Contraction and Reaction in Generalized Schrödinger Bridges

By Alexis M.H. Teter

This book marks a milestone in the theory and algorithms of the Schr dinger bridge and its generalizations--a topic of significant current interest across statistical mechanics, stochastic control theory, and generative AI. The topic had its genesis in physics, specifically in 1931 when Erwin Schr dinger conceived a thought experiment to interpret quantum mechanics through the lens of non-equilibrium Markov processes. Schr dinger's idea was much ahead of its time: neither the axiomatic theory of stochastic processes, nor the mathematics of large deviations principle were available then. The connections with stochastic control became clear only in the late twentieth century. In the current time, the development of diffusion models in generative AI have re-energized this area via vigorous interdisciplinary research. Simply put, Schr dinger bridge is a diffusion process that interpolates probability distributions observed at given times with maximum likelihood guarantee. This book contributes to two significant generalizations of the Schr dinger bridge: one where prior knowledge in underlying physics is incorporated as some prescribed drift and diffusion coefficients, the other where additional regularization is enforced on the paths of the diffusion process. For the former, this book establishes contraction guarantees for convergence of numerical algorithms. For the latter, this book shows how regularization manifests as reaction rate and derives corresponding Markov kernels for nontrivial models of practical interest. New points of contact with quantum mechanics, in the spirit of Schr dinger's original work, are also uncovered.

• Format: Hardcover
• Language: English
• ISBN: 3032372135
• ISBN13: 9783032372130
• Release Date: October 2026
• Publisher: Springer
• Length: 192 Pages