Variational structures, convergence to equilibrium and multiscale analysis for non-Markovian systems

A partial differential equation (PDE) is an equation that involves an unknown function of multiple (spatial and temporal) variables and its partial derivatives. PDEs appear in many branches of Mathematics and constitute a powerful mathematical framework for the modelling, analysis and computation/simulation of complex systems. A striking example is the Fokker-Planck equation which has been used intensively in statistical mechanics to describe the time evolution of the probability density function of the position of a particle moving under the influence of an external force, a friction and random forces. By solving a PDE and studying the qualitative and quantitative properties of its solutions, one gains important insights into the understanding and control of the underlying complex system/phenomenon.
The second law of thermodynamics, which was discovered almost 200 years ago, states that the entropy of an isolated system can never decrease over time. It has been of fundamental interest to build mathematical theories that preserve physical structures, in particular to incorporate the second law of thermodynamics to mathematical models and analysis. In 1998, Jordan, Kinderlehrer and Otto made a major breakthrough in mathematical analysis with the introduction of a theory of Wasserstein gradient flows, particularly proving that the diffusion equation is a gradient flow (steepest descent) of the (negative) Boltzmann entropy with respect to the Wasserstein metric, which was a seemingly unrelated concept coming from the theory of optimal transport. This work not only rigorously shows that entropy is the driving force of a diffusion process but also reveals explicitly the Wasserstein metric as a dissipation mechanism. Since then the theory of Wasserstein gradient flows has provided a unified framework and tools for analysing many Markovian (memoryless) systems. During the last twenty years, many evolutionary PDEs for models in biology, chemistry, mechanics, and physics have been studied via this framework, including the Fokker-Planck equation, porous medium equations, thin-film equations, nonlinear aggregation-diffusion equations, interface evolutions, as well as pattern formation and evolution. However, many real-life systems such as anomalous diffusion, plasma transport, chemotaxis movements and human travel, just to name a few, involve long-range interactions and have memory (history), thus are non-Markovian. The mathematical analysis of non-Markovian processes is often much more difficult than that of Markovian ones because describing a non-Markovian process requires an infinite set of multidimensional probability distributions and one cannot obtain the higher-dimensional distributions from lower dimensional ones as in the Markovian case. A generalisation of the theory of Wasserstein gradient flows to non-Markovian systems is challenging but desirable.

This proposal aims at developing variational theory and methods for studying a large class of nonlocal nonlinear partial differential equations describing non-Markovian systems. Important examples considered in the proposal include the fractional Fokker-Planck equation, the fractional porous medium equation and the space-time fractional diffusion equation. Firstly, we will introduce discrete approximation schemes that take into account the memory effects, constructively showing the existence and uniqueness of solutions to the PDEs. Secondly, we will generalise the celebrated hypocoercivity method introduced by C. Villani (Fields Medal in 2010) to obtain rate of convergence to the equilibrium of the PDEs. Thirdly, we will derive and exploit connections between variational structures and stochastic processes to characterise multi-scale behaviour of these systems, both deriving effective systems and obtaining a quantification of errors. The outcome of the proposal will significantly deepen our understanding of the complex systems modelled by the PDEs.