Microscopic interpretation of Wasserstein gradient flows

Abstract

The discovery of Wasserstein gradient flows provided a mathematically precise formulation for the way in which thermodynamic systems are driven by entropy. Since the entropy of a system in equilibrium can be related to the large deviations of stochastic particle systems, the question arises whether a similar relation exists between Wasserstein gradient flows and stochastic particle systems. In the work presented in this thesis, such relation is studied for a number of systems. As explained in the introduction chapter, central to this research is the study of two types of large deviations for stochastic particle systems. The first type is related to the probability of the empirical measure of the particle system, after a fixed time step, conditioned on the initial empirical measure. The large-deviation rate provides a variational formulation of the transition between two macroscopic measures, in fixed time. This formulation can then be related to a discrete-time formulation of a gradient flow, known as minimising movement. To this aim, a specific small-time development of the rate is used, based on the concept of Mosco- or Gamma-convergence. The other type of large deviations concerns the probability of the trajectory of the empirical measure in a time interval. For these large deviations, the rate provides a variational formulation for the trajectory of macroscopic measures. For some systems, this rate can be related to a continuous-time formulation of gradient flows, known as an entropy-dissipation inequality. Chapter 2 serves as background, where a number of results from particle system theory and large deviations are proven in a general setting. In particular, the generator of the empirical measure is calculated, the many-particle limit of the empirical measure is proven, and the discrete-time large deviation principle is proven. In Chapter 3, the discrete-time large-deviation rate is studied for a system of independent Brownian particles in a force field, which yields the Fokker-Planck equation in the many-particle limit. Based on an estimate for the fundamental solution of the Fokker-Planck equation, it is proven that the small-time development of the rate functional coincides with the minimising movement of the Wasserstein gradient flow, under the conjecture that a similar relation holds if there is no force field. The Fokker-Planck equation is studied further in Chapter 4, but with a different technique. Here, an alternative formulation of the discrete-time large-deviation rate is derived from the continuous-time large-deviation rate. With this alternative formulation, the small-time development is proven in a much more general context, so that the conjecture of the previous chapter can indeed be dropped. To complete the discussion, it is mentioned that the continuous-time large deviations can be coupled to a Wasserstein gradient flow more directly, via an entropy-dissipation inequality. Chapter 5 discusses two related systems, whose evolutions are described by a system of reaction-diffusion equations, and by the diffusion equation with decay. In order to deal with the fact that the diffusion equation with decay is not mass-conserving, the decayed mass is added back to the system as decayed matter, which results in a system of reactiondiffusion equations. This allows for the construction of a system of independent particles whose probability evolves according to the reaction-diffusion equations. Similar to the previous chapters, a small-time development of the discrete-time large-deviation rate is proven. It turns out that in general the resulting functional can not be associated with a minimising movement formulation of a Wasserstein gradient flow. Nevertheless, it is proven that this functional defines a variational scheme that approximates the system of reaction-diffusion equations. As such, the functional, which involves entropy terms and Wasserstein distances, can be interpreted as a generalisation of a minimising movement. Chapter 6 is concerned with the effect of trivial Dirichlet boundary conditions on the discrete-time rate functional and its small-time development. Since the diffusion equation with trivial Dirichlet boundary conditions loses mass at the boundaries, the system is first transformed into a mass-conserving system by adding the amount of lost mass back to the system in two delta measures at the boundaries. This again allows for the construction of a corresponding microscopic particle system. For this particle system, the discrete-time large-deviation rate is calculated, and its small-time development is proven. It is still unclear whether the resulting functional can be used to define an variational approxiation scheme for the diffusion equation with Dirichlet boundary conditions. In Chapter 7, general Markov chains on a finite state space are studied. The macroscopic equation is then a linear system of ordinary differential equations, and the microscopic particle system consists of independent Markovian particles on the finite state space. First, the discrete-time rate functional for this particle system is calculated, and its small-time asymptotic development is proven for a two-state system. Although the resulting function looks like a minimising movement, it is still unknown whether this function defines a variational formulation for the system of linear differential equations. Next, the continuous-time rate functional is calculated, at least formally, using the Feng-Kurtz formalism. It is then shown that the continous-time rate function can be rewritten into an entropy-dissipation inequality. However, this inequality may imply unphysical behaviour unless the Markov chain satisfies detailed balance. Finally, the used methods and techniques and the general results are evaluated in Chapter8. It is shown that one can expect to find a gradient-flow structure via large-deviations of microscopic particle systems, if the macroscopic equation satisfies a generalised version of the detailed balance condition. Some important implications of detailed balance are proven for both the discrete-time approach, as well as for the continuous-time approach. The chapter ends with a discussion of the results for the different systems, and the general methods that are used in this thesis.