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A Schr\"{o}dinger bridge establishes a dynamic transport map between two target distributions via a reference process, simultaneously solving an associated entropic optimal transport problem. We consider the setting where samples from the…

Machine Learning · Computer Science 2025-01-22 Stefano Peluchetti

The Skorokhod embedding problem is to represent a given probability as the distribution of Brownian motion at a chosen stopping time. Over the last 50 years this has become one of the important classical problems in probability theory and a…

Probability · Mathematics 2016-05-16 Mathias Beiglboeck , Alexander M. G. Cox , Martin Huesmann

In this paper, we show a large deviation principle for certain sequences of static Schr\"{o}dinger bridges, typically motivated by a scale-parameter decreasing towards zero, extending existing large deviation results to cover a wider range…

Probability · Mathematics 2025-06-23 Viktor Nilsson , Pierre Nyquist

In this work, we revisit the discrete-time Schr\"{o}dinger Bridge (SB) and Density Steering (DS) problems for Gaussian mixture model (GMM) boundary distributions. Building on the existing literature, we construct a set of feasible Markovian…

Systems and Control · Electrical Eng. & Systems 2026-04-02 George Rapakoulias , Fengjiao Liu , Panagiotis Tsiotras

We study the Schr\"odinger bridge problem when the endpoint distributions are available only through samples. Classical computational approaches estimate Schr\"odinger potentials via Sinkhorn iterations on empirical measures and then…

Machine Learning · Statistics 2026-02-10 Denis Belomestny , Alexey Naumov , Nikita Puchkin , Denis Suchkov

The Lambert problem originated in orbital mechanics. It concerns with determining the initial velocity for a boundary value problem involving the dynamical constraint due to gravitational potential with additional time horizon and endpoint…

Optimization and Control · Mathematics 2024-10-04 Alexis M. H. Teter , Iman Nodozi , Abhishek Halder

Fish migration is a mass movement that affects the hydrosphere and ecosystems. While it occurs on multiple temporal scales, including daily and intraday fluctuations, the latter remains less studied. In this study, for a stochastic…

Optimization and Control · Mathematics 2026-03-27 H. Yoshioka

We study the problem of stopping a Brownian motion at a given distribution $\nu$ while optimizing a reward function that depends on the (possibly randomized) stopping time and the Brownian motion. Our first result establishes that the set…

Probability · Mathematics 2020-04-15 Mathias Beiglböck , Marcel Nutz , Florian Stebegg

Monge-Kantorovich optimal mass transport (OMT) provides a blueprint for geometries in the space of positive densities -- it quantifies the cost of transporting a mass distribution into another. In particular, it provides natural options for…

Optimization and Control · Mathematics 2015-06-16 Yongxin Chen , Tryphon T. Georgiou , Michele Pavon

The martingale optimal transport aims to optimally transfer a probability measure to another along the class of martingales. This problem is mainly motivated by the robust superhedging of exotic derivatives in financial mathematics, which…

Probability · Mathematics 2016-08-04 Gaoyue Guo , Xiaolu Tan , Nizar Touzi

A Schr\"odinger bridge is the most probable time-dependent probability distribution that connects an initial probability distribution $w_{i}$ to a final one $w_{f}$. The problem has been solved and widely used for the case of simple…

Statistical Mechanics · Physics 2025-07-02 Henri Orland

We show that the minimum effort control of colloidal self-assembly can be naturally formulated in the order-parameter space as a generalized Schr\"{o}dinger bridge problem -- a class of fixed-horizon stochastic optimal control problems that…

Optimization and Control · Mathematics 2023-10-17 Iman Nodozi , Charlie Yan , Mira Khare , Abhishek Halder , Ali Mesbah

Over the last several years, there has been significant progress in developing neural solvers for the Schr\"odinger Bridge (SB) problem and applying them to generative modelling. This new research field is justifiably fruitful as it is…

In this article we investigate entropic interpolations. These measure valued curves describe the optimal solutions of the Schr{\"o}dinger problem [Sch31], which is the problem of finding the most likely evolution of a system of independent…

Analysis of PDEs · Mathematics 2021-07-09 Gauthier Clerc , Giovanni Conforti , Ivan Gentil

In 1931/32, Schroedinger studied a hot gas Gedankenexperiment, an instance of large deviations of the empirical distribution and an early example of the so-called maximum entropy inference method. This so-called Schroedinger bridge problem…

Optimization and Control · Mathematics 2020-11-30 Yongxin Chen , Tryphon T. Georgiou , Michele Pavon

We study the convergence of an $N$-particle Markovian controlled system to the solution of a family of stochastic McKean-Vlasov control problems, either with a finite horizon or Schr\"odinger type cost functional. Specifically, under…

Probability · Mathematics 2024-05-22 Francesco C. De Vecchi , Chiara Rigoni

In this paper, we investigate the multi-marginal Schrodinger bridge (MSB) problem whose marginal constraints are marginal distributions of a stochastic differential equation (SDE) with a constant diffusion coefficient, and with time…

Probability · Mathematics 2025-07-15 Rentian Yao , Young--Heon Kim , Geoffrey Schiebinger

In a two-period financial market where a stock is traded dynamically and European options at maturity are traded statically, we study the so-called martingale Schr\"odinger bridge Q*; that is, the minimal-entropy martingale measure among…

Mathematical Finance · Quantitative Finance 2022-04-27 Marcel Nutz , Johannes Wiesel , Long Zhao

Transporting between arbitrary distributions is a fundamental goal in generative modeling. Recently proposed diffusion bridge models provide a potential solution, but they rely on a joint distribution that is difficult to obtain in…

Machine Learning · Computer Science 2025-03-03 Jun Hyeong Kim , Seonghwan Kim , Seokhyun Moon , Hyeongwoo Kim , Jeheon Woo , Woo Youn Kim

For a fixed flow-based generative model under a small inference budget, sample quality can depend strongly on where the sampler spends its few function evaluations. Flow matching and Schr\"odinger bridges define probability paths, yet their…

Machine Learning · Computer Science 2026-05-18 Bruno Trentini , Dejan Stancevic , Michael M. Bronstein , Alexander Tong , Luca Ambrogioni