Second-order operators with degenerate coefficients

Abstract

We consider properties of second-order operators H = − ∑ i , j = 1 d ∂ i c i j ∂ j on ℝd with bounded real symmetric measurable coefficients. We assume that C = (cij) ⩾ 0 almost everywhere, but allow for the possibility that C is degenerate. We associate with H a canonical self-adjoint viscosity operator H0 and examine properties of the viscosity semigroup S(0) generated by H0. The semigroup extends to a positive contraction semigroup on the Lp-spaces with p ∈ [1, ∞]. We establish that it conserves probability and satisfies L2 off-diagonal bounds, and that the wave equation associated with H0 has finite speed of propagation. Nevertheless, S(0) is not always strictly positive because separation of the system can occur even for subelliptic operators. This demonstrates that subelliptic semigroups are not ergodic in general and their kernels are neither strictly positive nor Hölder continuous. In particular, one can construct examples for which both upper and lower Gaussian bounds fail even with coefficients in C2−ε(ℝd) with ε > 0.