Dirichlet form

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1) The domain $D$ is a dense subset of $\mathbb{R}^n$
1) The domain $D$ is a dense subset of $\mathbb{R}^n$
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2) $\mathcal{E}$ is symmetric, that is $\mathcal{E}(f,g)=\mathcal{E}(g,f)$ for any $f,g \in D$.
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2) $\mathcal{E}$ is symmetric, that is $\mathcal{E}(u,v)=\mathcal{E}(v,u)$ for any $u,v \in D$.
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3) $\mathcal{E}(f,f)\geq 0$ for any $f \in D$.
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3) $\mathcal{E}(u,u) \geq 0$ for any $u \in D$.
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4) The set $D$ equipped with the inner product defined by $(f,g)_{\mathcal{E}} := (f,g)_{L^2(\mathbb{R}^n)} + \mathcal{E}(f,g)$ is a real Hilbert space.
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4) The set $D$ equipped with the inner product defined by $(u,v)_{\mathcal{E}} := (u,v)_{L^2(\mathbb{R}^n)} + \mathcal{E}(u,v)$ is a real Hilbert space.
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5) For any $f \in D$ we have that $f_* = (f\vee 0) \wedge 1 \in D$ and $\mathcal{E}(f_*,f_*)\leq \mathcal{E}(f,f)$
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5) For any $u \in D$ we have that $u_* = (u\vee 0) \wedge 1 \in D$ and $\mathcal{E}(u_*,u_*)\leq \mathcal{E}(u,u)$
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A particular case of a Dirichlet form are defined by  integrals of the form
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An example of a Dirichlet form is given by  any integral of the form
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\[ \iint_{\R^n \times \R^n} (u(y)-u(x))^2 k(x,y)\, \dd x \dd y, \]
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\mathcal{E}(u,v) = \[ \iint_{\R^n \times \R^n} (u(y)-u(x))(v(y)-v(x))k(x,y)\, \dd x \dd y, \]
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for some nonnegative kernel $K$.
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where $K$ is some non-negative symmetric kernel.
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If the kernel $K$ satisfies the bound $K(x,y) \leq \Lambda |x-y|^{-n-s}$, then the quadratic form is bounded in $\dot H^{s/2}$. If moreover, $\lambda |x-y|^{-n-s} \leq K(x,y)$, then the form is comparable to the norm in $\dot H^{s/2}$ squared.
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If the kernel $K$ satisfies the bound $K(x,y) \leq \Lambda |x-y|^{-n-s}$, then the quadratic form is bounded in $\dot H^{s/2}$ . If moreover, $\lambda |x-y|^{-n-s} \leq K(x,y)$, then the form is comparable to the norm in $\dot H^{s/2}$ squared and in that case the set $D \subset L^2(\mathbb{R}^n)$ defined above is given by  $H^{s/2}(\mathbb{R}^n)$
Dirichlet forms are natural generalizations of the Dirichlet integrals
Dirichlet forms are natural generalizations of the Dirichlet integrals

Revision as of 21:57, 18 November 2012

$$ \newcommand{\dd}{\mathrm{d}} \newcommand{\R}{\mathbb{R}} $$

A Dirichlet form in $\mathbb{R}^n$ is a bilinear function

\begin{equation*} \mathcal{E}: D\times D \to \mathbb{R} \end{equation*}

with the following properties

1) The domain $D$ is a dense subset of $\mathbb{R}^n$

2) $\mathcal{E}$ is symmetric, that is $\mathcal{E}(u,v)=\mathcal{E}(v,u)$ for any $u,v \in D$.

3) $\mathcal{E}(u,u) \geq 0$ for any $u \in D$.

4) The set $D$ equipped with the inner product defined by $(u,v)_{\mathcal{E}} := (u,v)_{L^2(\mathbb{R}^n)} + \mathcal{E}(u,v)$ is a real Hilbert space.

5) For any $u \in D$ we have that $u_* = (u\vee 0) \wedge 1 \in D$ and $\mathcal{E}(u_*,u_*)\leq \mathcal{E}(u,u)$


An example of a Dirichlet form is given by any integral of the form \mathcal{E}(u,v) = \[ \iint_{\R^n \times \R^n} (u(y)-u(x))(v(y)-v(x))k(x,y)\, \dd x \dd y, \] where $K$ is some non-negative symmetric kernel.

If the kernel $K$ satisfies the bound $K(x,y) \leq \Lambda |x-y|^{-n-s}$, then the quadratic form is bounded in $\dot H^{s/2}$ . If moreover, $\lambda |x-y|^{-n-s} \leq K(x,y)$, then the form is comparable to the norm in $\dot H^{s/2}$ squared and in that case the set $D \subset L^2(\mathbb{R}^n)$ defined above is given by $H^{s/2}(\mathbb{R}^n)$

Dirichlet forms are natural generalizations of the Dirichlet integrals \[ \int a_{ij}(x) \partial_i u \partial_j u \dd x, \] where $a_{ij}$ is elliptic.

The Euler-Lagrange equation of a Dirichlet form is a fractional order version of elliptic equations in divergence form. They are studied using variational methods and they are expected to satisfy similar properties [1][2][3].

References

(There should be a lot more references here)

  1. Barlow, Martin T.; Bass, Richard F.; Chen, Zhen-Qing; Kassmann, Moritz (2009), "Non-local Dirichlet forms and symmetric jump processes", Transactions of the American Mathematical Society 361 (4): 1963–1999, doi:10.1090/S0002-9947-08-04544-3, ISSN 0002-9947, http://dx.doi.org/10.1090/S0002-9947-08-04544-3 
  2. Kassmann, Moritz (2009), "A priori estimates for integro-differential operators with measurable kernels", Calculus of Variations and Partial Differential Equations 34 (1): 1–21, doi:10.1007/s00526-008-0173-6, ISSN 0944-2669, http://dx.doi.org/10.1007/s00526-008-0173-6 
  3. Caffarelli, Luis; Chan, Chi Hin; Vasseur, Alexis (2011), Journal of the American Mathematical Society (24): 849–869, doi:10.1090/S0894-0347-2011-00698-X, ISSN 0894-0347 


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