  In this article, we introduce Brownian motion on stable looptrees using
resistance techniques. We prove an invariance principle characterising it as
the scaling limit of random walks on discrete looptrees, and prove precise
local and global bounds on its heat kernel. We also conduct a detailed
investigation of the volume growth properties of stable looptrees, and show
that the random volume and heat kernel fluctuations are locally
log-logarithmic, and globally logarithmic around leading terms of $r^{\alpha}$
and $t^{\frac{-\alpha}{\alpha + 1}}$ respectively. These volume fluctuations
are the same order as for the Brownian continuum random tree, but the upper
volume fluctuations (and corresponding lower heat kernel fluctuations) are
different to those of stable trees.
