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The submartingale problem for a class of degenerate elliptic operators
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  • Published: 12 December 2006

The submartingale problem for a class of degenerate elliptic operators

  • Richard F. Bass1 &
  • Alexander Lavrentiev2 

Probability Theory and Related Fields volume 139, pages 415–449 (2007)Cite this article

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  • 3 Citations

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Abstract

We consider the degenerate elliptic operator acting on \({C^2_b}\) functions on [0,∞)d:

$$\mathcal{L}f(x)=\sum_{i=1}^d a_i(x) x_i^{\alpha_i} \frac{\partial^2 f}{\partial x_i^2} (x) +\sum_{i=1}^d b_i(x) \frac{\partial f}{\partial x_i}(x), $$

where the a i are continuous functions that are bounded above and below by positive constants, the b i are bounded and measurable, and the \({\alpha_i\in (0,1)}\) . We impose Neumann boundary conditions on the boundary of [0,∞)d. There will not be uniqueness for the submartingale problem corresponding to \({\mathcal{L}}\) . If we consider, however, only those solutions to the submartingale problem for which the process spends 0 time on the boundary, then existence and uniqueness for the submartingale problem for \({\mathcal{L}}\) holds within this class. Our result is equivalent to establishing weak uniqueness for the system of stochastic differential equations

$$ {\rm d}X_t^i=\sqrt{2a_i(X_t)} (X_t^i)^{\alpha_i/2}{\rm d}W^i_t + b_i(X_t) {\rm d}t + {\rm d}L_t^{X^i},\quad X^i_t \geq 0, $$

where \({W_t^i}\) are independent Brownian motions and \({L^{X_i}_t}\) is a local time at 0 for X i.

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Author information

Authors and Affiliations

  1. Department of Mathematics, University of Connecticut, Storrs, CT, 06269-3009, USA

    Richard F. Bass

  2. Department of Mathematics, University of Wisconsin, Fox Valley Menasha, WI, 54952, USA

    Alexander Lavrentiev

Authors
  1. Richard F. Bass
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  2. Alexander Lavrentiev
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Corresponding author

Correspondence to Richard F. Bass.

Additional information

Research partially supported by NSF grant DMS-0244737.

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Cite this article

Bass, R.F., Lavrentiev, A. The submartingale problem for a class of degenerate elliptic operators. Probab. Theory Relat. Fields 139, 415–449 (2007). https://doi.org/10.1007/s00440-006-0047-9

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  • Received: 15 December 2005

  • Revised: 08 November 2006

  • Published: 12 December 2006

  • Issue Date: November 2007

  • DOI: https://doi.org/10.1007/s00440-006-0047-9

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Keywords

  • Martingale problem
  • Stochastic differential equations
  • Degenerate elliptic operators
  • Speed measure
  • Perturbation
  • Bessel process
  • Littlewood-Paley

Mathematics Subject Classification (2000)

  • Primary 60H10
  • Secondary 60H30
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