Dirichlet forms and degenerate elliptic operators

Abstract

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 \[ |(ϕ_A,S_tϕ_B)|\leq e^{-d(A;B)^2(4t)^{-1}}\|ϕ_A\|_2\|ϕ_B\|_2 \] for all $t>0$ and all $ϕ_A\in L_2(A)$, $ϕ_B\in L_2(B)$. 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 $A$.