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

偏微分方程(PDE)是一个涉及多个(空间和时间)变量及其偏导数的未知函数的方程。偏微分方程组出现在数学的许多分支中，并构成了复杂系统的建模、分析和计算/仿真的强大数学框架。一个明显的例子是统计力学中广泛使用的Fokker-Planck方程，它描述了粒子在外力、摩擦力和随机力的影响下运动的位置的概率密度函数的时间演化。通过求解偏微分方程组并研究其解的定性和定量性质，人们可以获得对潜在复杂系统/现象的理解和控制的重要见解。近200年前发现的热力学第二定律指出，孤立系统的熵永远不会随着时间的推移而减小。建立保持物理结构的数学理论，特别是将热力学第二定律纳入数学模型和分析，一直是人们的根本兴趣。1998年，Jordan、Kinderlehrer和Otto在数学分析方面取得了重大突破，引入了Wasserstein梯度流理论，特别证明了扩散方程是(负)Boltzmann熵相对于Wasserstein度规的梯度流(最陡下降)，这是来自最优传输理论的一个看似无关的概念。这项工作不仅严格地证明了熵是扩散过程的驱动力，而且还明确地揭示了Wasserstein度规作为一种耗散机制。从那时起，沃瑟斯坦梯度流理论为分析许多马尔科夫(无记忆)系统提供了一个统一的框架和工具。在过去的二十年里，许多生物、化学、力学和物理模型的演化偏微分方程组都通过这个框架被研究，包括Fokker-Planck方程、多孔介质方程、薄膜方程、非线性聚集-扩散方程、界面演化以及图案的形成和演化。然而，许多现实生活中的系统，如异常扩散、等离子体运输、趋化运动和人类旅行，仅举几例，涉及远程相互作用和具有记忆(历史)，因此是非马尔可夫的。非马尔可夫过程的数学分析通常比马尔可夫过程的数学分析困难得多，因为描述非马尔可夫过程需要无限多维概率分布的集合，并且不能像马尔可夫过程那样从低维分布中获得高维分布。将沃瑟斯坦梯度流理论推广到非马尔科夫系统是具有挑战性的，但也是可取的。这一建议旨在发展变分理论和方法来研究一大类描述非马尔科夫系统的非局部非线性偏微分方程组。文中考虑的重要例子包括分数阶Fokker-Planck方程、分数阶多孔介质方程和时空分数阶扩散方程。首先，我们将引入考虑记忆效应的离散逼近格式，建设性地证明了偏微分方程解的存在唯一性。其次，我们将推广C.Villani(2010年菲尔兹奖)提出的著名的亚矫顽力方法，以获得收敛到偏微分方程组平衡的速度。第三，我们将推导和利用变分结构和随机过程之间的联系来刻画这些系统的多尺度行为，既得到有效的系统，又获得误差的量化。提案的结果将大大加深我们对PDE所模拟的复杂系统的理解。

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.