Homogenisation for degenerate nonlinear Partial Differential Equations

The project is in the area of stochastic homogenization for nonlinear PDEs (Partial Differential Equations) associated to a low regularity condition called the Hormander condition. In particular I am interested in those cases where, even starting from a stochastic microscopic model, the effective problem (= PDE modelling the macroscopic behaviour) is deterministic. Where the microscopic stochastic model is related to Hormander-type PDEs, the rescaling becomes usually anisotropic. The regularity theory of such PDEs mimics, at least in the linear case, the standard one (H ¨ormander’s Theorem ‘69), but deriving finer properties like asymptotics in the presence of randomness requires an intricate combination of ideas from analysis, probability and geometry. The influence of the geometry on the associated analysis problems, is particularly evident in the case of homogenisation where the rescaling of the microscopic model is strictly connected to the underlying geometry. In fact, in the standard uniformly elliptic/parabolic case (respectively coercive for the first order case), one takes the limit as e tends to zero of an equation depending on e.g. x/e (where x is a point in Rn) i.e. isotropic rescaling. When considering a degenerate PDE related to H ¨ormander vector fields, the rescaling becomes anisotropic: for example in the first Heisenberg group a point (x, y, z) scales as (x/e, y/e, z/e2). Moreover points (elements on the manifold) and velocities of curves (elements in the tangent space) scale now in a very different way and any identification between objects with different natures (which is often a well-hidden key point in the standard elliptic/parabolic case) is no longer allowed.

The challenge in the study of these limit theorems is to find approaches which do not rely on the commutativity of the Euclidean structure or on the identification between manifold (points) and tangent space (velocities), and dealing with the difficulty of using very degenerate underlying geometries which are not even locally isomorphic to the Euclidean Rn.The mathematical questions arising in the project are not only interesting from the pure mathematical point of view but they also have important applications in many different fields, for example in the study of the visual cortex (see Citti-Sarti model of the visual cortex) and financial models e.g. Asian options. One of the goals of the project is to provide error estimates for homogenisation problems, i.e. in cases where it is already known that solutions of the e-problem converge to solutions of the homogenised problem we want to estimate the rate of convergence.