# Extremal flows in Wasserstein space

@article{Conforti2017ExtremalFI, title={Extremal flows in Wasserstein space}, author={Giovanni Conforti and Michele Pavon}, journal={Journal of Mathematical Physics}, year={2017} }

We develop an intrinsic geometric approach to calculus of variations on Wasserstein space. We show that the flows associated to the Schroedinger bridge with general prior, to Optimal Mass Transport and to the Madelung fluid can all be characterized as annihilating the first variation of a suitable action. We then discuss the implications of this unified framework for stochastic mechanics: It entails, in particular, a sort of fluid-dynamic reconciliation between Bohm's and Nelson's stochastic…

## 8 Citations

### Wasserstein Hamiltonian flow with common noise on graph

- Mathematics
- 2022

. We study the Wasserstein Hamiltonian ﬂow with a common noise on the density manifold of a ﬁnite graph. Under the framework of stochastic variational principle, we ﬁrst de-velop the formulation of…

### What is a stochastic Hamiltonian process on finite graph? An optimal transport answer

- Mathematics, Computer ScienceJournal of Differential Equations
- 2021

### Stochastic Control Liaisons: Richard Sinkhorn Meets Gaspard Monge on a Schrödinger Bridge

- MathematicsSIAM Rev.
- 2021

The zero-temperature problem in the continuous-time and space setting turns out to be the celebrated Benamou-Brenier characterization of the McCann displacement interpolation flow in OMT.

### Hopf-Cole Transformation and Schrödinger Problems

- MathematicsGSI
- 2019

This work presents two examples of canonical transformations, including a Schrodinger problem associated with a quadratic Renyi entropy, and study generalized Hopf–Cole transformations motivated by theSchrodinger bridge problem.

### On the Stochastic Mechanics Foundation of Quantum Mechanics

- PhysicsUniverse
- 2021

Among the famous formulations of quantum mechanics, the stochastic picture developed since the middle of the last century remains one of the less known ones. It is possible to describe quantum…

### Stochastic control liasons: Richard Sinkhorn meets Gaspard Monge on a Schroedinger bridge

- Computer ScienceArXiv
- 2020

A multifacet and versatile framework, intertwining SBP and OMT, provides the substrate for a historical and technical overview of the field taken up in this paper.

### Hessian metric via transport information geometry

- Computer ScienceArXiv
- 2020

It is proposed to study the Hessian metric of given functional in the space of probability space embedded with $L^2$--Wasserstein (optimal transport) metric, which contains and extends the classical Wasserstein metric.

## References

SHOWING 1-10 OF 86 REFERENCES

### Extremal Curves in Wasserstein Space

- MathematicsGSI
- 2017

It is shown that known Newton-type laws for Optimal Mass Transport, Schrodinger Bridges and the classic Madelung fluid can be derived from variational principles on Wasserstein space by annihilating the first variation of a suitable action.

### Hamilton-Jacobi equations in the Wasserstein space

- Mathematics
- 2008

We introduce a concept of viscosity solutions for Hamilton-Jacobi equations (HJE) in the Wasserstein space. We prove existence of solutions for the Cauchy problem for certain Hamil- tonians defined…

### A Geometric Perspective on Regularized Optimal Transport

- MathematicsJournal of Dynamics and Differential Equations
- 2018

We present new geometric intuition on dynamical versions of regularized optimal transport. We introduce two families of variational problems on Riemannian manifolds which contain analogues of the…

### Stochastic control, entropic interpolation and gradient flows on Wasserstein product spaces

- Mathematics
- 2016

Since the early nineties, it has been observed that the Schroedinger bridge problem can be formulated as a stochastic control problem with atypical boundary constraints. This in turn has a fluid…

### Optimal Transport for Applied Mathematicians: Calculus of Variations, PDEs, and Modeling

- Mathematics
- 2015

Preface.- Primal and Dual Problems.- One-Dimensional Issues.- L^1 and L^infinity Theory.- Minimal Flows.- Wasserstein Spaces.- Numerical Methods.- Functionals over Probabilities.- Gradient Flows.-…

### Entropic and Displacement Interpolation: A Computational Approach Using the Hilbert Metric

- MathematicsSIAM J. Appl. Math.
- 2016

The purpose of this paper is to show that a similar approach can be taken in the context of diffusion processes which leads to a new proof of a classical result on Schroedinger bridges and provides an efficient computational scheme for both, Schroedingers bridges and OMT.

### On the Relation Between Optimal Transport and Schrödinger Bridges: A Stochastic Control Viewpoint

- MathematicsJ. Optim. Theory Appl.
- 2016

A new look at the relation between the optimal transport problem and the Schrödinger bridge problem from a stochastic control perspective is taken and a generalization of optimal mass transport in the form of a (fluid dynamic) problem of optimal transport with prior is considered.

### Quantization of Dynamical Systems and Stochastic Control Theory

- Physics
- 1983

In the general framework of stochastic control theory we introduce a suitable form of stochastic action associated to the controlled process. A variational principle gives all the main features of…

### A stochastic control approach to reciprocal diffusion processes

- Mathematics
- 1991

The problem of forcing a nondegenerate diffusion process to a given final configuration is considered. Using the logarithmic transformation approach developed by Fleming, it is shown that the…