## Abstract

Let E ⊂ ℝ ^{d}, d ≥ 2, be an unbounded domain that is either open or closed. If it is closed, we assume that the boundary is locally the boundary of an extension domain. We present conservativeness criteria for (possibly reflected) diffusions with state space E and generator L which in the interior of E is given in the following suggestive form: Lf =1/2 ∑ _{i,j=1} ^{d}∂ _{j}(a _{ij}∂ _{i}f) + ∑ _{i=1} ^{d} B _{i}∂ _{i} f. Here the diffusion matrix (a _{ij}) is allowed to be non-symmetric, is merely assumed to consist of measurable functions, and satisfies locally a strict ellipticity condition. Moreover, B = (B _{1}; : : : ;B _{d}) is a divergence free vector field that satisfies some sector condition. Our main tool is a recently extended forward and backward martingale decomposition, which reduces to the well-known Lyons-Zheng decomposition in the symmetric case.

Original language | English |
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Pages (from-to) | 419-444 |

Number of pages | 26 |

Journal | Forum Mathematicum |

Volume | 24 |

Issue number | 2 |

DOIs | |

Publication status | Published - 2012 Mar 1 |

## Keywords

- Conservativeness criteria
- Diffusion processes
- Divergence form operators
- Lyons-Zheng decomposition
- Non-explosion test
- Non-symmetric Dirichlet form

## ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics