Decomposition of backward SLE in the capacity parameterization

2017/10/15 by Mackey, Benjamin, Zhan, Dapeng
#60G #FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.1710.05376

Abstract

We prove that, for κ≤ 4, backward chordal SLE_κ admits backward chordal SLE_κ(-4,-4) decomposition for the capacity parametrization. This means that, for any bounded measurable subset U⊂ Q₄:=\mathbb R₊×\mathbb 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↦ (d_t,c_t) spends in U, where d_t>0>c_t 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 ([1] κ≤ 4, [10] κ<8), and for the capacity parametrization ([10] κ< ∞).

Related

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