TY - JOUR

T1 - Conservativeness of non-symmetric diffusion processes generated by perturbed divergence forms

AU - Takeda, Masayoshi

AU - Trutnau, Gerald

N1 - Funding Information:
The first author was supported in part by Grant-in-Aid for Scientific Research (No. 22340024 (B)), Japan Society for the Promotion of Science. The second author was supported by the Research Settlement Fund for New Faculty and the research project “Advanced Research and Education of Financial Mathematics” at Seoul National University. The corresponding author is Gerald Trutnau.

PY - 2012/3

Y1 - 2012/3

N2 - 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∂ if) + ∑ 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.

AB - 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∂ if) + ∑ 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.

KW - Conservativeness criteria

KW - Diffusion processes

KW - Divergence form operators

KW - Lyons-Zheng decomposition

KW - Non-explosion test

KW - Non-symmetric Dirichlet form

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U2 - 10.1515/FORM.2011.111

DO - 10.1515/FORM.2011.111

M3 - Review article

AN - SCOPUS:84858626630

VL - 24

SP - 419

EP - 444

JO - Forum Mathematicum

JF - Forum Mathematicum

SN - 0933-7741

IS - 2

ER -