Optimal Transport (OT) is the problem of transporting one distribution or shape of mass to another as efficiently as possible, relative to a given cost of transport. For two sets of discrete data, the problem is related to that of optimal matching between the two; think about matching students to colleges, or patients to hospitals in order to minimize some overall cost. This simply stated problem is pervasive in many areas; for instance, when data needs to be matched to models in a way that respects the geometry of the underlying space, as is often the case in data science and econometrics. OT is also important in models of continuous time evolution of cells, governed by the underlying geometry of the Waddington landscape. This is where we see PDEs and developmental biology intersect. Key areas for connections to optimal transport are machine learning, data science and statistics, generative models in AI, economics, biology, and importantly, intersections of these. On the other hand, many of these problems lead to new problems in mathematics itself and drive the subject into new theoretical horizons.

Our focus in this workshop is twofold: one is the increasing intersection between stochastic processes and OT, and the other is the problem of Wasserstein projections. Let us describe what these problems are, how they are related, and how this unique choice of objective puts our workshop apart from other workshops and conferences in OT and related areas.

(a) Stochastic transport. OT is related to stochastic processes in a multitude of ways. The general idea being that instead of thinking of transport/matching as an immediate transfer of one probability measure to another, one may trace various paths of this transfer and minimize some energy function over these paths. These optimal paths over probability measures are solutions to PDEswhichare related to various diffusion models. An example of this is Brownian motion which is related to the heat equation. The solution of the latter may be interpreted as the most efficient way to increase entropy over probability measures equipped with the 2-Wasserstein metric. This idea of relating stochastic processes as optimal paths in the geometry of the Wasserstein space has had an enormous impact. On the one hand one may analyze convergence rates of diffusion via tools from optimization. Furthermore, one may interpret various processes arising from applications, such as training evolution in neural networks, cell developments over time, and behaviors of algorithms over iterations. A particularly important process in application is called the (dynamic) Schr¨odinger bridge which has gained prominence in entropic regularization of OT which is used in sampling and statistical estimation of OT.

More recently, a new direction in optimal transport has emerged within a dynamic framework, driven by the need to account for the flow of information. Crucially, the resulting causal and adapted transport distances respect the intrinsic temporal structure of stochastic processes, ensuring that decisions are based solely on available information. This leads to the concept of Wasserstein space of stochastic processes, which enjoys geometric properties analogous to those of the usual Wasserstein space of probability measures. These concepts have already found a range of applications in robust finance, stochastic optimization, risk management, and optimal control, where sequential decision-making is essential; see e.g. [4, 1, 2]. In fact, the adapted counterpart of the Wasserstein distance proved to be a robust distance for a wide range of dynamic optimization problems, in the sense that two models which are close in this distance do perform similarly when facing all those problems; see [5]. Importantly, such distance metrizes a topology which has emerged from several other areas of mathematics (from Aldous’ extended weak topology to Hellwig’s information topology); see [6, 7]. Considering the wide scope of application of classical transport in a static setting, this line of transport theory has the potential for groundbreaking applications in a dynamic stochastic context.

(b) Wasserstein projections. Imagine there is a class of probability models and given our data we would like to find the model that fits best with our data. This is a problem of Wasserstein projection. We are given a subset of the Wasserstein space of probability measures, and given another probability, we wish to find the closest point in that subset measured in 2-Wasserstein distance. This kind of problems are ubiquitous in applications where robustness or model uncertainty is prominent. It may seem like a distinct problem from topic (a) stochastics and OT but recentresults have shown an intriguing connection between the two that merits further investigation. An example is in a paper by Fathi, Gozlan and Prodhomme [8] where the authors show the following connection between Caffarelli contraction theorem, entropic regularization and Wasserstein projections. Caffarelli’s contraction theorem gives a sufficient condition when the OT map is Lipschitz. This result plays an important role in the Functional Inequality literature, as it enables to transfer geometric inequalities such as Log-Sobolev or Gaussian Isoperimetric inequalities from the Gaussian measure to probability measures with a uniformly log-concave density. Fathi et al shows that Caffarelli’s theorem follows as a corollary of a certain property of Wasserstein projections on the cone generated by convex stochastic ordering. Moreover, their proof critically depends on the Sinkhorn algorithm which is used to compute entropy-regularized OT.

Stochastic ordering between probability distributions is a classical topic in economics, finance, risk management, and probability. Stochastic orders provide preferences over sets of distributions as well as their (possible) causal relations, including time ordering of marginals of stochastic processes. A few recent works [3, 9] considered statistical problems about such ordering, where Wasserstein projection to the cone of convex orders has been crucially used, demonstrating versatility of such projection methods.

Here we see the emergence of the theme of our workshop. Both areas, stochastics and OT and Wasserstein projections, are emergent theories with a lot of possible applications. But they also seem to intersect in interesting ways that are not fully understood. The key novelty of this workshop is to bring these mostly distinct communities together in one workshop, to facilitate the perfect platform for exchanges, starting new collaborations, including early career researchers and graduate students, which will lead to breakthroughs in classical areas of probability, PDE and their modern applications.

One possible area of impact that we are particularly excited about is model uncertainty. Model uncertainty has a long history with a multitude of applications in finance, economics, and decision making. OT, and in particular the 2-Wasserstein case, has been extensively used to model distributional robustness in optimization problems and decision making. Recently, new asymmetric OT divergences such as the Bregman-Wasserstein divergence have been introduced to allow to quantify uncertainty between distributions in an asymmetric fashion. Indeed asymmetry is desirable when decisions are based on monetary outcomes. Furthermore, recent advances in stochastic OT allow for the modeling of uncertainty over time. Finally, stochastic ordering is of central importance in economic decision making, thus also for quantifying decisions that perform best in the worst case, a classical distributional robust approach.