Abstract:

In the talk I look at the limit behaviour for a family of Cauchy problems for Hamilton-Jacobi equations describing a stochastic microscopic model.  The Hamiltonian considered is not coercive in the total gradient.  The Hamiltonian depends on a lower dimensional gradient variable which is associated to a Carnot group structure.  The rescaling is adapted to the Carnot group structure, therefore it is anisotropic.  Under suitable stationary-ergodic assumptions on the Hamiltonian, the solutions of the stochastic microscopic models will converge to a function independent of the random variable: the limit function can be characterised as the unique viscosity solution of a deterministic PDE.  The key step will be to introduce suitable lower-dimensional constrained variational problems.  In collaboration with Claudio Marchi and Paola Mannucci.