Dirichlet form

Summary

A Dirichlet form is a closed, symmetric, nonnegative bilinear form on a space $L^2(X, m)$ that does not increase under truncation of functions. The model is the classical Dirichlet energy $\int \abs{\nabla u}^2 \, dx$. Dirichlet forms correspond one to one to symmetric Markovian semigroups on $L^2(X, m)$. By a theorem of Fukushima, a regular Dirichlet form is the energy of a symmetric Hunt process, so the theory joins analysis and probability.