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Evolution by horizontal mean curvature flow: a stochastic approach

ARGOMENTI: Seminari

SEMINARI DI EQUAZIONI DIFFERENZIALI E APPLICAZIONI
Lunedi' 15 dicembre alle ore 11 in aula 2BC/60 Federica Dragoni (Universita' di Padova) terra' un seminario dal titolo "Evolution by horizontal mean curvature flow: a stochastic approach".

-Abstract
We study the phenomenon of evolution by horizontal mean curvature flow in Carnot-Caratheodory spaces. We use the stochastic approach introduced by Soner and Touzi (2002) and Buckdahn, Cardaliaguet and Quincampoix (2001) in the euclidean case. We first explain the geometric evolution and point out why the classical "smooth" equation is not able to describe the phenomenon. In euclidean spaces, the classical equation "normal velocity = mean curvature vector" can describe the evolution of regular surfaces at least for small times, until topological singularities appear. Instead, in Carnot-Caratheodory spaces, the topological singularites lie in the space itself, and there are very few regular surfaces (for example, all the spheres are not). The main difficulty occurs because of the existence of characteristic points, which are points where the horizontal normal is not defined. We introduce a definition corresponding to the euclidean generalized evolution by mean curvature flow for the level sets. Then we give a geometric interpretation for the characteristic points, comparing them with similar phenomena arising in the euclidean evolution. In the second part we introduce a stochastic
control problem: we show that, looking at the initial surface evolving as a target set for a suitable controlled Brownian motion, the associated reachability set solves (in the viscosity sense) the PDE for the level sets evolution by horizontal mean curvature flow. Joint work with N. Dirr (University of Bath) and M. von Renesse (Technische University of Berlin).

Rif. int. M. Bardi, P. Mannucci, A. Cesaroni

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