Large scales analysis of SPDEs

The study of the KPZ equation and the KPZ fixed point have witnessed remarkable progresses in the last decade. Since its introduction by Kardar, Parisi and Zhang [KPZ86] in 1986, the KPZ equation has been the default model to capture the dynamics of a large variety of discrete physical models and random interfaces growth. The equation is ill-posed already in dimension d=1, being one of the main examples of singular SPDEs. This type of nonlinear SPDEs for long had been intractable due to the irregularity of the noise and nonlinear terms involved in the equations. This until the seminal work of Hairer [H13] and his subsequent development of Regularity Structures, which now provide a robust framework to study virtually all type of locally subcritical singular SPDEs, allowing to make sense of the equations and their solution through appropriate renormalization. In the meantime various efforts recently culminated with an exact description of KPZ fixed point and proved the large scales convergence of the KPZ solution. On the other hand, the study of large scales fluctuations has recently seen various progresses [CSZ20, MU18] also in higher dimensions d >= 2, where the Edwards-Wilkinson (Gaussian) universality class is the attracting fixed point in weak disorder regimes. Here (critical/super-critical settings) the pioneering theories of Regularity Structures and Paracontrolled Distributions no longer readily apply, hence to make sense of the equations the study has focused with driving white noise appropriately regularized.

Aims and Objectives:
Typically this large scales analysis has considered Gaussian driving noise with finite range correlations at the microscopic level (coming from compactly supported mollifiers). We want to investigate the impact of long range correlations of the noise (either in time or space, or jointly) on the large scales dynamics/statistics. Hence understand if the same universality class behaviour is displayed at large scales, and understand whether there may occur phase transitions depending on the noise correlations decay and the spatial dimension d.

Novelty of the methodology:
At present there is some understanding and expectations for the KPZ equation coming from numerical simulations and works from the physics literature (in d <= 2), these point in contrasting directions at times and lack a fully mathematical treatment. Hence new methodologies will be required to investigate analytically long range correlations regimes, going beyond the short range correlations settings in the existing mathematical literature.

The project is aligned with the following EPSRC research areas: Mathematical Analysis, Mathematical Physics, Statistics and Applied Probability.

References:
[KPZ86] Kardar, M., Parisi, G. and Zhang, Y.C., 1986. Dynamic scaling of growing interfaces. Physical Review Letters, 56(9), p.889.
[H13] Hairer, M., 2013. Solving the KPZ equation. Annals of mathematics, pp.559-664.
[CSZ20] Caravenna, F., Sun, R. and Zygouras, N., 2020. The two-dimensional KPZ equation in the entire subcritical regime. The Annals of Probability, 48(3), pp.1086-1127.
[MU18] Magnen, J. and Unterberger, J., 2018. The scaling limit of the KPZ equation in space dimension 3 and higher. Journal of Statistical Physics, 171(4), pp.543-598

Probabilistic modelling permeates the Financial services, healthcare, technology and other Service industries crucial to the UK's continuing social and economic prosperity, which are major users of stochastic algorithms for data analysis, simulation, systems design and optimisation. There is a major and growing skills shortage of experts in this area, and the success of the UK in addressing this shortage in cross-disciplinary research and industry expertise in computing, analytics and finance will directly impact the international competitiveness of UK companies and the quality of services delivered by government institutions.
By training highly skilled experts equipped to build, analyse and deploy probabilistic models, the CDT in Mathematics of Random Systems will contribute to
- sharpening the UK's research lead in this area and
- meeting the needs of industry across the technology, finance, government and healthcare sectors

MATHEMATICS, THEORETICAL PHYSICS and MATHEMATICAL BIOLOGY

The explosion of novel research areas in stochastic analysis requires the training of young researchers capable of facing the new scientific challenges and maintaining the UK's lead in this area. The partners are at the forefront of many recent developments and ideally positioned to successfully train the next generation of UK scientists for tackling these exciting challenges.
The theory of regularity structures, pioneered by Hairer (Imperial), has generated a ground-breaking approach to singular stochastic partial differential equations (SPDEs) and opened the way to solve longstanding problems in physics of random interface growth and quantum field theory, spearheaded by Hairer's group at Imperial. The theory of rough paths, initiated by TJ Lyons (Oxford), is undergoing a renewal spurred by applications in Data Science and systems control, led by the Oxford group in conjunction with Cass (Imperial). Pathwise methods and infinite dimensional methods in stochastic analysis with applications to robust modelling in finance and control have been developed by both groups.
Applications of probabilistic modelling in population genetics, mathematical ecology and precision healthcare, are active areas in which our groups have recognized expertise.

FINANCIAL SERVICES and GOVERNMENT

The large-scale computerisation of financial markets and retail finance and the advent of massive financial data sets are radically changing the landscape of financial services, requiring new profiles of experts with strong analytical and computing skills as well as familiarity with Big Data analysis and data-driven modelling, not matched by current MSc and PhD programs. Financial regulators (Bank of England, FCA, ECB) are investing in analytics and modelling to face this challenge. We will develop a novel training and research agenda adapted to these needs by leveraging the considerable expertise of our teams in quantitative modelling in finance and our extensive experience in partnerships with the financial institutions and regulators.

DATA SCIENCE:

Probabilistic algorithms, such as Stochastic gradient descent and Monte Carlo Tree Search, underlie the impressive achievements of Deep Learning methods. Stochastic control provides the theoretical framework for understanding and designing Reinforcement Learning algorithms. Deeper understanding of these algorithms can pave the way to designing improved algorithms with higher predictability and 'explainable' results, crucial for applications.
We will train experts who can blend a deeper understanding of algorithms with knowledge of the application at hand to go beyond pure data analysis and develop data-driven models and decision aid tools
There is a high demand for such expertise in technology, healthcare and finance sectors and great enthusiasm from our industry partners. Knowledge transfer will be enhanced through internships, co-funded studentships and paths to entrepreneurs