Abstract:
This thesis is concerned with probabilistic variants of the transport problem, in particular weak optimal transport and adapted optimal transport.
Weak optimal transport was introduced by Gozlan, Roberto, Samson and Tetali as a nonlinear relaxation of classical optimal transport. On the one hand, this framework of weak optimal transport still retains many characteristics of usual optimal transport, allowing for a compelling theory. On the other hand, this type of relaxation is suitable to cover a number of problems that lie outside the scope of the classical theory such as entropic optimal transport, martingale optimal transport and transport problems with barycentric costs.
In [12] we establish a general duality theorem for weak optimal transport together with dual attainment as well as complementary slackness conditions which characterize primal and dual optimizers, i.e. we establish a fundamental theorem (in the sense of [4, Theorem 1.13]).
As applications we provide concise derivations of the Brenier–Strassen theorem, the convex Kantorovich–Rubinstein formula and the structure theorem of entropic optimal transport. We also extend Strassen’s theorem in the direction of Gangbo–McCann’s transport problem for convex costs. Moreover, we determine the optimizers for a new family of transport problems which contains the Brenier–Strassen, the martingale Benamou–Brenier and the entropic martingale transport problem as extreme cases.
The second main topic of the thesis is adapted optimal transport and adapted weak topologies. Researchers from different areas have independently defined extensions of the usual weak convergence of laws of stochastic processes with the goal of adequately accounting for the flow of information. Natural approaches are convergence of the Aldous–Knight prediction process, Hellwig’s information topology, convergence in adapted distribution in the sense of Hoover–Keisler, and the weak topology induced by optimal stopping problems.

Online:
univienna.zoom.us/j/63409418072
Meeting ID: 634 0941 8072
Passcode: 227324