Anomalous diffusion via self-interaction and reflection
Lead Research Organisation:
Durham University
Department Name: Mathematical Sciences
Abstract
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Organisations
Publications
Avena L
(2024)
Gaussian, stable, tempered stable and mixed limit laws for random walks in cooling random environments
in Annales de l'Institut Henri Poincaré, Probabilités et Statistiques
Avena L
(2023)
Laws of Large Numbers for Weighted Sums of Independent Random Variables: A Game of Mass
in Journal of Theoretical Probability
Barrera G
(2025)
Gradual convergence for Langevin dynamics on a degenerate potential
in Stochastic Processes and their Applications
Bhattacharjee C
(2025)
Perimeter length of the convex hull of Brownian motion in the hyperbolic plane
Bresar M
(2025)
Superdiffusive limits for Bessel-driven stochastic kinetics
Brešar M
(2025)
Brownian motion with asymptotically normal reflection in unbounded domains: From transience to stability
in The Annals of Probability
Cygan W
(2024)
Iterated-logarithm laws for convex hulls of random walks with drift
in Transactions of the American Mathematical Society
Da Costa C
(2025)
Long-range one-dimensional internal diffusion-limited aggregation
Da Costa C
(2024)
Limiting distributions for RWCRE in the sub-ballistic regime and in the critical Gaussian regime
in Electronic Journal of Probability
| Description | As a result of the research funded through this grant, we have discovered novel phenomena and mechanisms for anomalous diffusion in mathematical models. In particular, for reflecting diffusions, in which a diffusion process is confined to a domain via reflections from the boundary, we have shown how, in funnel-like domains, increasingly rapid reflections can accelerate a process much faster than standard diffusion, even, in some cases, to cause a process with no intrinsic drift to reach infinity in finite time. We have also established a detailed description of the limit theory and fluctuations for some of these models, which again demonstrates how reflections can lead to a rich collection of phenomena broadly described as anomalous diffusion. We have also obtained quantitative results for anomalous diffusion in stochastic-kinetic models, in which one stochastic component plays the role of velocity in another stochastic evolution. This funding also enabled us to support the research programme "Stochastic systems for anomalous diffusion", which ran July to December 2024 at the Isaac Newton Institute for Mathematical Science, Cambridge. The programme, organized by Codina Cotar (University College London), Aleksandar Mijatovic (University of Warwick and The Alan Turing Institute), Anastasia Papavasiliou (University of Warwick), Ellen Powell (Durham University), Kavita Ramanan (Brown University), Kilian Raschel (CNRS and Université d'Angers), Perla Sousi (University of Cambridge), and Andrew Wade (Durham University), hosted a large number of international experts on anomalous diffusion and related areas and stimulated much activity and interaction. In particular, a series of workshops explored the state of the art of many aspects of relevant mathematics, together with applications in the natural world, and in computational statistics and machine learning. |
| Exploitation Route | We believe that our work on stochastic reflection has drawn attention of mathematicians to novel phenomena in reflecting processes, and we have developed and deployed tools for the analysis of these processes that could lead to further advances. Our work on quantifying rates of convergence, using a combination of ideas developed within the research team over several years, have already been applied further in the context of stochastic optimization and simulation algorithms, where such questions are central. |
| Sectors | Digital/Communication/Information Technologies (including Software) |
| URL | https://www.newton.ac.uk/event/ssd/ |
| Description | The scientific outputs of the project are still in progress, and it is early days to assess economic and societal impact. As part of the "Stochastic systems for anomalous diffusion" research program that we jointly organized at the Isaac Newton Institute for Mathematical Sciences, July - December 2024, this award was directly used to support the workshop "Modelling and Applications of Anomalous Diffusions", 4 September 2024 to 6 September 2024, including and "Open for Business Day" organized with the Newton Gateway to Mathematics (5th September 2024). It aimed to provide a meeting ground to facilitate interactions and exchanges between representatives of academia, research and industry, relevant to the theme, with the objective of identifying points of mutual interest and possible co-activity. Moreover, associated to the programme there is to be held an art competition, "Anomalous Mathematical Patterns Sci-Art Contest", December 2024 - March 2025, organized by Professor Codina Cotar of University College London. The goals of the contest are to: showcase the interplay of mathematics (in particular of probability theory) and art in the modern world; explore the different ways in which artists and mathematicians approach mathematical concepts; find ways to communicate research in mathematics to as wide an audience as possible; inspire new generations towards mathematical education and training; potentially spark collaborations among and between mathematicians and artists. |
| Impact Types | Cultural |
| Description | Clay Lecturer - Balint Toth - Stochastic systems for anomalous diffusion |
| Amount | $7,200 (USD) |
| Organisation | Clay Mathematical Institute |
| Sector | Charity/Non Profit |
| Country | United States |
| Start | 09/2024 |
| End | 11/2024 |
| Description | Clay Lecturer - Gordon Slade - Stochastic systems for anomalous diffusion |
