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

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