  The two-dimensional Loewner exploration process is generalized to the case
where the random force is self-similar with positively correlated increments.
We model this random force by a fractional Brownian motion with Hurst exponent
$H\geq \frac{1}{2}\equiv H_{\text{BM}}$, where $H_{\text{BM}}$ stands for the
one-dimensional Brownian motion. By manipulating the deterministic force, we
design a scale-invariant equation describing self-similar traces which lack
conformal invariance. The model is investigated in terms of the "input
diffusivity parameter" $\kappa$, which coincides with the one of the ordinary
Schramm-Loewner evolution (SLE) at $H=H_{\text{BM}}$. In our numerical
investigation, we focus on the scaling properties of the traces generated for
$\kappa=2,3$, $\kappa=4$ and $\kappa=6,8$ as the representatives, respectively,
of the dilute phase, the transition point and the dense phase of the ordinary
SLE. The resulting traces are shown to be scale-invariant. Using two equivalent
schemes, we extract the fractal dimension, $D_f(H)$, of the traces which
decrease monotonically with increasing $H$, reaching $D_f=1$ at $H=1$ for all
$\kappa$ values. The left passage probability (LPP) test demonstrates that, for
$H$ values not far from the uncorrelated case (small $\epsilon_H\equiv
\frac{H-H_{\text{BM}}}{H_{\text{BM}}}$) the prediction of the ordinary SLE is
applicable with an effective diffusivity parameter $\kappa_{\text{eff}}$. Not
surprisingly, the $\kappa_{\text{eff}}$'s do not fulfill the prediction of SLE
for the relation between $D_f(H)$ and the diffusivity parameter.
