  Non-differentiable potentials, such as the V-shaped (linear) potential,
appear in various areas of physics. For example, the effective action for
branons in the framework of the brane world scenario contains a Liouville-type
interaction, i.e., an exponential of the V-shaped function. Another example is
coming from particle physics when the standard model Higgs potential is
replaced by a periodic self-interaction of an N-component scalar field which
depends on the length, thus it is O(N) symmetric. We first compare classical
and quantum dynamics near non-analytic points and discuss in this context the
role of quantum fluctuations. We then study the renormalisation of such
potentials, focusing on the Exact Wilsonian Renormalisation approach, and we
discuss how quantum fluctuations smoothen the bare singularity of the
potential. Applications of these results to the non-differentiable effective
branon potential and to the O(N) models when the spatial dimension is varied
and to the O(N) extension of the sine-Gordon model in (1+1) dimensions are
presented.
