# Schroedinger's problem and Optimal Transport :a multidisciplinary perspective.

PTDC/MAT-STA/28812/2017

## Members

- Jean-David Benamou
- Guillaume Carlier
- Ana Bela Cruzeiro (Co-PI)
- Ana Patricia Gonçalves
- Christian Léonard
- Gabriel Peyré
- Hugo Tavares
- Pierre Vuillermot
- Jean-Claude Zambrini (PI)

## Description

The international community of Optimal Transportation (OT) has re-discovered in recent years a problem posed by E. Schroedinger in 1932, concerning the curious role of probability theory in Quantum Mechanics. A solution of this problem, in terms of stochastic processes (diffusions), was given in 1986 by the PI of this project, without ideas or techniques of OT, but the problem and its solution can and have been re-interpreted since then in these terms. This approach (also known today as "entropic regularization") has improved considerably , in the last few years, the speed of numerical computations of the solution of OT problems in various applied fields, from medical imaging to machine learning, fluids models or mathematical economics. In this way, it has also introduced new and promising theoretical connexions between the theory of stochastic processes, Fluids dynamics, Quantum Physics and OT. Our project aims to join forces of three research groups in Lisbon, Paris and New-York in order to explore further the theoretical, numerical and applied consequences of "Schroedinger's problem" and its generalisations.

1/01/2019-1/01/2021

## Relevant publications

- J.C. Zambrini
- Variational processes and stochastic versions of mechanics

*J. Math. Phys.*27 (9), September 1986, p.2307. - A.B. Cruzeiro, Wu Liming,J.C. Zambrini
- Bernstein processes associated with a Markov process

*Stochastic Analysis and Mathematical Physics*ANESTOC98, Proc. of the third International Workshop, Ed R. Rebolledo, Trends in Mathematical Series, Birkhauser, Boston (2000). - C. Leonard, S. Roelly, J.C. Zambrini
- Reciprocal processes : a measure theoretic point of view

*Probability Surveys*11 (2014), p 237 J.G. - J. D. Benamou, Y. Brenier
- A computational fluid mechanics solution to the Monge-Kantorovich
mass transfer problem

*Numerische Mathematik*Vol 84, Issue 3, Jan 2000, p 375. - A. Galichon
- Optimal transport methods in Economics

Princeton Univ. Press (2016). - M. Arnaudon, A.B. Cruzeiro, S. Fang
- Generalized stochastic Lagrangian paths for the Navier-Stokes
equation

*Ann. Scuola Norm. Sup. Pisa*pp 24, DOI Number: 10.2422/2036-2145.201602_006 (2017).