The energy of a deterministic Loewner chain: Reversibility and interpretation via SLE${}_{0+}$

Abstract

We study some features of the energy of a deterministic chordal Loewner chain, which is defined as the Dirichlet energy of its driving function. In particular, using an interpretation of this energy as a large deviation rate function for SLE${}_{\kappa }$ as $\kappa$ tends to 0 and the known reversibility of the SLE${}_{\kappa }$ curves for small $\kappa$, we show that the energy of a deterministic curve from one boundary point A of a simply connected domain D to another boundary point B, is equal to the energy of its time-reversal ie. of the same curve but viewed as going from B to A in D.

28 pages, 5 figures, minor changes in Sec. 2.2., to appear in J. Europ. Math. Soc

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• A support theorem for SLE curves
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• On Loewner chains driven by semimartingales and complex Bessel-type SDEs
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