2015/09/16 by Zhan, Dapeng
#60D #FOS: Mathematics #Probability (math.PR)
paper · doi:10.48550/arxiv.1509.05015
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 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 \mathbb R against a measure with density on \mathbb 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.