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Optimal Transport and Stochastic Dynamics

Periodic Reporting for period 4 - OPTRASTOCH (Optimal Transport and Stochastic Dynamics)

Okres sprawozdawczy: 2021-08-01 do 2022-07-31

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.
The research is divided into three directions: (A) geometric aspects of optimal transport, (B) applications to stochastic dynamics, and (C) optimal transport in non-commutative probability. Major results have been obtained in all of these directions.

A) One of the highlights is an investigation of the geodesics of discrete optimal transport metrics using discrete Hamilton–Jacobi equations. Another key contribution is a trajectorial approach to entropy dissipation and discrete gradient flows.

B) A main line of research deals with the large-scale behaviour of dynamical optimal transport on graphs. Several major results have been obtained, including convergence theorems for optimal transport on finite-volume meshes and homogenisation results for optimal transport on periodic graphs.

C) We developed a rich theory of dynamical optimal transport for density matrices with applications to entropy inequalities in non-commutative probability and variational structures for quantum Markov semigroups.

Other highlights include a study of evolutionary Gamma-convergence using discrete transport metrics, a gradient flow formulations of dissipative PDE on metric graphs, and a treatment of discrete and continuous models for chemical reaction networks using optimal transport methods.

The primary impact of the project is in mathematics, but several of the obtained results may have future applications in other sciences, including physics, chemistry, biology, and computer science.
The project uses new mathematical methods to analyse stochastic processes and the associated geometric structures. The obtained results are significantly beyond the state of the art.