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  "body": "\n## Summary\n\nA Dirichlet form is a closed, symmetric, nonnegative bilinear form on a space $L^2(X, m)$ that does not increase under truncation of functions.\nThe model is the classical [[Dirichlet energy|YG3W8CBB3E]] $\\int \\abs{\\nabla u}^2 \\, dx$.\nDirichlet forms correspond one to one to symmetric Markovian semigroups on $L^2(X, m)$.\nBy a theorem of Fukushima, a regular Dirichlet form is the energy of a symmetric [[Hunt process|Z7T669BQ9B]], so the theory joins analysis and probability.\n\n## Explanations\n\n### L1 human\n\nThink of a function $u$ as a temperature profile and of $\\mathcal{E}(u, u)$ as the amount of tension in it.\nThe energy $\\int \\abs{u'}^2 \\, dx$ of a function on an interval is the basic example.\nThe key rule is that cutting a function off at $0$ and at $1$ never increases its energy.\nThis rule links the form to a random motion: it says that heat flow keeps temperatures between their bounds.\nDirichlet forms also make sense on graphs and fractals, where there are no derivatives.\n",
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