a Dirichlet form
/ DEER-ish-lay /
A Dirichlet form is the variational, energy-based way to encode a symmetric (reversible) Markov process: instead of describing the dynamics by a generator or transition semigroup, you describe it by a quadratic 'energy' functional on functions, and a remarkable correspondence (the Beurling-Deny / Fukushima theory) translates between the two. This viewpoint constructs and analyses Markov processes on rough spaces — fractals, graphs, infinite-dimensional spaces — where writing a differential generator is hopeless, and it is the natural home of Poincaré and log-Sobolev inequalities.
Definition: on L^2(E, mu) a Dirichlet form is a bilinear map E(f, g), densely defined on a domain F that is a Hilbert space under the norm E(f, f) + ||f||^2_(L^2), which is symmetric (E(f, g) = E(g, f)), non-negative (E(f, f) >= 0), closed (F is complete), and crucially MARKOVIAN: the energy does not increase under normal contractions — if you clip f to the unit interval, say replace f by its truncation f* = min(max(f, 0), 1), then E(f*, f*) <= E(f, f). The correspondence: a closed symmetric form E is in one-to-one relation with a non-positive self-adjoint operator A on L^2(mu) via E(f, g) = - integral (A f) g dmu (so A is the generator), hence with a strongly continuous symmetric contraction semigroup P_t = e^(tA); the Markovian (contraction) property of the form is EXACTLY what makes that semigroup positivity-preserving and sub-Markov, and Fukushima's theorem then associates to a regular Dirichlet form an actual mu-symmetric Hunt (càdlàg strong Markov) process. The prototype is the Dirichlet energy E(f, f) = (1/2) integral |grad f|^2 dx, whose generator is the Laplacian and whose process is Brownian motion.
Why it matters: the Dirichlet-form route is how one rigorously builds diffusions and analyses their spectral gaps and functional inequalities without any smoothness — on fractals, on path/loop spaces, on configuration spaces of particle systems — and it makes the reversibility/self-adjointness structure manifest and computable. Honest scope: the theory as cleanly stated is for SYMMETRIC (reversible) processes; non-symmetric Dirichlet forms exist but are substantially more delicate, and 'regularity' of the form (a compatibility with the topology of E) is a genuine hypothesis needed before a nice associated process is guaranteed, not a free lunch.
The classical Dirichlet form E(f, f) = (1/2) integral_(R^d) |grad f|^2 dx on H^1(R^d) corresponds to the generator A = (1/2) Delta via integration by parts E(f, g) = -integral (A f) g dx, hence to the heat semigroup and Brownian motion. On a discrete graph, the analogous form E(f, f) = (1/2) sum over edges (f(x) - f(y))^2 yields the graph Laplacian and a reversible random walk.
The Dirichlet energy is the variational fingerprint of Brownian motion; the same recipe on a graph gives a reversible walk.
The clean correspondence is for symmetric (reversible) processes; the Markovian (normal-contraction) property and regularity of the form are essential hypotheses, not automatic from being a quadratic form.