Interplay of partial differential equations and stochastic processes, March 2023

Many complex systems in natural and applied sciences are often described by partial differential equations and/or stochastic processes. In this workshop, we bring together researchers working in the two fields to foster the interaction between them, aiming to join forces to develop new methods and techniques to advance our understanding of complex systems in physics, biology and social sciences.

This workshop is funded by EPSRC Grants EP/W008041/1 and EP/V038516/1.

Tentative schedule

13:00-13:50, Karen Habermann (University of Warwick): A polynomial expansion for Brownian motion and the associated fluctuation process.

13:50-14:40, Valeria Giunta (University of Sheffield): Multi-stability in non-local advection-diffusion models

14:40-15:20, coffee break

15:20-16:10, Tommaso Rosati (University of Warwick): Global in time solutions to perturbations of the 2D stochastic Navier-Stokes equations

16:10-17:00 Khanh Duy Trinh (Waseda University): Mean-field limit for stochastic processes related to classical beta ensembles.

Title and abstracts 

Dr Karen Habermann (Warwick University)

Title: A polynomial expansion for Brownian motion and the associated fluctuation process

Abstract: We start by deriving a polynomial expansion for Brownian motion expressed in terms of shifted Legendre polynomials by considering Brownian motion conditioned to have vanishing iterated time integrals of all orders. We further discuss the fluctuations for this expansion and show that they converge in finite dimensional distributions to a collection of independent zero-mean Gaussian random variables whose variances follow a scaled semicircle. We then link the asymptotic convergence rates of approximations for Brownian Lévy area which are based on the Fourier series expansion and the polynomial expansion of the Brownian bridge to these limit fluctuations. We close with a general study of the asymptotic error arising when approximating the Green's function of a Sturm-Liouville problem through a truncation of its eigenfunction expansion, both for the Green's function of a regular Sturm-Liouville problem and for the Green's function associated with the classical orthogonal polynomials.

Dr Valeria Giunta (University of Sheffield)

Title: Multi-stability in non-local advection-diffusion models

Abstract: In many biological systems, it is essential for individuals to gain information from their local environment before making decisions. In particular, through sight, hearing or smell, animals detect the presence of other individuals and adjust their behaviour accordingly. Interestingly, this feature is not only restricted to higher-level species, such as animals, but is also found in cells. For example, some human immune cells are able to interact non-locally by extending long thin protrusions to detect the presence of chemicals or signaling molecules. Indeed, the process of gaining information about the surrounding environment is intrinsically non-local and mathematically this leads to non-local advection terms in continuum models.

In this seminar, I will focus on a class of nonlocal advection-diffusion equations modeling population movements generated by inter- and intra-species interactions. I will show that the model supports a great variety of complex spatio-temporal patterns, including stationary aggregations, segregations, oscillatory patterns, and irregular spatio-temporal solutions.

However, if populations respond to each other in a symmetric fashion, the system admits an energy functional that is decreasing and bounded below, suggesting that patterns will be asymptotically stable. I will describe novel techniques for using this functional to gain insight into the analytic structure of the stable steady-state solutions. This process reveals a range of possible stationary patterns, including regions of multi-stability. These will be validated via comparison with numerical simulations.

Dr Tommaso Rosati (University of Warwick)

Title: Global in time solutions to perturbations of the 2D stochastic Navier-Stokes equations

Abstract: We prove global in time well-posedness for perturbations of the 2D Navier-Stokes equations driven by a perturbation of additive space-time white noise. The proof relies on a dynamic high-low frequency decomposition, tools from paracontrolled calculus and an L2 energy estimate for low frequencies. Our argument requires the solution to the linear equation to be a log-correlated field. We do not rely on (or have) explicit knowledge of the invariant measure: the perturbation is not restricted to the Cameron--Martin space of the noise. Our approach allows for anticipative and critical (L2) initial data. Joint work with Martin Hairer.

Dr Khanh Duy Trinh (Waseda University)

Title: Mean-field limit for stochastic processes related to classical beta ensembles

Abstract: Gaussian beta ensembles, Laguerre beta ensembles and Jacobi beta ensembles are the three classical beta ensembles on the real line. Related to them are stochastic processes known as beta Dyson's Brownian motions, beta Laguerre processes and beta Jacobi processes, respectively. This talk introduces a moment method to study the mean-field limit of those three stochastic processes. A moment method can also be used to study fluctuations around the limit as well.