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For every $a>0$, if $(K_t)$ is ${\mathrm{SLE}}_{\kappa }$, then $(a^{-1}{K}_{a^{2}t})$ has the same law. Its driving function is $a^{-1}{U}_{a^{2}t}$, which has the same law as $U_t$ by Brownian scaling.

A scaling limit of a random curve is a weak limit of laws after rescaling the curves and choosing the space in which they are compared. One first specifies a topology; tightness is then proved in that topology, rather than determining the topology by itself.

For continuous curves on $[0,1]$ considered up to increasing reparameterization, a common comparison starts fromwhere $\varphi$ ranges over increasing homeomorphisms of $[0,1]$ and curves at zero distance are identified. This compares geometric traces with their order of traversal while allowing different traversal speeds. A parametrized process limit may instead use uniform distance or a Skorokhod J1 topology, so the intended notion of a curve matters.

For example, Brownian scaling gives $a^{-1/2}{B}_{at}$ the same law as Brownian motion for every $a>0$. It is already invariant under this rescaling. For lattice models a scale change can instead produce a different continuum law; SLE is a central family of candidate limits. In an appropriate curve space, tightness supplies subsequential weak limits, and identifying every subsequential law with the same law establishes convergence of the full sequence. Finite-dimensional distributions alone do not provide the required control over entire curves.

Standard Brownian motion $B$ in ${\mathbb{R}}^d$ is transient for $d\ge 3$: $|B_t|\to \infty$ almost surely. In dimension three, start away from zero and use the positive local martingale $1/|B_t|$. It is a supermartingale, hence converges to a finite limit. Brownian scaling and the normal distribution of $B_t/\sqrt{t}$ show that $1/|B_t|\to 0$ in probability, so its almost-sure limit must be zero. Thus $|B_t|\to \infty$.

For a start at zero, apply the same argument from any positive deterministic time; the position there is nonzero almost surely. In higher dimensions, project onto three coordinates: the full radius dominates a radius already tending to infinity. This differs from recurrence of planar Brownian motion, where every neighbourhood is visited repeatedly.