Stochastic Partial Differential Equations and Interacting particle systems
Lead Research Organisation:
Heriot-Watt University
Department Name: S of Mathematical and Computer Sciences
Abstract
In broad terms, as far as the subject of my PhD is concerned, it involves an interplay of analysis, stochastic analysis and calculus of variations.
To be precise, the project will be mainly focused on the topic of Stochastic Partial Differential Equations (SPDEs). Field of Stochastic Partial Differential Equations is quite new, the necessity of studying Stochastic Partial Differential Equations has been strongly motivated by applications. For example, in physics from problems such as wave propagation in random media and turbulence. Significant stimuli have also come from biology and neurobiology, in particular from population genetics.
Recent research efforts conducted in 2014 led to major discoveries in the field of Stochastic Partial Differential Equations, and this has opened to a wide range of new problems to be studied.
The analysis of Stochastic Partial Differential Equations is also related to interacting particle systems and my PhD is between these two areas. Throughout my PhD we will look into non-linear Stochastic Partial Differential Equations and related interacting particle systems and we will see how these two disciplines come together to describe the long-time behavior of the solution.
To be precise, the project will be mainly focused on the topic of Stochastic Partial Differential Equations (SPDEs). Field of Stochastic Partial Differential Equations is quite new, the necessity of studying Stochastic Partial Differential Equations has been strongly motivated by applications. For example, in physics from problems such as wave propagation in random media and turbulence. Significant stimuli have also come from biology and neurobiology, in particular from population genetics.
Recent research efforts conducted in 2014 led to major discoveries in the field of Stochastic Partial Differential Equations, and this has opened to a wide range of new problems to be studied.
The analysis of Stochastic Partial Differential Equations is also related to interacting particle systems and my PhD is between these two areas. Throughout my PhD we will look into non-linear Stochastic Partial Differential Equations and related interacting particle systems and we will see how these two disciplines come together to describe the long-time behavior of the solution.
Organisations
Studentship Projects
| Project Reference | Relationship | Related To | Start | End | Student Name |
|---|---|---|---|---|---|
| EP/V520044/1 | 01/10/2020 | 31/10/2025 | |||
| 2438206 | Studentship | EP/V520044/1 | 01/10/2020 | 31/03/2024 | Martin Kolodziejczyk |
| Description | Following Martin's Hairer work the interest in Stochastic Partial Differential Equations (SPDEs) has grown in the mathematical community. In this project we have studied a particular class of SPDEs the so-called McKean Vlasov SPDEs. In nature, McKean-Vlasov systems originally arise when describing systems of many particles. One could think systems of planets as particles under which acts the gravitational force or flocks of birds (i.e. the particles) flying around the planet which interacts with one another in a particular way or, alternatively, herds of animals such as horses, cows, sheep. It is known that in the long time behaviour many McKean-Vlasov systems presents many equilibria points. (Think of equilibria points as configurations that do not change through time e.g. a mass m in valley under gravity g, the mass m stays at the bottom of the valley independently of time). In this project we have added to McKean-Vlasov systems an external force F (highly irregular) which acts on each particle. In this regime we have shown that the resulting systems converges to a unique equilibrium point. |
| Exploitation Route | The outcomes may help scientists in understanding how specific groups of animals interact with one other. |
| Sectors | Aerospace, Defence and Marine,Digital/Communication/Information Technologies (including Software),Electronics,Financial Services, and Management Consultancy |
| URL | https://arxiv.org/abs/2211.08004 |