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Uniqueness in law for a class of degenerate diffusions with continuous covariance
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  • Published: 16 November 2011

Uniqueness in law for a class of degenerate diffusions with continuous covariance

  • Gerard Brunick1 

Probability Theory and Related Fields volume 155, pages 265–302 (2013)Cite this article

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Abstract

We study the martingale problem associated with the operator

$$ Lu(s, x) = \partial_su(s, x) + \frac{1}{2} \sum_{i,j=1}^{d_0} a^{ij}(s, x) \partial_{ij}u(s, x) + \sum_{i,j=1}^d B^{ij} x^j \partial_iu(s, x), $$

where d 0 ≤  d. We show that the martingale problem is well-posed when the function a is continuous and strictly positive definite on \({\mathbb{R}^{d_0}}\) and the matrix B takes a particular lower-diagonal, block form. We then localize this result to show that the martingale problem remains well-posed when B is replaced by a sufficiently smooth vector field whose Jacobian matrix satisfies a nondegeneracy condition.

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Authors and Affiliations

  1. Department of Statistics and Applied Probability, University of California, Santa Barbara, CA, 93106-3110, USA

    Gerard Brunick

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  1. Gerard Brunick
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Correspondence to Gerard Brunick.

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Brunick, G. Uniqueness in law for a class of degenerate diffusions with continuous covariance. Probab. Theory Relat. Fields 155, 265–302 (2013). https://doi.org/10.1007/s00440-011-0398-8

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  • Received: 09 May 2011

  • Revised: 17 September 2011

  • Published: 16 November 2011

  • Issue Date: February 2013

  • DOI: https://doi.org/10.1007/s00440-011-0398-8

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Keywords

  • Martingale problem
  • Stochastic differential equations
  • Degenerate parabolic operators
  • Homogeneous groups

Mathematics Subject Classification (2000)

  • 60H10
  • 35K65
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