Conformal restriction: The chordal case

We characterize and describe all random subsets $K$of a given simply connected planar domain (the upper half-plane ${\mathbb H}$, say) which satisfy the ``conformal restriction'' property, i.e., $K$ connects two fixed boundary points ($0$ and $\infty$, say) and the law of $K$ conditioned to remain in a simply connected open subset $H$ of ${\mathbb H}$is identical to that of $\Phi(K)$, where $\Phi$ is a conformal map from ${\mathbb H}$ onto $H$ with $\Phi(0)=0$ and $\Phi(\infty)=\infty$. The construction of this family relies on the stochastic Loewner evolution processes with parameter $\kappa \le 8/3$ and on their distortion under conformal maps. We show in particular that SLE$_{8/3}$ is the only random simple curve satisfying conformal restriction and we relate it to the outer boundaries of planar Brownian motion and SLE$_6$.