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CPAA

We study evolution by horizontal mean curvature flow in sub-
Riemannian geometries by using stochastic approach to prove the existence of
a generalized evolution in these spaces. In particular we show that the value
function of suitable family of stochastic control problems solves in the viscosity
sense the level set equation for the evolution by horizontal mean curvature flow.

NHM

We consider a so-called random
obstacle model for the motion of a hypersurface through a field of
random obstacles,
driven by a constant driving field.
The resulting semi-linear parabolic PDE with random coefficients does not
admit a global nonnegative stationary solution,
which implies that an interface that was flat originally cannot get
stationary.
The absence of global stationary solutions is shown by
proving lower bounds on the growth of stationary solutions on
large domains with Dirichlet boundary conditions.
Difficulties arise because the random
lower order part of the equation cannot be bounded uniformly.

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