# From a Large-Deviations Principle to the Wasserstein Gradient Flow: A New Micro-Macro Passage

@article{Adams2010FromAL, title={From a Large-Deviations Principle to the Wasserstein Gradient Flow: A New Micro-Macro Passage}, author={Stefan Adams and Nicolas Dirr and Mark A. Peletier and Johannes Zimmer}, journal={Communications in Mathematical Physics}, year={2010}, volume={307}, pages={791-815} }

We study the connection between a system of many independent Brownian particles on one hand and the deterministic diffusion equation on the other. For a fixed time step h > 0, a large-deviations rate functional Jh characterizes the behaviour of the particle system at t = h in terms of the initial distribution at t = 0. For the diffusion equation, a single step in the time-discretized entropy-Wasserstein gradient flow is characterized by the minimization of a functional Kh. We establish a new…

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## References

SHOWING 1-10 OF 40 REFERENCES

### Contractions in the 2-Wasserstein Length Space and Thermalization of Granular Media

- Mathematics
- 2006

An algebraic decay rate is derived which bounds the time required for velocities to equilibrate in a spatially homogeneous flow-through model representing the continuum limit of a gas of particles…

### Scaling Limits of Interacting Particle Systems

- Mathematics
- 1998

1. An Introductory Example: Independent Random Walks.- 2. Some Interacting Particle Systems.- 3. Weak Formulations of Local Equilibrium.- 4. Hydrodynamic Equation of Symmetric Simple Exclusion…

### A variational principle for the Kramers equation with unbounded external forces

- Mathematics
- 2000

A time discrete variational principle is developed for the Cauchy problem of the Kramers equation with unbounded external force fields. The variational scheme is based on the idea of maximizing a…

### A Convexity Principle for Interacting Gases

- Mathematics
- 1997

A new set of inequalities is introduced, based on a novel but natural interpolation between Borel probability measures on R d . Using these estimates in lieu of convexity or rearrangement…

### A Family of Nonlinear Fourth Order Equations of Gradient Flow Type

- Mathematics
- 2009

Global existence and long-time behavior of solutions to a family of nonlinear fourth order evolution equations on R d are studied. These equations constitute gradient flows for the perturbed…

### A large deviation approach to optimal transport

- Mathematics
- 2007

A probabilistic method for solving the Monge-Kantorovich mass transport problem on $R^d$ is introduced. A system of empirical measures of independent particles is built in such a way that it obeys a…

### Well-posedness of a parabolic moving-boundary problem in the setting of Wasserstein gradient flows

- Mathematics
- 2008

We develop a gradient-flow framework based on the Wasserstein metric for a parabolic moving-boundary problem that models crystal dissolution and precipitation. In doing so we derive a new weak…

### Optimal Transport: Old and New

- Mathematics
- 2008

Couplings and changes of variables.- Three examples of coupling techniques.- The founding fathers of optimal transport.- Qualitative description of optimal transport.- Basic properties.- Cyclical…

### Theory of anomalous chemical transport in random fracture networks

- Physics
- 1998

We show that dominant aspects of chemical (particle) transport in fracture networks\char22{}non-Gaussian propagation\char22{}result from subtle features of the steady flow-field distribution through…