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Decomposition of backward SLE in the capacity parametrization

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  • Mackey, Benjamin
  • Zhan, Dapeng

Abstract

We prove that, for κ≤4, backward chordal SLEκ admits backward chordal SLEκ(−4,−4) decomposition for the capacity parametrization. This means that, for any bounded measurable subset U⊂Q4≔R+×R−, if we integrate the laws of extended backward chordal SLE κ(−4,−4) with different pairs of force points (x,y) against some suitable density function G(x,y) restricted to U, then we get a measure, which is absolutely continuous with respect to the law of backward chordal SLE κ, and the Radon–Nikodym derivative is a constant depending on κ times the capacity time that the generated welding curve t↦(dt,ct) spends in U, where dt>0>ct are the pair of points that are swallowed by the process at time t. For the forward SLE curve, a similar analysis has been done for SLE in the natural parametrization (Field, 0000) κ≤4, (Zhan, 0000) κ<8), and for the capacity parametrization (Zhan, 0000) κ<∞).

Suggested Citation

  • Mackey, Benjamin & Zhan, Dapeng, 2019. "Decomposition of backward SLE in the capacity parametrization," Statistics & Probability Letters, Elsevier, vol. 146(C), pages 27-35.
    • Handle: RePEc:eee:stapro:v:146:y:2019:i:c:p:27-35
    DOI: 10.1016/j.spl.2018.10.021
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