Dirichlet form

In the branch of mathematics known as potential theory, a Dirichlet form is a generalization of the Laplacian that can be defined on any measure space, without the need for mentioning partial derivatives. This allows mathematicians to study the Laplace equation and heat equation on spaces that are not manifolds: for example, fractals. To accomplish this generalization, one focuses not on the Laplacian itself but on the quantity

$\mathcal{E}(u) = \int_{\mathbb{R}^n} |\nabla u|^2\;dx$

that is minimized when the Laplacian vanishes.

Technically, a Dirichlet form is a Markovian closed symmetric form on an L²-space.^[1] 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 Beurling and Deny (1958, 1959) on Dirichlet spaces.

A Dirichlet form on a measure space $(X, \mu)$ is a bilinear function

$\mathcal{E}: D\times D \to \mathbb{R}$

such that

1) The domain $D$ is a dense subset of $L^2(X, \mu)$

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 $(u,v)_{\mathcal{E}} := (u,v)_{L^2(X,\mu)} + \mathcal{E}(u,v)$ is a real Hilbert space.

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

In other words, a Dirichlet form is nothing but a positive symmetric bilinear form defined on a dense subset of $L^2(X, \mu)$ such that 4) and 5) hold. Alternatively, the quadratic form $u \to \mathcal{E}(u,u)$ itself is known as the Dirichlet form and it is still denoted by $\mathcal{E}$, so $\mathcal{E}(u):=\mathcal{E}(u,u)$.

The best known Dirichlet form is the Dirichlet energy of functions on $\mathbb{R}^n$

$\mathcal{E}(u) = \int_{\mathbb{R}^n} |\nabla u|^2\;dx$

which gives rise to the space $H^1(\mathbb{R}^n)$. Another example of a Dirichlet form is given by

$\mathcal{E}(u) = \iint_{\mathbb{R}^n \times \mathbb{R}^n} (u(y)-u(x))^2 k(x,y)\, \mathrm{d}x \mathrm{d} y$

where $k:\mathbb R^n\times \mathbb R^n\to \mathbb R$ is some non-negative symmetric integral 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 $H^{s/2}(\mathbb{R}^n)$. Thus Dirichlet forms are natural generalizations of the Dirichlet integrals

$\mathcal{E}(u) = \int (A\nabla u,\nabla u)\; \mathrm{d} x,$

where $A(x)$ 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.^[2][3][4]

References

1. Fukushima, M, Oshima, Y., & Takeda, M. (1994). Dirichlet forms and symmetric Markov processes. Walter de Gruyter & Co , ISBN 3-11-011626-X
2. 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
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
4. Caffarelli, Luis; Chan, Chi Hin; Vasseur, Alexis (2011), "Regularity theory for parabolic nonlinear integral operators", Journal of the American Mathematical Society 24 (24): 849–869, doi:10.1090/S0894-0347-2011-00698-X, ISSN 0894-0347

• Beurling, Arne; Deny, J. (1958), "Espaces de Dirichlet. I. Le cas élémentaire", Acta Mathematica 99 (1): 203–224, doi:10.1007/BF02392426, ISSN 0001-5962, MR 0098924
• Beurling, Arne; Deny, J. (1959), "Dirichlet spaces", Proceedings of the National Academy of Sciences of the United States of America 45 (2): 208–215, doi:10.1073/pnas.45.2.208, ISSN 0027-8424, JSTOR 90170, MR 0106365
• Fukushima, Masatoshi (1980), Dirichlet forms and Markov processes, North-Holland Mathematical Library 23, Amsterdam: North-Holland, ISBN 978-0-444-85421-6, MR 569058
• Jost, Jürgen; Kendall, Wilfrid; Mosco, Umberto; Röckner, Michael; Sturm, Karl-Theodor (1998), New directions in Dirichlet forms, AMS/IP Studies in Advanced Mathematics 8, Providence, RI: American Mathematical Society, p. xiv+277, ISBN 0-8218-1061-8, MR 1652277.
• Hazewinkel, Michiel, ed. (2001), "Abstract potential theory", Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4