# Random Perturbations of Ultraparabolic Partial Differential Equations under rescaling

### Abstract

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.

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.

### Planned Impact

Ordinary and partial differential equations have been used across the centuries to describe almost everything around us.

In this context the general contribution of pure mathematicians is to provide a rigorous analysis of the equations used and their properties in order to gain an understanding and knowing instead of merely "guessing" the solution.

Even if no one can precisely say which will be the impact of this research on a very long-term time scale,

there are certainly significant and realistic short-term, mid-term and long-term benefits and I will outline some of them.

The major beneficiary is the community of mathematicians working in related subject areas (e.g. homogenisation, nonlinear degenerate PDEs, mean field equations, analysis in manifolds, curvature flows). They will gain from both the results proved and the new techniques developed. This will be delivered mainly by publications in high-standard mathematical journals, participations as speaker at international conferences and workshops, seminars and talks in universities in the UK and overseas, but also through informal conversations and exchange of ideas.

The research will have a significant direct impact on the PDRA.

The educational impact of this multidisciplinary mathematical proposal will extend to the whole UK. In fact the PDRA, postgraduates and undergraduates trained to work at the interface of a broad range of mathematical areas (analysis, probability and geometry) will bring their interdisciplinary background and attitude in all the institutions (inside and outside the academic world) were they will work in the future.

Besides the already mentioned benefits for the academics whose research interests are directly connected with the results of the project, there are several other mathematical communities which will benefit from the new techniques developed in the research.

Partial differential equations connected to sub-Riemannian manifolds and Lie groups have a very broad range of possible applications: some very well-established (robotic, image processing, etc) but some others still to be discovered:

-Problem 3: visual cortex and image processing is motivated by models of the visual cortex, or rather how an image is processed by the first layer of the visual cortex. While it is at this stage a pure mathematics project answering mathematical questions, this project can in the future contribute to a better understanding of the visual cortex and improve and/or explain numerical algorithms used in image processing.

-Problems 1 and 2: homogenisation and in particular stochastic homogenisation has important applications to all those systems which have a random microscopic structure while the observed macroscopic system looks deterministic. As example one could mention material science or biology. A better understanding of the connection between microscopic and macroscopic properties in the long run help to design better materials.

-The special case of degenerate PDEs under the Hormander condition has applications in finance (option pricing, e.g. Asian options but also pricing pension plans).

In this context the general contribution of pure mathematicians is to provide a rigorous analysis of the equations used and their properties in order to gain an understanding and knowing instead of merely "guessing" the solution.

Even if no one can precisely say which will be the impact of this research on a very long-term time scale,

there are certainly significant and realistic short-term, mid-term and long-term benefits and I will outline some of them.

The major beneficiary is the community of mathematicians working in related subject areas (e.g. homogenisation, nonlinear degenerate PDEs, mean field equations, analysis in manifolds, curvature flows). They will gain from both the results proved and the new techniques developed. This will be delivered mainly by publications in high-standard mathematical journals, participations as speaker at international conferences and workshops, seminars and talks in universities in the UK and overseas, but also through informal conversations and exchange of ideas.

The research will have a significant direct impact on the PDRA.

The educational impact of this multidisciplinary mathematical proposal will extend to the whole UK. In fact the PDRA, postgraduates and undergraduates trained to work at the interface of a broad range of mathematical areas (analysis, probability and geometry) will bring their interdisciplinary background and attitude in all the institutions (inside and outside the academic world) were they will work in the future.

Besides the already mentioned benefits for the academics whose research interests are directly connected with the results of the project, there are several other mathematical communities which will benefit from the new techniques developed in the research.

Partial differential equations connected to sub-Riemannian manifolds and Lie groups have a very broad range of possible applications: some very well-established (robotic, image processing, etc) but some others still to be discovered:

-Problem 3: visual cortex and image processing is motivated by models of the visual cortex, or rather how an image is processed by the first layer of the visual cortex. While it is at this stage a pure mathematics project answering mathematical questions, this project can in the future contribute to a better understanding of the visual cortex and improve and/or explain numerical algorithms used in image processing.

