Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 203 / 2 / a

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- Solution a

Solution

If

g_{t}

is driven by

U_{t} = κ B_{t}

, then the mapping-out functions of

γ (t) = r γ (t / r^{2})

are

g_{t} (z) = r g_{t / r^{2}} (z / r),

and their driver is

U_{t} = r U_{t / r^{2}}

. By Brownian scaling,

(r B_{t / r^{2}})_{t \geq 0}

is Brownian. Thus

U

has the same law as

U

, and uniqueness of the Chordal Loewner equation proves

(r γ (t / r^{2}))_{t \geq 0} = d (γ (t))_{t \geq 0} .

This is the Scaling invariance of SLE.

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