Schramm–Loewner evolution

Remarks

The definition in $D$ does not depend on the choice of $\phi$, because two choices differ by a scaling $z \mapsto cz$ of $\mathbb{H}$ with $c > 0$. Brownian scaling shows that such a scaling only changes the time parameter of the hulls. Almost surely the hulls are generated by a continuous curve $\gamma$: $\mathbb{H} \setminus K_t$ is the unbounded component of $\mathbb{H} \setminus \gamma[0, t]$. This was proved by Rohde and Schramm for $\kappa \ne 8$, and by Lawler, Schramm and Werner for $\kappa = 8$. The curve is simple for $\kappa \le 4$, touches itself for $4 < \kappa < 8$, and fills space for $\kappa \ge 8$. Example: for $\kappa = 2$, $\mathrm{SLE}_2$ is the scaling limit of loop-erased random walk, a theorem of Lawler, Schramm and Werner. For $\kappa = 0$ the driving function is $0$ and the hulls are deterministic vertical segments; this degenerate case is sometimes included as $\mathrm{SLE}_0$. Radial $\mathrm{SLE}_\kappa$ is defined in the same way from the radial Loewner equation.