Convergence to closed-form distribution for the backward $SLE_{\kappa}$ at some random times and the phase transition at $\kappa=8$

We study a one-dimensional SDE that we obtain by performing a random time change of the backward Loewner dynamics in $\mathbb{H}$. The stationary measure for this SDE has a closed-form expression. We show the convergence towards its stationary measure for this SDE, in the sense of random ergodic averages. The precise formula of the density of the stationary law gives a phase transition at the value $\kappa=8$ from integrability to non-integrability, that happens at the same value of $\kappa$ as the change in behavior of the $SLE_{\kappa}$ trace from non-space filling to space-filling curve. Using convergence in total variation for the law of this diffusion towards stationarity, we identify families of random times on which the law of the arguments of points under the backward $SLE_{\kappa}$ flow converge to a closed form expression measure. For $\kappa=4,$ this gives precise characterization for the random times on which the law of the arguments of points under the backward $SLE_{\kappa}$ flow converge to the uniform law.