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In [[mathematics]], a '''Dirichlet form''' is a [[Markovian]] [[closed linear operator|closed]] [[symmetric form]] on an [[L2-space|''L''<sup>2</sup>-space]].<ref>Fukushima, M, Oshima, Y., & Takeda, M. (1994). ''Dirichlet forms and symmetric Markov processes.'' Walter de Gruyter & Co ,  ISBN 3-11-011626-X</ref> 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 {{harvs|txt|last1=Beurling|last2=Deny|year=1958|year2=1959}} on  Dirichlet spaces.
 
A Dirichlet form on a [[measure space]]  <math>(X, \mu)</math> is a bilinear function
 
:<math>\mathcal{E}: D\times D \to \mathbb{R}</math>
 
such that
 
1) The domain <math>D</math> is a dense subset of <math>L^2(X, \mu)</math>
 
2) <math>\mathcal{E}</math> is symmetric, that is <math>\mathcal{E}(u,v)=\mathcal{E}(v,u)</math> for any <math>u,v \in D</math>.
 
3) <math>\mathcal{E}(u,u) \geq 0</math> for any <math>u \in D</math>.
 
4) The set <math>D</math> equipped with the inner product defined by <math>(u,v)_{\mathcal{E}} := (u,v)_{L^2(X,\mu)} + \mathcal{E}(u,v)</math> is a real Hilbert space.
 
5) For any <math>u \in D</math> we have that <math>u_* = \min (\max(u,  0) , 1) \in D</math> and <math>\mathcal{E}(u_*,u_*)\leq \mathcal{E}(u,u)</math>
 
In other words, a Dirichlet form is nothing but a positive symmetric bilinear form defined on a dense subset of <math>L^2(X, \mu)</math> such that 4) and 5) hold.  
Alternatively, the quadratic form <math>u \to \mathcal{E}(u,u)</math> itself is known as the Dirichlet form and it is still denoted by <math>\mathcal{E}</math>,
so <math>\mathcal{E}(u):=\mathcal{E}(u,u)</math>.
 
The best known Dirichlet form is the Dirichlet energy of functions on <math>\mathbb{R}^n</math>
:<math>\mathcal{E}(u) = \int_{\mathbb{R}^n} |\nabla u|^2\;dx</math>
 
which gives rise to the space <math>H^1(\mathbb{R}^n)</math>. Another example of a Dirichlet form is given by  
:<math>\mathcal{E}(u) = \iint_{\mathbb{R}^n \times \mathbb{R}^n} (u(y)-u(x))^2 k(x,y)\, \mathrm{d}x \mathrm{d} y </math>
 
where <math>k:\mathbb R^n\times \mathbb R^n\to \mathbb R</math> is some non-negative symmetric [[integral kernel]].
 
If the kernel <math>k</math> satisfies the bound <math>k(x,y) \leq \Lambda |x-y|^{-n-s}</math>, then the quadratic form is bounded in <math>\dot H^{s/2}</math>.  
If moreover, <math>\lambda |x-y|^{-n-s} \leq k(x,y)</math>, then the form is comparable to the norm in <math>\dot H^{s/2}</math> squared and in that case the set
<math>D \subset L^2(\mathbb{R}^n)</math> defined above is given by  <math>H^{s/2}(\mathbb{R}^n)</math>. Thus Dirichlet forms are natural generalizations of the [[Dirichlet integral]]s
:<math>\mathcal{E}(u) = \int  (A\nabla u,\nabla u)\; \mathrm{d} x, </math>
 
where <math>A(x)</math> 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.<ref name="BBCK"/><ref name="K"/><ref name="CCV"/>
 
== References ==
(There should be a lot more references here)
{{reflist|refs=
<ref name="CCV">{{Citation | last1=Caffarelli | first1=Luis | last2=Chan | first2=Chi Hin | last3=Vasseur | first3=Alexis | title= | doi=10.1090/S0894-0347-2011-00698-X | year=2011 | journal=[[Journal of the American Mathematical Society]] | issn=0894-0347 | issue=24 | pages=849–869}}</ref>
<ref name="BBCK">{{Citation | last1=Barlow | first1=Martin T. | last2=Bass | first2=Richard F. | last3=Chen | first3=Zhen-Qing | last4=Kassmann | first4=Moritz | title=Non-local Dirichlet forms and symmetric jump processes | url=http://dx.doi.org/10.1090/S0002-9947-08-04544-3 | doi=10.1090/S0002-9947-08-04544-3 | year=2009 | journal=[[Transactions of the American Mathematical Society]] | issn=0002-9947 | volume=361 | issue=4 | pages=1963–1999}}</ref>
<ref name="K">{{Citation | last1=Kassmann | first1=Moritz | title=A priori estimates for integro-differential operators with measurable kernels | url=http://dx.doi.org/10.1007/s00526-008-0173-6 | doi=10.1007/s00526-008-0173-6 | year=2009 | journal=Calculus of Variations and Partial Differential Equations | issn=0944-2669 | volume=34 | issue=1 | pages=1–21}}</ref>
}}
{{Reflist}}
*{{Citation | last1=Beurling | first1=Arne | last2=Deny | first2=J. | title=Espaces de Dirichlet. I. Le cas élémentaire | doi=10.1007/BF02392426 | mr=0098924 | year=1958 | journal=[[Acta Mathematica]] | issn=0001-5962 | volume=99 | pages=203–224}}
*{{Citation | last1=Beurling | first1=Arne | last2=Deny | first2=J. | title=Dirichlet spaces | jstor=90170 | mr=0106365 | year=1959 | journal=[[Proceedings of the National Academy of Sciences|Proceedings of the National Academy of Sciences of the United States of America]] | issn=0027-8424 | volume=45 | pages=208–215}}
*{{Citation | last1=Fukushima | first1=Masatoshi | title=Dirichlet forms and Markov processes | publisher=North-Holland | location=Amsterdam | series=North-Holland Mathematical Library | isbn=978-0-444-85421-6 | mr=569058 | year=1980 | volume=23}}
*{{citation
| last1 = Jost | first1 = Jürgen
| last2 = Kendall | first2 = Wilfrid
| last3 = Mosco | first3 = Umberto
| last4 = Röckner | first4 = Michael
| last5 = Sturm | first5 = Karl-Theodor | author5-link = Karl-Theodor Sturm
| mr = 1652277
| isbn = 0-8218-1061-8
| location = Providence, RI
| page = xiv+277
| publisher = American Mathematical Society
| series = AMS/IP Studies in Advanced Mathematics
| title = New directions in Dirichlet forms
| volume = 8
| year = 1998}}.
*{{eom|id=p/p074150|title=Abstract potential theory}}
 
{{DEFAULTSORT:Dirichlet Form}}
[[Category:Markov processes]]

Latest revision as of 07:38, 7 May 2014

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