Reversibility of Backward SLE Lamination

Dapeng Zhan
Michigan State University

The backward chordal SLE$_\kappa$ is defined by solving the backward chordal Loewner equation $\partial_t f_t(z)=\frac{-2}{f_t(z)-\sqrt{\kappa} B_t}$, $f_0(z)=z$, where $B_t$ is a standard Brownian motion. We are especially interested in the case $\kappa\in(0,4]$, in which every $f_t$ maps the upper half plane conformally onto the upper half plane without a simple curve, and the continuation of $f_t$ maps two real intervals onto the two sides of the curve. Moreover, the family $(f_t)$ induces a conformal lamination, i.e., a random homemorphism $\phi$ between $[0,\infty)$ and $(-\infty,0]$ such that every $x>0$ is glued together with $\phi(x)<0$ by $f_t$ when $t$ is big enough. Our main result is that such lamination satisfies the following type of reversibility: if we define $\psi(x)=1/\phi(1/x)$, then $\psi$ has the same distribution as $\phi$. The talk is based on the joint work with Steffen Rohde.

Presentation (PDF File)

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