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1. Global hypocoercivity of kinetic Fokker-Planck-Alignment equations

   https://doi.org/10.3934/krm.2022005  

   Shvydkoy, Roman (January 2022, Kinetic and Related Models)

   In this note we establish hypocoercivity and exponential relaxation to the Maxwellian for a class of kinetic Fokker-Planck-Alignment equations arising in the studies of collective behavior. Unlike previously known results in this direction that focus on convergence near Maxwellian, our result is global for hydrodynamically dense flocks, which has several consequences. In particular, if communication is long-range, the convergence is unconditional. If communication is local then all nearly aligned flocks quantified by smallness of the Fisher information relax to the Maxwellian. In the latter case the class of initial data is stable under the vanishing noise limit, i.e. it reduces to a non-trivial and natural class of traveling wave solutions to the noiseless Vlasov-Alignment equation.The main novelty in our approach is the adaptation of a mollified Favre filtration of the macroscopic momentum into the communication protocol. Such filtration has been used previously in large eddy simulations of compressible turbulence and its new variant appeared in the proof of the Onsager conjecture for inhomogeneous Navier-Stokes system. A rigorous treatment of well-posedness for smooth solutions is provided. Lastly, we prove that in the limit of strong noise and local alignment solutions to the Fokker-Planck-Alignment equation Maxwellialize to solutions of the macroscopic hydrodynamic system with the isothermal pressure. 

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2. A TRAJECTORIAL APPROACH TO THE GRADIENT FLOW PROPERTIES OF LANGEVIN–SMOLUCHOWSKI DIFFUSIONS

   https://doi.org/10.4213/tvp5505  

   KARATZAS, I.; SCHACHERMAYER, W.; TSCHIDERER, B. (January 2022, Theory of probability and its applications)

   We revisit the variational characterization of conservative di↵usion as entropic gra- dient flow and provide for it a probabilistic interpretation based on stochastic calculus. It was shown by Jordan, Kinderlehrer, and Otto that, for diffusions of Langevin–Smoluchowski type, the Fokker–Planck probability density flow maximizes the rate of relative entropy dissipation, as mea- sured by the distance traveled in the ambient space of probability measures with finite second moments, in terms of the quadratic Wasserstein metric. We obtain novel, stochastic-process ver- sions of these features, valid along almost every trajectory of the dffusive motion in the backwards direction of time, using a very direct perturbation analysis. By averaging our trajectorial results with respect to the underlying measure on path space, we establish the maximal rate of entropy dissipation along the Fokker–Planck flow and measure exactly the deviation from this maximum that corresponds to any given perturbation. A bonus of our trajectorial approach is that it derives the HWI inequality relating relative entropy (H), Wasserstein distance (W), and relative Fisher information (I). 

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3. Clustering Dynamics on Graphs: From Spectral Clustering to Mean Shift Through Fokker–Planck Interpolation.

   https://doi.org/10.1007/978-3-030-93302-9_4  

   Craig, Katy; Garcia Trillos, Nicolas; Slepcev, Dejan (December 2021, Modeling and simulation in science engineering technology)

   Bellomo, N.; Carrillo, J.A.; Tadmor, E. (Ed.)

   In this work, we build a unifying framework to interpolate between density-driven and geometry-based algorithms for data clustering and, specifically, to connect the mean shift algorithm with spectral clustering at discrete and continuum levels. We seek this connection through the introduction of Fokker–Planck equations on data graphs. Besides introducing new forms of mean shift algorithms on graphs, we provide new theoretical insights on the behavior of the family of diffusion maps in the large sample limit as well as provide new connections between diffusion maps and mean shift dynamics on a fixed graph. Several numerical examples illustrate our theoretical findings and highlight the benefits of interpolating density-driven and geometry-based clustering algorithms. 

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4. A deep learning method for solving Fokker-Planck equations

   Jiayu Zhai, Matthew Dobson (January 2022, Proceedings of Machine Learning Research)

   Joan Bruna, Jan S (Ed.)

   The time evolution of the probability distribution of a stochastic differential equation follows the Fokker-Planck equation, which usually has an unbounded, high-dimensional domain. Inspired by Li (2019), we propose a mesh-free Fokker-Planck solver, in which the solution to the Fokker-Planck equation is now represented by a neural network. The presence of the differential operator in the loss function improves the accuracy of the neural network representation and reduces the demand of data in the training process. Several high dimensional numerical examples are demonstrated. 

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5. Data-Driven Computational Methods for Quasi-Stationary Distribution and Sensitivity Analysis

   https://doi.org/10.1007/s10884-022-10137-2  

   Li, Yao; Yuan, Yaping (January 2022, Journal of Dynamics and Differential Equations)

   This paper studies computational methods for quasi-stationary distributions (QSDs). We first proposed a data-driven solver that solves Fokker–Planck equations for QSDs. Similar to the case of Fokker–Planck equations for invariant probability measures, we set up an optimization problem that minimizes the distance from a low-accuracy reference solution, under the constraint of satisfying the linear relation given by the discretized Fokker–Planck operator. Then we use coupling method to study the sensitivity of a QSD against either the change of boundary condition or the diffusion coefficient. The 1-Wasserstein distance between a QSD and the corresponding invariant probability measure can be quantitatively estimated. Some numerical results about both computation of QSDs and their sensitivity analysis are provided. 

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