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Kurdyka-Lojasiewicz-Simon inequality for gradient flows in metric spaces

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Speaker: 
Daniel Hauer
Affiliation: 
University of Sydney
Date: 
Thu, 2017-09-28 17:00 - 17:30
Location: 
MPIM Lecture Hall

In this talk, I will present new tools and methods in the study of  the trend to equilibrium of gradient flows in metric spaces  $(\mathcal{M},d)$ in the entropy and metric sense, to establish decay rates, finite time of extinction, and to characterise Lyapunov stable equilibrium points. More precisely,

  • I begin by introducing a gradient inequality in the metric space framework, which in the Euclidean space $\mathbb{R}^{N}$ is due to Lojasiewicz [Éditions du C.N.R.S., 87-89, Paris, 1963] and Kurdyka [Ann. Inst. Fourier, 48 (3), 769-783, 1998]. In the Hilbert space framework, this inequality is known under the name Kurdyka-Lojasiewicz gradient inequality and I will outline its connection to the classical entropy-entropy production inequality used in kinetic theory to study the long time asymptotic behaviour of flows (e.g., time-dependent probability distribution on the phase space of particles).
  • I show that the validity of the Kurdyka-Lojasiewicz gradient inequality in a neighbourhood of an equilibrium point yields the trend to equilibrium in the entropy sense and metric sense and provides decay rates and finite time of extinction of gradient flows.
  • I explain the construction of a talweg curve and show that it yields the validity of a Kurdyka-Lojasiewicz inequality with optimal growth function $\theta$.
  • For $1<p<\infty$, I outline that on the $p$-Wasserstein space $\mathcal{P}_{p}(\mathbb{R}^{N})$ or on $\mathcal{P}_{p,d}(X)$ provided $(X,d,\nu)$ is a (compact) measure  length spaces satisfying a \emph{$(p,\infty)$-Ricci curvature bounded from below by $K\in \mathbb{R}$}, the Kurdyka-Lojasiewicz inequality becomes the celebrated Talagrand entropy transportation inequality. We show that this one is equivalent to $p$-logarithmic Sobolev inequalities. On $\mathcal{P}_{p,d}(X)$, this problem is connected with an interesting regularity problem. We note that our notion of Ricci curvature is consistent in the case $p=2$ with the one introduced by Lott & Villani [Ann. Math.  (2),169(3):903-991, 2009] and Sturm [Acta Math., 196(1):133-177, 2006].


The results presented in this talk are obtain in joint work with Prof José Mazón (Universitat de Valéncia, Valencia, Spain) and available at
http://www.maths.usyd.edu.au/u/pubs/publist/preprints/2017/hauer-7.html
https://arxiv.org/abs/1707.03129.

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