A note on Loewner energy, conformal restriction and Werner's measure on self-avoiding loops

In this note, we establish an expression of the Loewner energy of a Jordan curve on the Riemann sphere in terms of Werner’s measure on simple loops of SLE${}_{8\mathrm{/}3}$ type. The proof is based on a formula for the change of the Loewner energy under a conformal map that is reminiscent of the restriction properties derived for SLE processes.