Wasserstein gradient flows from large deviations of many-particle limits

Duong, Manh Hong; Laschos, Vaios; Renger, D. R. Michiel

doi:10.1051/cocv/2013049

Cited by 43 publications

(45 citation statements)

References 14 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In the case of stochastic processes, this leads to a large-deviations rate function J that is defined on a suitable space of curves. It is now known that many gradient flows and large-deviations principles are strongly connected [2,3,16,30]. In abstract terms, the rate function J of the large-deviations principle simultaneously figures as the defining quantity of the gradient flow, in the sense that…”

Section: Large Deviations Gradient Flows and Variational Formulationsmentioning

confidence: 99%

Quadratic and rate-independent limits for a large-deviations functional

Bonaschi

Peletier

2015

Continuum Mech. Thermodyn.

View full text Add to dashboard Cite

We construct a stochastic model showing the relationship between noise, gradient flows and rateindependent systems. The model consists of a one-dimensional birth-death process on a lattice, with rates derived from Kramers' law as an approximation of a Brownian motion on a wiggly energy landscape. Taking various limits, we show how to obtain a whole family of generalized gradient flows, ranging from quadratic to rate-independent ones, connected via 'L log L' gradient flows. This is achieved via Mosco-convergence of the renormalized large-deviations rate functional of the stochastic process.

show abstract

Section: Large Deviations Gradient Flows and Variational Formulationsmentioning

confidence: 99%

Quadratic and rate-independent limits for a large-deviations functional

Bonaschi

Peletier

2015

Continuum Mech. Thermodyn.

View full text Add to dashboard Cite

show abstract

“…in the sense of Gamma-convergence (see also [ADPZ11,DLR13,PR11]). Written informally, this result states that…”

Section: Brownian Particles and Wasserstein Dissipationmentioning

confidence: 99%

“…This was later generalized to a larger class of systems in [DLR13]. The function on the right-hand side of (5.7) is well known in the theory of gradient flows, as the basis for a discrete-time approximation of a gradient flow.…”

Section: Brownian Particles and Wasserstein Dissipation Takementioning

confidence: 99%

Preface – Acknowledgements

2016

Oral Poetics and Cognitive Science

View full text Add to dashboard Cite

“…This direction recently has been received a lot of attention. In [1,7,8,19,28], the authors address this question, for the case of the linear diffusion equation, i.e., for q = 1, by establishing an intriguing connection between a microscopic many-particle model and the macroscopic gradient flow structure of the diffusion equation. They show that the functional K h in (4) is asymptotically equivalent, as h → 0, to a discrete rate functional J h that comes from the large deviation principle of the microscopic model.…”

Section: The Porous Medium Equationmentioning

confidence: 99%

Asymptotic equivalence of the discrete variational functional and a rate-large-deviation-like functional in the Wasserstein gradient flow of the porous medium equation

Duong

2015

Asymptotic Analysis

Self Cite

View full text Add to dashboard Cite

In this paper, we study the Wasserstein gradient flow structure of the porous medium equation restricted to q-Gaussians. The JKO-formulation of the porous medium equation gives a variational functional K h , which is the sum of the (scaled-) Wasserstein distance and the internal energy, for a time step h. We prove that, for the case of q-Gaussians on the real line, K h is asymptotically equivalent, in the sense of Γ -convergence as h tends to zero, to a rate-large-deviation-like functional. The result explains why the Wasserstein metric as well as the combination of it with the internal energy play an important role.1 E 1 is the Boltzmann-Shannon entropy. 0921-7134/15/$35.00 © 2015 -IOS Press and the authors. All rights reserved 86 M.H. Duong / Asymptotic equivalence of the discrete variational functional with respect to the Wasserstein distance W 2 on the space of probability measures with finite second moment P 2 (R d ),where Γ (μ, ν) is the set of all probability measures on R d × R d with marginals μ and ν. This statement can be understood in a variety of different ways. In [27], Otto shows this from a differential geometry point of view by considering (P 2 (R d ), W 2 ) as an infinite dimensional Riemannian manifold. However, for the purpose of this paper, the most useful way is that the solution t → ρ(t, x) can be approximated by the JKO-discretization scheme as follows [17,31].

show abstract

Wasserstein gradient flows from large deviations of many-particle limits

Cited by 43 publications

References 14 publications

Quadratic and rate-independent limits for a large-deviations functional

Quadratic and rate-independent limits for a large-deviations functional

Preface – Acknowledgements

Asymptotic equivalence of the discrete variational functional and a rate-large-deviation-like functional in the Wasserstein gradient flow of the porous medium equation

Contact Info

Product

Resources

About