  Standard Schramm-Loewner evolution (SLE) is driven by a continuous Brownian
motion which then produces a trace, a continuous fractal curve connecting the
singular points of the motion. If jumps are added to the driving function, the
trace branches. In a recent publication [1] we introduced a generalized SLE
driven by a superposition of a Brownian motion and a fractal set of jumps
(technically a stable L\'evy process). We then discussed the small-scale
properties of the resulting L\'evy-SLE growth process. Here we discuss the same
model, but focus on the global scaling behavior which ensues as time goes to
infinity. This limiting behavior is independent of the Brownian forcing and
depends upon only a single parameter, $\alpha$, which defines the shape of the
stable L\'evy distribution. We learn about this behavior by studying a
Fokker-Planck equation which gives the probability distribution for endpoints
of the trace as a function of time. As in the short-time case previously
studied, we observe that the properties of this growth process change
qualitatively and singularly at $\alpha =1$. We show both analytically and
numerically that the growth continues indefinitely in the vertical direction
for $\alpha > 1$, goes as $\log t$ for $\alpha = 1$, and saturates for $\alpha<
1$. The probability density has two different scales corresponding to
directions along and perpendicular to the boundary. In the former case, the
characteristic scale is $X(t) \sim t^{1/\alpha}$. In the latter case the scale
is $Y(t) \sim A + B t^{1-1/\alpha}$ for $\alpha \neq 1$, and $Y(t) \sim \ln t$
for $\alpha = 1$. Scaling functions for the probability density are given for
various limiting cases.
