In classical optimal transport, the contributions of Benamou-Brenier and Mc- Cann regarding the time-dependent version of the problem are cornerstones of the field and form the basis for a variety of applications in other mathematical areas.

Stretched Brownian motion provides an analogue for the martingale version of this problem. We provide a characterization in terms of gradients of convex functions, similar to the characterization of optimizers in the classical transport problem for quadratic distance cost.

Based on joint work with Julio Backhoff-Veraguas, Walter Schachermayer and Bertram Tschiderer.

Motivated by Knightian uncertainty, S. Peng introduced his celebrated G–Brownian motion. Intuitively speaking, it corresponds to a dynamic worst case expectation in a model where volatility is uncertain but postulated to take values in a bounded interval. Natural extensions of the G–Brownian motion are nonlinear diffusions, whose volatility (and drift) takes values in a random set that is allowed to depend on the canonical process in a Markovian way. Nonlinear diffusions satisfy the dynamic programming principle, which entails the semigroup property of a corresponding family of sublinear operators. In this talk, we discuss regularity properties of these semigroups that allow us to relate them to evolution equations. In particular, we explain a novel type of smoothing property and a stochastic representation result for general sublinear semigroups with pointwise generators of Hamilton-Jacobi-Bellman type. Latter also implies a unique characterization theorem for such semigroups.

The talk is based on joint work with Lars Niemann (University of Freiburg).