An invariance principle for gradient flows in the space of probability measures
type: Item
instance of: scholarly article
title: An invariance principle for gradient flows in the space of probability measures
author: Rishabh S. Gvalani
author: Jeremy S.-H. Wu
author: J. A. Carrillo
published in: Journal of Differential Equations
publication date: 2022-12-16T00:00:00Z
full work available at URL: https://arxiv.org/abs/2010.00424
review text: This article studied the qualitative convergence results of a general class of evolution PDEs described by p-Wasserstein gradient flows. These qualitative convergence results come from dynamical systems known as the LaSalle Invariance Principle. A weaker form of this principle is a well known result in finite dimensional gradient flows: Let $F: \mathbb{R}^d \to \mathbb{R}$ be a sufficiently smooth function and $x: [0 , \infty)\to \mathbb{R}^d$ be a curve solving the ODE \[ \frac{d}{dt}x(t) = -\nabla F(x(t)), \ \ x(0) = x_0 \in \mathbb{R}^d. \] Assume further that the set $\{x(t) : t \geq 0\}$ is relatively compact in $\mathbb{R}^d$. Then, the distance between $x(t)$ and the set of critical points of $F$ (i.e. $\{ y \in \mathbb{R}^n \mid \nabla F(y) = 0\}$) converges to $0$ as $t \rightarrow \infty.$ The authors studied this principle in the context of gradient flows in the metric spaces $(P(\Omega),W_p)$ and $(P_p(\Omega), W_p)$ where $\Omega$ is either $\mathbb{R}^d$ or the torus $ \mathbb{T}^d$ and $P(\Omega)$ (respectively, $P_p(\Omega)$) is the set of all Borel probability measures on $\Omega$ (respectively, with finite $p$-th moment) and $W_p$ denotes the $p$-Wasserstein distance defined by \[ W_p(\mu, \nu)^p = \inf \left\{ \int_{\mathbb{R}^d\times \mathbb{R}^d} |x-y|^p \ d\pi(x,y) \mid \pi \in P(\mathbb{R}^d \times \mathbb{R}^d)\text{ has marginals }\mu\text{ and }\nu\right\}. \] There are multiple notions of gradient flows on a general metric space: \begin{itemize} \item[(a)] $p$-curves of maximal slope. This is a very weak notion founded on an appropriately defined metric derivative of a curve $\rho(t): [0,\infty)\to P_p(\Omega),$ weak upper gradient of the energy $E : P_p(\Omega) \to \mathbb{R} \cup \{+\infty\}$ and Young's inequality. In this formalism, it is rather easier to establish the existence theory but does not guarantee the uniqueness of solutions. \item[(b)] Evolutionary Variational Inequality (EVI), which requires a notion of convexity assumptio…
reviewed by: Debabrata Karmakar
Mathematics Subject Classification ID: 35B40
Mathematics Subject Classification ID: 35K90
Mathematics Subject Classification ID: 37L15
Mathematics Subject Classification ID: 49Q22
Mathematics Subject Classification ID: 35Q70
zbMATH DE Number: 7633392
zbMATH Keywords: gradient flows in metric spaces
zbMATH Keywords: Wasserstein gradient flow
zbMATH Keywords: La Salle invariance principle
MaRDI profile type: Publication
OpenAlex ID: W4310067445
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DOI: 10.1016/J.JDE.2022.11.028