UC Santa Cruz

A bridge is a diffusion process that connects two given points in a finite dimensional vector space that are pinned at two given times. A Schrödinger bridge is a diffusion process that lifts this concept to the infinite dimensional manifold of probability measures. This dissertation contributes to the rapidly growing research area of Schrödinger bridge that is undergoing an explosion of mathematical and algorithmic developments across several disciplines: stochastic control, non-equilibrium statistical mechanics, and generative AI. This burgeoning interest is partly due to the fact that a Schrödinger bridge comes with a large deviation guarantee: it is the most-likely measure-valued path connecting the given endpoint measures. In this sense, the Schrödinger bridge is the most parsimonious model consistent with observed snapshots at two times. Part of its modern popularity is also because both the theory and algorithm for Schrödinger bridge are nonparametric, i.e., dispense the form or even the existence of finite dimensional sufficient statistic for the underlying measures. The computation is done directly on samples, without gridding or parameterizing the underlying state space. Starting from the classical Schrödinger bridge--a diffusive version of the optimal mass transport--this work considers several generalizations, and make progress in two fronts: contraction and reaction. The issue of contraction concerns with the convergent numerical solution in the generalized Schrödinger bridge problems. The standard approach for solving such problems is to deploy dynamic Sinkhorn recursions that are contractive with respect to Hilbert's projective metric. We quantify the worst-case contraction coefficient for the linear Schrödinger bridge in terms of the problem data. The issue of reaction concerns with the presence of an additive state cost in the objective that regularizes the controlled sample paths for all times, in addition to enforcing the exact steering between the endpoint statistics. Solving such problems via dynamic Sinkhorn recursions require Markov kernels associated with certain reaction-advection-diffusion partial differential equations, where the state costs play the role of the reaction rates. Such Markov kernels are not transition probability kernels since the reaction terms contribute to the non-conservation of probability mass, making direct application of dynamic Sinkhorn recursion challenging. For both classical and linear quadratic Schrödinger bridge problems, we overcome this challenge by deriving the associated Markov kernels in closed form, and showing how the dynamic Sinkhorn recursions with the derived kernels still apply. We show that the probabilistic Lambert problem--a problem of current interest in astrodynamics--is an instance of the generalized Schrödinger bridge where the state cost, and thus the reaction rate arises not from regularization, but from the nonlinear gravitational potential. This connects two hitherto disparate research areas, and enables nonparametric solution of the probabilistic Lambert problem. We conclude with several ongoing and future directions of research, including new equivalence between the Schrödinger bridge problems and the two point boundary value problems involving the quantum mechanical Schrödinger equation with generalized Bohm potential that we derive. Surprisingly, we show that such generalized Bohm potentials are necessarily complex-valued wherein the real part of the potential encodes elastic scattering (transmission of wave function), and the imaginary part encodes inelastic scattering (absorption of wave function).