The Conformal Invariance Of SLE(κ,ρ) With Force Points

Stochastic Loewner evolution (SLE) is a one-parameter family of random planargrowth processes obtained by solving the Loewner equation when the driving function isa one-dimensional Browner motion. In this paper the conformal invariance of SLE(κ, ρ)with force point are studied. First, the Mo¨bius transformation of chordal and radialSLE(κ, ρ) are discussed. It is proved that the image of Mo¨bius transformation underunit circle D of radial SLE(κ, ρ) starting from D to the upper half-plane H has the samelaw as a time change of chordal SLE(κ, ρ) in H; at the same time, we deduce that theradial SLE(κ, ρ) under D has the same law as the image of Mo¨bius coordinate trans-formation starting from unit circle to itself; and the chordal SLE(κ, ρ) under H has thesame law as the image of Mo¨bius transformation starting from upper half-plane to itself;and establish associated martingales. Second, the conformal invariance of SLE(κ, ρ) instrip domain are discussed. we derive that the image of conformal mapping of SLE(κ, ρ)with force point under Strip domain S has the same law as radial SLE(κ, ρ) with forcepoint in D; we obtain that the image of conformal mapping under D starting from Dto S has the same law as Strip-SLE(κ, ρ) with force point in S; we conclude that theSLE(κ, ρ) with force point under S has the same law as a time change of the image forthe conformal mapping from S to itself.