Heat content asymptotics of some random Koch type snowflakes

We consider the short time asymptotics of the heat content $E$ of a domain $D$ of ${\mathbb{R}}^d$. The novelty of this paper is that we consider the situation where $D$ is a domain whose boundary $\mathrm{\partial }D$ is a random Koch type curve. When $\mathrm{\partial }D$ is spatially homogeneous, we show that we can recover the lower and upper Minkowski dimensions of $\mathrm{\partial }D$ from the short time behaviour of $E(s)$. Furthermore, in some situations where the Minkowski dimension exists, finer geometric fluctuations can be recovered and the heat content is controlled by $s^{\alpha }{e}^{f(\log (1\mathrm{/}s))}$ for small $s$, for some $\alpha \in (0,\mathrm{\infty })$ and some regularly varying function $f$. The function $f$ is not constant is general and carries some geometric information. When $\mathrm{\partial }D$ is statistically self-similar, then the Minkowski dimension and content of $\mathrm{\partial }D$ typically exist and can be recovered from $E(s)$. Furthermore, the heat content has an almost sure expansion $E(s) = c s^{\alpha} N_\infty + o(s^\alpha)$forsmall$s$,forsome$c$and$\alpha \in (0, \infty)$andsomepositiverandomvariable$N_\infty$ with unit expectation arising as the limit of some martingale.