Dirichlet forms and degenerate elliptic operators

It is shown that the theory of real symmetric second-order elliptic operators in divergence form on $Ri^d$ 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 $S_t$ can be described in terms of a function $(A,B) mapsto d(A ;B)in[0,infty]$ over pairs of measurable subsets of $Ri^d$. Then [ |(phi_A,S_tphi_B)|leq e^{-d(A;B)^2(4t)^{-1}}|phi_A|_2|phi_B|_2 ] for all $t>0$ and all $phi_Ain L_2(A)$, $phi_Bin L_2(B)$. Moreover $S_tL_2(A)subseteq L_2(A)$ for all $t>0$ if and only if $d(A ;A^c)=infty$ where $A^c$ denotes the complement of $A$.