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I'm currentely studing the Dirichlet form and to be honest, I really don't see in what they are useful. I don't really get the point with them. I recall the definition :

Definition Let $(H,\left<\cdot ,\cdot \right>)$ a Hilbert space. Set $E=E^s+E^a$ a bilinear form defined on a dense subset $D$ of $B$ where $E^s$ is symmetric and $E^a$ antisymmetric. We Then $E$ is a Dirichlet form if

  • $E^s$ is positive definite on $D$

  • $(E^s+\left<\cdot ,\cdot \right>,D)$ is a Hilbert space,

  • $(E,D)$ is coercive, i.e. there is $K>0$ s.t. $$|E(x,y)+\left<x,y\right>|^2\leq K |E^s(x,x)+\left<x,x\right>||E^s(y,y)+\left<y,y\right>|$$

  • for all $x\in D$, we have $x^*=\min(x^+,1)\in D$ where $x^+=\max\{x,0\}$ and $$E(x+x^*,x-x^*)\geq 0\quad \text{and}\quad E(x-x^*,x+x^*)\geq 0.$$

Seeing this definition, what is the motivation behind ? Because as written, it looks a bit barbarous for me. I can accept the first point of the definition, but the 3 other assumption looks to arise from nowhere. Maybe someone knows a very good small introduction to get the point with these Dirichlet form ?

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  • $\begingroup$ I' don't know much about Dirichlet forms specifically, but that looks suspiciously like a retroactive definition: a definition that's been written precisely so that some theorem or other is true about the things that it's defining. $\endgroup$ Jan 16 '19 at 13:41
  • $\begingroup$ There is a typo in the fourth bullet point. It should be $x^\ast=\min(x^+,1)$. $\endgroup$
    – MaoWao
    Jan 16 '19 at 13:41
  • $\begingroup$ @MaoWao: thank you :) $\endgroup$
    – idm
    Jan 16 '19 at 13:43
  • $\begingroup$ One more quibble: Dirichlet forms are not defined on arbitrary Hilbert spaces, but on $L^2$ spaces (otherwise, the $1$ in the fourth condition would not make sense). $\endgroup$
    – MaoWao
    Jan 16 '19 at 14:57
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1) Don't get caught up in the technical details too much. I will focus on symmetric Dirichlet forms because I think the situation is somewhat more transparent in this case (the coercivity condition is automatically satisfied for symmetric forms).

Condition 2 is there to ensure the existence of a generator of the form. More precisely, there is a bijective correspondence between positive self-adjoint operators and symmetric bilinear forms satisfying the first two conditions assigning to the form $E$ the operator $$ D(L)=\{u\in D(E)\mid\exists v\in H\,\forall w\in D(E)\colon E(u,w)=\langle v,w\rangle\},\,Lu=v. $$ One particularly important example is the Dirichlet energy $$ D(E)=W^{1,2}(\mathbb{R^n}),\,E(u,v)=\int_{\mathbb{R^n}}\nabla u\cdot\nabla v\,dx. $$ The corresponding operator is $L=-\Delta$ with domain $D(L)=W^{2,2}(\mathbb{R}^n)$.

For non-symmetric forms one needs the coercivity condition to extend this correspondence (with a wider class of operators).

2) The fourth condition is the most important one. In the symmetric case it can be reformulated as $E(C(u),C(u))\leq E(u,u)$ for $C\in C^1(\mathbb{R})$ with $C(0)=0$ and $|C'|\leq 1$. As you see, this is satisfied for the Dirichlet energy above.

There is the following intuition behind: $E(u)$ is supposed to measure the oscillation of $u$ (if you know a little quantum mechanics, think of $u$ as wave function and of the oscillation of $u$ as measure of the energy of a paricle in state $u$). The fourth condition then says that the oscillation decreases if one damps the function $u$.

This property has several interesting consequences. The first (and easiest to prove) is that the semigroup generated by the generator of a Dirichlet form is Markovian. Markovian semigroups play a role in many places in mathematics, most notably in connection with Markov processes, which brings me to the second consequence.

Fukushima (later extended to the non-symmetric case by Ma and Röckner) showed that there is bijective correspondence between so-called regular Dirichlet forms and a class of symmetric Markov processes. This allows one to define Markov processes on non-smooth spaces like fractals or limit spaces of Riemannian manifolds.

Finally, there is another nice characterization of Dirichlet forms. As you see, the Dirichlet energy is of the form $E(u,v)=\langle \nabla u,\nabla v\rangle$ and $\nabla$ satisfies the Leibniz rule. Cipriani and Sauvageot showed that every Dirichlet form can be written in the form $E(u,v)=\langle \partial u,\partial v\rangle_{\mathcal{H}}$, where $\partial$ is a derivation with values in the Hilbert space $\mathcal{H}$. One can use this characterization for example to define the gradient of a function on metric spaces where you don't have a good geometric notion of tangent space.

To wrap it up, Dirichlet forms are related to a lot of interesting mathematical objects at the intersection of analysis, geometry and probability, and, what is nore, they often provide a technically easier approach.

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  • $\begingroup$ Thank you for your very nice answer. Just a questions : What do you mean by "generator of the form" ? In what $L=-\Delta $ is relevant ? $\endgroup$
    – idm
    Jan 16 '19 at 16:12

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