-Problems 1 and 2: homogenisation and in particular stochastic homogenisation has important applications to all those systems which have a random microscopic structure while the observed macroscopic system looks deterministic. As example one could mention material science or biology. A better understanding of the connection between microscopic and macroscopic properties in the long run help to design better materials.

-The special case of degenerate PDEs under the Hormander condition has applications in finance (option pricing, e.g. Asian options but also pricing pension plans).

## People |
## ORCID iD |

Federica Dragoni (Principal Investigator) |

Description | 1. We consider a stochastic homogenization problem, i.e. the question whether solutions to partial differential equations with random coefficients on a small scale converge on a larger scale to solutions of a "homogenized" equation with constant coefficients. The novelty here lies in the fact that the equations are degenerate (non-coercive), a case which could not be treated by previous techniques. We are able to show that if the degeneracy is related to a geometric structure, then this structure can be exploited to show a homogenization result by variational methods. This is joint work with N. Dirr from Cardiff and P. Mannucci and C. Marchi from Padova, to be found in a forthcoming preprint. 2. Symmetry is one of the most important concepts in mathematics. For solutions of partial differential equations, it is important to know whether they "inherit" symmetries from the data. We were able to answers this question in a specific case: Solutions to certain degenerate equations in a star-shaped domain (a domain where one "central "point can be connected to any other point by a line which stays in the domain) have star-shaped level sets, if "star-shaped" is interpreted in a way connected to the geometry that is underlying the degeneracy of the equation. (Lines are here so-called X-lines, as previously introduced by the PI and M. Bardi) This is joint work with N. Garofalo from Padova and P. Salani from Firenze, to be found in a forthcoming preprint. 3. Mean-field games as introduced by J.-M. Lasry and P.-L. lions as effective models for interactions of very many rational agents have recently generated a lot of interest. We are able to show existence and uniqueness for a degenerate version of these mean field equations. This is joint work with E. Feleqi. |

Exploitation Route | While the primary beneficiaries are other mathematicians working in the area of ultraparabolic/sub-elliptic partial differential equations, Sub-riemannian geometries have found applications e.g. in robotics, finance or image processing, so in the long term there may be applications. |

Sectors |
Other |

Description | Geometric property of level sets for nonlinear degenerate PDEs |

Organisation | University of Florence |

Department | Department of Mathematics and Informatics 'Ulisse Dini' |

Country | Italy, Italian Republic |

Sector | Academic/University |

PI Contribution | Research discussions on ongoing project with Professor Nicola Garofalo and Paolo Salani. Preliminary result for starshaped sets w.r.t. the origin. |

Collaborator Contribution | Research discussions on ongoing project with Professor Nicola Garofalo and Paolo Salani. Preliminary result for starshaped sets w.r.t. the origin. |

Impact | Ongoing project |

Start Year | 2015 |

Description | Geometric property of level sets for nonlinear degenerate PDEs |

Organisation | University of Padova |

Department | Department of Mathematics |

Country | Italy, Italian Republic |

Sector | Academic/University |

PI Contribution | Research discussions on ongoing project with Professor Nicola Garofalo and Paolo Salani. Preliminary result for starshaped sets w.r.t. the origin. |

Collaborator Contribution | |

Impact | Ongoing project |

Start Year | 2015 |

Description | Stochastic homogenization of degenerate PDEs and invariant measure |

Organisation | University of Padova |

Department | Department of Mathematics |

Country | Italy, Italian Republic |

Sector | Academic/University |

PI Contribution | We have concluded the mathematical proofs for a paper with Claudio Marchi and Paola Mannucci, and made significant progress on the final draft of the paper. We have discussed future projects to start in the next months (error estimates and existence of a stationary ergodic measure). |

Collaborator Contribution | We have concluded the mathematical proofs for a paper with Claudio Marchi and Paola Mannucci, We have discussed future projects to start in the near future (error estimates and existence of a stationary ergodic measure). In mathematics the contributions are mutual since research advance through scientific discussions among collaborators. |

Impact | Preprint almost ready for submission. |

Start Year | 2014 |