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- Dirichlet_form abstract "In mathematics, a Dirichlet form is a Markovian closed symmetric form on an L2-space. Such objects are studied in abstract potential theory, based on the classical Dirichlet's principle. The theory of Dirichlet forms originated in the work of Beurling and Deny (1958, 1959) on Dirichlet spaces.A Dirichlet form on a measure space is a bilinear functionsuch that1) The domain is a dense subset of 2) is symmetric, that is for any .3) for any .4) The set equipped with the inner product defined by is a real Hilbert space.5) For any we have that and In other words, a Dirichlet form is nothing but a positive symmetric bilinear form defined on a dense subset of such that 4) and 5) hold. Alternatively, the quadratic form itself is known as the Dirichlet form and it is still denoted by , so .The best known Dirichlet form is the Dirichlet energy of functions on which gives rise to the space . Another example of a Dirichlet form is given by where is some non-negative symmetric integral kernel.If the kernel satisfies the bound , then the quadratic form is bounded in . If moreover, , then the form is comparable to the norm in squared and in that case the set defined above is given by . Thus Dirichlet forms are natural generalizations of the Dirichlet integralswhere is a positive symmetric matrix. The Euler-Lagrange equation of a Dirichlet form is a non-local analogue of an elliptic equations in divergence form. Equations of this type are studied using variational methods and they are expected to satisfy similar properties.".
- Dirichlet_form wikiPageID "25002118".
- Dirichlet_form wikiPageRevisionID "591211199".
- Dirichlet_form hasPhotoCollection Dirichlet_form.
- Dirichlet_form id "p/p074150".
- Dirichlet_form last "Beurling".
- Dirichlet_form last "Deny".
- Dirichlet_form title "Abstract potential theory".
- Dirichlet_form year "1958".
- Dirichlet_form year "1959".
- Dirichlet_form subject Category:Markov_processes.
- Dirichlet_form type Abstraction100002137.
- Dirichlet_form type Act100030358.
- Dirichlet_form type Activity100407535.
- Dirichlet_form type Event100029378.
- Dirichlet_form type MarkovProcesses.
- Dirichlet_form type Procedure101023820.
- Dirichlet_form type PsychologicalFeature100023100.
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- Dirichlet_form comment "In mathematics, a Dirichlet form is a Markovian closed symmetric form on an L2-space. Such objects are studied in abstract potential theory, based on the classical Dirichlet's principle.".
- Dirichlet_form label "Dirichlet form".
- Dirichlet_form sameAs m.09g9vjb.
- Dirichlet_form sameAs Q5280765.
- Dirichlet_form sameAs Q5280765.
- Dirichlet_form sameAs Dirichlet_form.
- Dirichlet_form wasDerivedFrom Dirichlet_form?oldid=591211199.
- Dirichlet_form isPrimaryTopicOf Dirichlet_form.