| Amount | $12,000 (USD) |
| Organisation | Clay Mathematical Institute |
| Sector | Charity/Non Profit |
| Country | United States |
| Start | 06/2024 |
| End | 08/2024 |
| Description | Dynamics of Random Systems (UK Easter Probability Meeting) |
| Amount | £5,500 (GBP) |
| Funding ID | 12353 |
| Organisation | London Mathematical Society |
| Sector | Academic/University |
| Country | United Kingdom |
| Start | 03/2025 |
| End | 04/2025 |
| Description | Geography Research Development Fund: Maths-Geography Working Group (Conrado da Costa) |
| Amount | £500 (GBP) |
| Organisation | Durham University |
| Department | Department of Geography |
| Sector | Academic/University |
| Country | United Kingdom |
| Start | 01/2025 |
| End | 06/2025 |
| Description | Stochastic Systems for Anomalous Diffusion |
| Amount | £250,000 (GBP) |
| Organisation | Isaac Newton Institute for Mathematical Sciences |
| Sector | Academic/University |
| Country | United Kingdom |
| Start | 06/2024 |
| End | 12/2024 |
| Description | "Chalkboard Ultra" Podcast Interview |
| Form Of Engagement Activity | A broadcast e.g. TV/radio/film/podcast (other than news/press) |
| Part Of Official Scheme? | No |
| Geographic Reach | Local |
| Primary Audience | Undergraduate students |
| Results and Impact | "Chalkboard Ultra" is a Maths-oriented Podcast created by two Durham University undergraduates. In Febraru 2025 they interviewed Andrew Wade on aspects of his research, including stochastic models of anomalous diffusion. The edited podcast will be made available within the coming weeks. |
| Year(s) Of Engagement Activity | 2025 |
| Description | Internet streaming channel (Conrado da Costa) |
| Form Of Engagement Activity | A broadcast e.g. TV/radio/film/podcast (other than news/press) |
| Part Of Official Scheme? | No |
| Geographic Reach | International |
| Primary Audience | Public/other audiences |
| Results and Impact | Conrado da Costa has an internet streaming channel where he discusses some of the mathematics of probability, based on a mix of classic texts and more modern topics, such as the subject of the current grant. |
| Year(s) Of Engagement Activity | 2023 |
| URL | https://www.twitch.tv/reader_of_mathematics |
| Description | Living Proof Podcast - Exploring anomalous diffusion: An Interview with Aleks Mijatovic and Codina Cotar |
| Form Of Engagement Activity | A broadcast e.g. TV/radio/film/podcast (other than news/press) |
| Part Of Official Scheme? | No |
| Geographic Reach | International |
| Primary Audience | Professional Practitioners |
| Results and Impact | Living Proof Podcast - Exploring anomalous diffusion: An Interview with Aleks Mijatovic and Codina Cotar In this episode of Living Proof, Dan Aspel speaks to Aleks Mijatovic (Warwick, Alan Turing Institute) and Codina Cotar (UCL) about the Stochastic systems for anomalous diffusion programme. |
| Year(s) Of Engagement Activity | 2024 |
| URL | https://www.newton.ac.uk/media/podcasts/post/63-exploring-anomalous-diffusion-an-interview-with-alek... |
| Description | Organized a 6 month programme on "Stochastic systems for anomalous diffusion" |
| Form Of Engagement Activity | Participation in an activity, workshop or similar |
| Part Of Official Scheme? | No |
| Geographic Reach | International |
| Primary Audience | Professional Practitioners |
| Results and Impact | Organized a 6 month programme at the Isaac Newton Institute for Mathematical Sciences, University of Cambridge, on "Stochastic systems for anomalous diffusion". The programme ran from 1 July--18 December 2024, with over 120 participants from Europe, the Americas, Asia, and Australasia, and a range of scientific backgrounds from probability theory, stochastic analysis, mathematical physics, statistics and machine learning. Thematic aspects were explored through four one-week workshops; two shorter workshops emphasized applications and industrial engagement. A regular seminar series, totalling over 40 talks, ran throughout the programme, along with numerous social activities. |
| Year(s) Of Engagement Activity | 2024 |
| URL | https://www.newton.ac.uk/event/ssd/ |
| Description | Organized workshop: Durham Probability Afternoon, Wednesday 6 September 2023 |
| Form Of Engagement Activity | Participation in an activity, workshop or similar |
| Part Of Official Scheme? | No |
| Geographic Reach | International |
| Primary Audience | Professional Practitioners |
| Results and Impact | Durham Probability Afternoon, research workshop, held at Durham University, Wednesday 6 September 2023. |
| Year(s) Of Engagement Activity | 2023 |
| URL | https://www.maths.dur.ac.uk/PiNE/2309Sep/ |
| Description | Organized workshop: Probability in the North East, 24 June 2024 |
| Form Of Engagement Activity | Participation in an activity, workshop or similar |
| Part Of Official Scheme? | No |
| Geographic Reach | International |
| Primary Audience | Professional Practitioners |
| Results and Impact | Organized "Probability in the North East" workshop, 24 June 2024, at Durham University. |
| Year(s) Of Engagement Activity | 2024 |
| URL | https://www.maths.dur.ac.uk/PiNE/2406Jun/ |
| Description | Prob-AM, a YouTube channel for dissemination of research outputs |
| Form Of Engagement Activity | Engagement focused website, blog or social media channel |
| Part Of Official Scheme? | No |
| Geographic Reach | International |
| Primary Audience | Professional Practitioners |
| Results and Impact | Short YouTube videos aimed at researchers and students (graduate and undergraduate). The main purpose is to extend awareness of my research by lowering the time/effort barrier for understanding and using the research output. |
| Year(s) Of Engagement Activity | 2023,2024,2025 |
| URL | https://www.youtube.com/channel/UCXSoLS_uKebYZ9GzgAF0ZsA |