Related papers: Extremal flows on Wasserstein space
We continue our investigation to the use of the variational method to derive flow relations for generalized Newtonian fluids in confined geometries. While in the previous investigations we used the straight circular tube geometry with eight…
The primary choice to summarize a finite collection of random objects is by using measures of central tendency, such as mean and median. In the field of optimal transport, the Wasserstein barycenter corresponds to the Fr\'{e}chet or…
Wasserstein gradient flows on probability measures have found a host of applications in various optimization problems. They typically arise as the continuum limit of exchangeable particle systems evolving by some mean-field interaction…
We present some exact solutions to the ideal hydrodynamics of a relativistic superfluid with an almost-conformal equation of state. The solutions have stress tensors which are invariant under Lorentz boosts in one direction, and represent…
We discuss the relation between the Wasserstein distance of order 1 between probability distributions on a metric space, arising in the study of Monge-Kantorovich transport problem, and the spectral distance of noncommutative geometry.…
We present new geometric intuition on dynamical versions of regularized optimal transport. We introduce two families of variational problems on Riemannian manifolds which contain analogues of the Schr\"odinger bridge problem and the Yasue…
Barotropic fluid flows with the same circulation structure as steady flows generically have comoving physical surfaces on which the vortex lines lie. These become Bernoullian surfaces when the flow is steady. When these surfaces are nested…
This paper is concerned by statistical inference problems from a data set whose elements may be modeled as random probability measures such as multiple histograms or point clouds. We propose to review recent contributions in statistics on…
The Wasserstein distance received a lot of attention recently in the community of machine learning, especially for its principled way of comparing distributions. It has found numerous applications in several hard problems, such as domain…
Multi-marginal optimal transport enables one to compare multiple probability measures, which increasingly finds application in multi-task learning problems. One practical limitation of multi-marginal transport is computational scalability…
We prove the following nonholonomic version of the classical Moser theorem: given a bracket-generating distribution on a connected compact manifold (possibly with boundary), two volume forms of equal total volume can be isotoped by the flow…
Optimal transport is a foundational problem in optimization, that allows to compare probability distributions while taking into account geometric aspects. Its optimal objective value, the Wasserstein distance, provides an important loss…
We study the Wasserstein space (with quadratic cost) of Euclidean spaces as an intrinsic metric space. In particular we compute their isometry groups. Surprisingly, in the case of the line, there exists a (unique) "exotic" isometric flow.…
We derive moment estimates and a strong limit theorem for space inverses of stochastic flows generated by jump SDEs with adapted coefficients in weighted H\"older norms using the Sobolev embedding theorem and the change of variable formula.…
An overview of recent results pertaining to the hydrodynamic description (both Newtonian and non-Newtonian) of granular gases described by the Boltzmann equation for inelastic Maxwell models is presented. The use of this mathematical model…
We present a framework for obtaining explicit bounds on the rate of convergence to equilibrium of a Markov chain on a general state space, with respect to both total variation and Wasserstein distances. For Wasserstein bounds, our main tool…
A planar superfluid is considered and interpreted in terms of electromagnetism and gravity. It has previously been suggested that the superfluid flow can be regarded as analogous to an electromagnetic field and that a non-vanishing density…
Vortex line and magnetic line representations are introduced for description of flows in ideal hydrodynamics and MHD, respectively. For incompressible fluids it is shown that the equations of motion for vorticity ${\bf \Omega}$ and magnetic…
Wasserstein distances are metrics on probability distributions inspired by the problem of optimal mass transportation. Roughly speaking, they measure the minimal effort required to reconfigure the probability mass of one distribution in…
We present a short overview on the strongest variational formulation for gradient flows of geodesically $\lambda$-convex functionals in metric spaces, with applications to diffusion equations in Wasserstein spaces of probability measures.…