Loewner--Kufarev entropy and large deviations of the Hastings--Levitov model

We consider the Hastings–Levitov HL(0) model in the small particle scaling limit and prove a large deviation principle. The rate function is given by the relative entropy of the driving measure $\rho$ for the Loewner–Kufarev equation: $$H(\rho )=\frac{1}{2\pi }\iint {\bar{\rho }}_t(\theta )\log {\bar{\rho }}_t(\theta )d\theta dt,$$ whenever $\rho ={\bar{\rho }}_{t}d\theta dt\mathrm{/}2\pi$ with ${\int }_{S^1}{\bar{\rho }}_{t}d\theta \mathrm{/}2\pi =1$. We investigate the class of shapes that can be generated by finite entropy Loewner evolution and show that it contains all Weil-Petersson quasicircles, all Becker quasicircles, a Jordan curve with a cusp, and a non-simple curve. We also consider the problem of finding a measure of minimal entropy generating a given shape as well as a simplified version of the problem for a related transport equation.