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 $\rho = \bar{\rho}_t d\theta dt/2\pi$ with $\int_{S^1} \bar{\rho}_t d\theta/2\pi = 1$. 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.