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Details of Grant 

EPSRC Reference: EP/N003209/1
Title: Random Perturbations of Ultraparabolic Partial Differential Equations under rescaling
Principal Investigator: Dragoni, Dr F
Other Investigators:
Researcher Co-Investigators:
Project Partners:
University of Bologna University of Padua (Padova)
Department: Sch of Mathematics
Organisation: Cardiff University
Scheme: First Grant - Revised 2009
Starts: 01 October 2015 Ends: 31 May 2017 Value (£): 99,896
EPSRC Research Topic Classifications:
Algebra & Geometry Mathematical Analysis
EPSRC Industrial Sector Classifications:
No relevance to Underpinning Sectors
Related Grants:
Panel History:
Panel DatePanel NameOutcome
16 Jun 2015 EPSRC Mathematics Prioritisation Panel June 2015 Announced
Summary on Grant Application Form
This proposal is in the area of nonlinear partial differential equations (PDEs). More precisely I am interesting in proving rigorous convergence for solutions of a randomly perturbed nonlinear PDE to the solution of an effective deterministic nonlinear PDE.

I look at different problems (both first-order and second-order) for nonlinear PDEs, associated to suitable Hoermander vector fields. The geometry of Hoermander vector fields (Carnot-Caratheodory spaces) is degenerate in the sense that some directions for the motion are forbidden (non admissible). A family of vector fields is said to satisfy the Hoermander condition (with step=k) if the vectors of the family together with all their commutators up to some order k-1 generate at any point the whole tangent space. If the Hoermander condition is satisfied, then one can always go everywhere by following only paths in the directions of the vector fields (admissible paths).

The natural scaling for PDE problems associated to these underlying geometries is anisotropic. For example, thinking of homogenisation of a standard uniformly elliptic/parabolic PDE, one usually takes the limit as epsilon (i.e. a small parameter) tends to zero of an equation depending for example on (x/epsilon,y/epsilon,z/epsilon), where (x,y,z) is a point in the 3-dimensional Euclidean space. This means that the equation is isotropically rescaled.

On the other end, when considering a degenerate PDE related to Hoermander vector fields, the rescaling needs to adapt to the new geometric underlying structure, e.g. a point (x,y,z) may scale as (x/epsilon,y/epsilon, z/epsilon^2).

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). Further complications come from the limited use of geodesic arguments due to the highly irregular nature of such curves.

Thus the proposed project requires an intricate combination of ideas and techniques from analysis, probability and geometry.
Key Findings
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