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

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 .

References

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