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Optimal Transport and Stochastic Dynamics

Periodic Reporting for period 2 - OPTRASTOCH (Optimal Transport and Stochastic Dynamics)

Reporting period: 2018-08-01 to 2020-01-31

The project deals with fundamental mathematical questions at the interface of analysis, probability theory, and geometry.
At the core of the project is the problem of optimal transport, an old problem that deals with the optimal allocation of resources.
In recent years, optimal transport has played a key role in important mathematical developments in areas such as non-smooth geometry, probability, and partial differential equations.
The project consists of a wide research program that aims to extend the scope of optimal transport significantly, focusing on applications to discrete geometry, stochastic processes, and quantum dynamics.
In the first two reporting periods significant results have been obtained in various areas of the project, and the groundwork has been laid for further progress.
Among the main results obtained so far are results on the geometry of geodesics in discrete optimal transport,
results on the asymptotic behavior of discrete optimal transport for large classes of meshes, and results on fluctuations in the totally asymmetric simple exclusion process.
The obtained results are significantly beyond the state of the art.
Areas of research in the remaining parts of the project include variational structures for non-reversible stochastic dynamics,
chemical reaction networks, and the geometry of non-commutative optimal transport and its applications to quantum dynamics.