Parabolicity, Brownian escape rate and properness of self-similar solutions of the direct and inverse Mean Curvature Flow

We study some potential theoretic properties of homothetic solitons ${\mathrm{\Sigma }}^n$ of the MCF and the IMCF. Using the analysis of the extrinsic distance function defined on these submanifolds in ${\mathbb{R}}^{n+m}$, we observe similarities and differences in the geometry of solitons in both flows. In particular, we show that parabolic MCF-solitons ${\mathrm{\Sigma }}^n$ with $n>2$ are self-shrinkers and that parabolic IMCF-solitons of any dimension are self-expanders. We have studied too the geometric behavior of parabolic MCF and IMCF-solitons confined in a ball, the behavior of the Mean Exit Time function for the Brownian motion defined on $\mathrm{\Sigma }$ as well as a classification of properly immersed MCF-self-shrinkers with bounded second fundamental form, following the lines of \cite{CaoLi}.