# Dirichlet form

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

    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

    * The domain $D$ is a dense subset of $\mathbb{R}^n$ and
* $\mathcal{E}$ is symmetric, that is $\mathcal{E}(f,g)=\mathcal{E}(g,f)$ for any $f,g \in D$.
* $\mathcal{E}(f,f)\geq 0$ for any $f \in D$.
* The set $D$ equipped with the inner product defined by

    \begin{equation*}
(f,g)_{\mathcal{E}} := (f,g)_{L^2(\mathbb{R}^n)} + \mathcal{E}(f,g)
\end{equation*}


A Dirichlet form refers to a quadratic functional defined by an integral 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 to fractional order 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
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
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