Evolution by mean curvature flow in sub-Riemannian geometries: A stochastic approach
Nicolas Dirr Federica Dragoni Max von Renesse
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.
keywords: stochastic processes and control. level set equation sub-Riemannian geometries Mean curvature flow
Non-existence of positive stationary solutions for a class of semi-linear PDEs with random coefficients
Jérôme Coville Nicolas Dirr Stephan Luckhaus
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.
keywords: Random obstacles Qualitative behavior of parabolic PDEs with random coefficients Interface evolution in Random media.

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