# Dirichlet form

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- | + | 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 $\mathcal{E}(f,g)=\mathcal{E}(g,f)$ for any $f,g \in D$. | |

- | + | ||

- | + | ||

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

+ | 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. | ||

- | A Dirichlet form | + | 5) For any $f \in D$ we have that $f_* = (f\vee 0) \wedge 1 \in D$ and $\mathcal{E}(f_*,f_*)\leq \mathcal{E}(f,f)$ |

+ | |||

+ | |||

+ | A particular case of a Dirichlet form are defined by integrals of the form | ||

\[ \iint_{\R^n \times \R^n} (u(y)-u(x))^2 k(x,y)\, \dd x \dd y, \] | \[ \iint_{\R^n \times \R^n} (u(y)-u(x))^2 k(x,y)\, \dd x \dd y, \] | ||

for some nonnegative kernel $K$. | for some nonnegative kernel $K$. | ||

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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. | 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. | ||

- | Dirichlet forms are natural generalizations | + | Dirichlet forms are natural generalizations of the Dirichlet integrals |

\[ \int a_{ij}(x) \partial_i u \partial_j u \dd x, \] | \[ \int a_{ij}(x) \partial_i u \partial_j u \dd x, \] | ||

where $a_{ij}$ is elliptic. | where $a_{ij}$ is elliptic. |

## Revision as of 21:53, 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 $\mathcal{E}(f,g)=\mathcal{E}(g,f)$ for any $f,g \in D$.

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

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.

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

A particular case of a Dirichlet form are defined by integrals of the form
\[ \iint_{\R^n \times \R^n} (u(y)-u(x))^2 k(x,y)\, \dd x \dd y, \]
for some nonnegative kernel $K$.

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.

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)

- ↑ 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 - ↑ 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 - ↑ 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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