Dirichlet form

Summary

In potential theory (the study of harmonic function) and functional analysis, Dirichlet forms generalize the Laplacian (the mathematical operator on scalar fields). Dirichlet forms 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. The benefit on these spaces is that one can do this without needing a gradient operator, and in particular, one can even weakly define a "Laplacian" in this manner if starting with the Dirichlet form.

Definition edit

When working on  , the "classical" Dirichlet form is given by:

 
where one often discusses   which is often referred to as the "energy" of the function  .

More generally, a Dirichlet form is a Markovian closed symmetric form on an L2-space.[1] In particular, a Dirichlet form on a measure space   is a bilinear function

 
such that
  1.   is a dense subset of  .
  2.   is symmetric, that is   for every  .
  3.   for every  .
  4. The set   equipped with the inner product defined by   is a real Hilbert space.
  5. For every   we have that   and  .

In other words, a Dirichlet form is nothing but a non negative symmetric bilinear form defined on a dense subset of   such that 4) and 5) hold.

Alternatively, the quadratic form   itself is known as the Dirichlet form and it is still denoted by  , so  .

Harmonic functions edit

Functions that minimize the energy given certain boundary conditions are called harmonic, and the associated Laplacian (weak or not) will be zero on the interior, as expected.

For example, let   be standard Dirichlet form defined for   as

 

Then a harmonic function in the standard sense, i.e. such that  , will have   as can be seen with integration by parts.

As an alternative example, the standard graph Dirichlet form is given by:

 
where   means they are connected by an edge. Let a subset of the vertex set be chosen, and call it the boundary of the graph. Assign a Dirichlet boundary condition (choose real numbers for each boundary vertex). One can find a function that minimizes the graph energy, and it will be harmonic. In particular, it will satisfy the averaging property, which is embodied by the graph Laplacian, that is, if   is a graph harmonic then
 
which is equivalent to the averaging property
 

Technically, 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.

Integral kernels edit

Another example of a Dirichlet form is given by

 
where   is some non-negative symmetric integral kernel.

If the kernel   satisfies the bound  , then the quadratic form is bounded in  . If moreover,  , then the form is comparable to the norm in   squared and in that case the set   defined above is given by  . Thus Dirichlet forms are natural generalizations of the Dirichlet integrals

 
where   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 edit

  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, arXiv:math/0609842, doi:10.1090/S0002-9947-08-04544-3, ISSN 0002-9947, S2CID 14411096
  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, S2CID 122914875
  4. ^ Caffarelli, Luis; Chan, Chi Hin; Vasseur, Alexis (2011), "Regularity theory for parabolic nonlinear integral operators", Journal of the American Mathematical Society, 24 (3): 849–869, doi:10.1090/S0894-0347-2011-00698-X, ISSN 0894-0347