## Abstract

We consider properties of second-order operators H = - σ ^{d}_{i,j=1} mathematicl equation presnted on ℝ^{d} with bounded real symmetric measurable coefficients. We assume that C = (c _{ij}) ≥ 0 almost everywhere, but allow for the possibility that C is degenerate. We associate with H a canonical self-adjoint viscosity operator H_{O} and examine properties of the viscosity semigroup S^{(0)} generated by HO. The semigroup extends to a positive contraction semigroup on the L_{p}-spaces with p [1,8]. We establish that it conserves probability and satisfies L_{2} off-diagonal bounds, and that the wave equation associated with HO 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 C^{2-e}(ℝ^{d}) with e < 0.

Original language | English |
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Pages (from-to) | 299-328 |

Number of pages | 30 |

Journal | Proceedings of the London Mathematical Society |

Volume | 95 |

Issue number | 2 |

DOIs | |

Publication status | Published - Sept 2007 |