Dirichlet forms and degenerate elliptic operators

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It is shown that the theory of real symmetric second-order elliptic operators in divergence form on ᵈ can be formulated in terms of a regular strongly local Dirichlet form irregardless of the order of degeneracy. The behaviour of the corresponding evolution semigroup Sₜ can be described in terms of a function (A,B) d(A;B)∈[0,∞] over pairs of measurable subsets of ᵈ. Then |(ϕₐ,SₜϕB)|≤ e⁻ᵈ⁽ᵃ;ᵇ⁾²⁽⁴ᵗ⁾⁻¹ϕₐ₂ϕB₂ for all t>0 and all ϕₐ∈ L₂(A), ϕB∈ L₂(B). Moreover SₜL₂(A) L₂(A) for all t>0 if and only if d(A;Aᶜ)=∞ where Aᶜ denotes the complement of A.