typically modeled as solutions to stochastic differential equations, that form the fundamental objects of study in Itô calculus and modern stochastic analysis.
-
E.
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.
- 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: Markov semigroup Target entity description: A Markov semigroup is a family of linear operators describing the time evolution of probability distributions in a Markov process, forming a semigroup under composition and preserving positivity and total mass.
-
A.
Markov processes
Markov processes are stochastic processes in which the future evolution depends only on the present state and not on the past history.
-
B.
Chapman–Kolmogorov equation
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.
-
C.
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.
-
D.
Itô processes
Itô processes are a class of stochastic processes, typically modeled as solutions to stochastic differential equations, that form the fundamental objects of study in Itô calculus and modern stochastic analysis.
-
E.
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.
- F. None of above. chosen
Statements (48)
| Predicate | Object |
|---|---|
| instanceOf |
family of linear operators
ⓘ
mathematical concept ⓘ semigroup of operators ⓘ |
| actsOn |
probability distributions
ⓘ
transition probabilities ⓘ |
| appearsIn |
mixing and convergence to equilibrium
ⓘ
study of invariant measures ⓘ |
| associatedWith | transition function of a Markov process ⓘ |
| definedOn |
L^p spaces
ⓘ
space of bounded measurable functions ⓘ space of probability measures ⓘ |
| describes |
time evolution of Markov processes
ⓘ
time evolution of probability distributions ⓘ |
| ensures |
Markov property in time evolution
ⓘ
conservation of total probability ⓘ |
| field |
functional analysis
ⓘ
probability theory ⓘ stochastic processes ⓘ |
| generalizes |
continuous-time Markov chain transition matrices
ⓘ
discrete-time Markov chain transition operators ⓘ |
| generatorCalled | Markov generator ⓘ |
| generatorProperty | d/dt T_t f |_{t=0} = A f for generator A ⓘ |
| hasParameter | time parameter t ≥ 0 ⓘ |
| hasVersion |
continuous-time Markov semigroup
ⓘ
discrete-time Markov semigroup ⓘ |
| property |
T_0 = identity operator
ⓘ
T_t 1 = 1 ⓘ T_t f ≥ 0 if f ≥ 0 ⓘ T_{s+t} = T_s ∘ T_t ⓘ contractive on L^∞ ⓘ mass preserving ⓘ positivity preserving ⓘ strongly continuous (in many settings) ⓘ |
| relatedTo |
Fokker–Planck equation
ⓘ
Kolmogorov backward equation ⓘ Fokker–Planck equation ⓘ
surface form:
Kolmogorov forward equation
infinitesimal generator of a Markov process ⓘ |
| specialCaseOf |
C_0-semigroup (in many analytic settings)
ⓘ
positive contraction semigroup ⓘ |
| typicalAssumption |
Feller property in Feller processes
ⓘ
measurability in time parameter ⓘ |
| usedIn |
Markov processes
ⓘ
surface form:
Markov chains
Itô processes ⓘ
surface form:
Markov diffusion processes
Markov processes ⓘ
surface form:
Markov process theory
ergodic theory ⓘ quantum probability ⓘ statistical physics ⓘ stochastic differential equations ⓘ |
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: Markov semigroup Description of subject: A Markov semigroup is a family of linear operators describing the time evolution of probability distributions in a Markov process, forming a semigroup under composition and preserving positivity and total mass.
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.