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Decomposition of Schramm–Loewner evolution along its curve

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  • Zhan, Dapeng

Abstract

We show that, for κ∈(0,8), the integral of the laws of two-sided radial SLEκ curves through different interior points against a measure with SLEκ Green’s function density is the law of a chordal SLEκ curve, biased by the path’s natural length. We also show that, for κ>0, the integral of the laws of extended SLEκ(−8) curves through different interior points against a measure with a closed formula density restricted in a bounded set is the law of a chordal SLEκ curve, biased by the path’s capacity length restricted in that set. Another result is that, for κ∈(4,8), if one integrates the laws of two-sided chordal SLEκ curves through different force points on R against a measure with density on R, then one also gets a law that is absolutely continuous w.r.t. that of a chordal SLEκ curve. To obtain these results, we develop a framework to study stochastic processes with random lifetime, and improve the traditional Girsanov’s Theorem.

Suggested Citation

  • Zhan, Dapeng, 2019. "Decomposition of Schramm–Loewner evolution along its curve," Stochastic Processes and their Applications, Elsevier, vol. 129(1), pages 129-152.
    • Handle: RePEc:eee:spapps:v:129:y:2019:i:1:p:129-152
    DOI: 10.1016/j.spa.2018.02.010
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