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Speaker:

Daniel Hauer
Affiliation:

University of Sydney
Date:

Thu, 2017-09-28 17:00 - 17:30
Location:

MPIM Lecture Hall
Parent event:

Metric Measure Spaces and Ricci Curvature 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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