On the occasion of the SIAM Journal on Mathematical Analysis’ 50th volume, a coauthor of the journal’s most popular article offers insight as to why the paper continues to spark so much interest.
My article with Richard Jordan and David Kinderlehrer highlighted the gradient flow structure of the Fokker-Planck equation describing overdamped Langevin dynamics of a particle in a potential. It did so via an (implicit) time discretization in the form of a sequence of variational problems. The insight that such time discretizations—called “minimizing movements” by Ennio De Giorgi’s school—are helpful in understanding dissipative dynamics became popular in geometric analysis through Fred Almgren, Jean Taylor, and Lihe Wang’s work on the flow of a hypersurface by its mean curvature. My Ph.D. advisor Stephan Luckhaus’ convergence result with fellow Ph.D. student Thomas Sturzenhecker provided me with an understanding of this research.
Surprisingly, our minimizing movement scheme made the Wasserstein metric (as recognized by Richard Jordan)—the transportation cost between two particle densities—appear. Our analysis relied on Yann Brenier’s work with optimal transportation, which was motivated by Vladimir Arnold’s geometric view of Euler’s equations for an inviscid incompressible fluid. Incidentally, I learned of Brenier’s study in Luis Caffarelli’s inspiring class on the regularity of the Monge-Ampère equation, observable as the Euler-Lagrange equation of optimal transportation.
Our article fed into two recent developments, the first classical theme being the thermodynamically-consistent modelling of quasistatic dissipative evolutions. This process properly separates driving energetics from limiting dissipation — an area somewhere between applied partial differential equations and rational mechanics. In particular, Alexander Mielke and his collaborators developed a general theory partly inspired by rate-independent evolutions like crack propagation. More recently, Mark Peletier and his colleagues showed that large deviation principles of the underlying stochastic process should guide the proper separation between the energy functional and the dissipation metric. A beautiful extension of optimal transportation suitable for reaction-diffusion equations has recently come to light. The proper extension of the minimizing movement structure to discrete Markov chains—and somewhat related dissipative quantum evolutions—is an active area of research.
Perhaps our article had its broadest impact in geometric analysis: companion articles put Robert McCann’s displacement convexity of entropy-functionals and ensuing contractivity properties of (nonlinear) diffusions into an infinite-dimensional geometric perspective. The intimate connection between these and the Ricci curvature of the underlying manifold (which on the infinitesimal level was already present in the Bakry-Emery calculus) eventually allowed for a “synthetic” definition of Ricci curvature on metric measure by John Lott, Cédric Villani, and Karl-Theodor Sturm. The work of Luigi Ambrosio, Nicola Gigli, and Giuseppe Savaré again made the connection to De Giorgi’s program.
Table 1. The 10 most downloaded articles from the SIAM Journal on Mathematical Analysis.
Table 1 displays the 10 most-downloaded articles from the SIAM Journal on Mathematical Analysis, all of which are freely accessible through the end of the year. This information is also available online.