  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.
