EPSRC Reference: 
EP/N003209/1 
Title: 
Random Perturbations of Ultraparabolic Partial Differential Equations under rescaling 
Principal Investigator: 
Dragoni, Dr F 
Other Investigators: 

Researcher CoInvestigators: 

Project Partners: 

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 Date  Panel Name  Outcome 
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 firstorder and secondorder) for nonlinear PDEs, associated to suitable Hoermander vector fields. The geometry of Hoermander vector fields (CarnotCaratheodory 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 k1 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 3dimensional 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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Potential use in nonacademic contexts 
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Impacts 
Description 
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Summary 

Date Materialised 


Sectors submitted by the Researcher 
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Project URL: 

Further Information: 

Organisation Website: 
http://www.cf.ac.uk 