This thesis consists of four chapters. In Chapter 1, we focus on a class of transportation inequalities known as the transportation-information inequalities. These inequalities bound optimal transportation costs in terms of relative Fisher information, and are known to characterize certain concentration properties of Markov processes around their invariant measures. We provide a characterization of the quadratic transportation-information inequality in terms of a dimension-free concentration property for i.i.d. copies of the underlying Markov process, identifying the precise high-dimensional concentration property encoded by this inequality. We also illustrate how this result is an instance of a general convex-analytic tensorization principle.
In Chapter 2, we study the entropic gradient flow property of McKean--Vlasov diffusions via a stochastic analysis approach. We formulate a trajectorial version of the relative entropy dissipation identity for these interacting diffusions, which describes the rate of relative entropy dissipation along every path of the diffusive motion. As a first application, we obtain a new interpretation of the gradient flow structure for the granular media equation. Secondly, we show how the trajectorial approach leads to a new derivation of the HWBI inequality.
In Chapter 3, we further extend the trajectorial approach to a class of degenerate diffusion equations that includes the porous medium equation. These equations are posed on a bounded domain and are subject to no-flux boundary conditions, so that their corresponding probabilistic representations are stochastic differential equations with normal reflection on the boundary. Our stochastic analysis approach again leads to a new derivation of the Wasserstein gradient flow property for these nonlinear diffusions, as well as to a simple proof of the HWI inequality in the present context.
Finally, in Chapter 4, we turn our attention to mean field approximation -- a method widely used to study the behavior of large stochastic systems of interacting particles. We propose a new approach to deriving quantitative mean field approximations for any strongly log-concave probability measure. Our framework is inspired by the recent theory of nonlinear large deviations, for which we offer an efficient non-asymptotic perspective in log-concave settings based on functional inequalities. We discuss three implications, in the contexts of continuous Gibbs measures on large graphs, high-dimensional Bayesian linear regression, and the construction of decentralized near-optimizers in high-dimensional stochastic control problems.
| Identifer | oai:union.ndltd.org:columbia.edu/oai:academiccommons.columbia.edu:10.7916/wngk-vh75 |
| Date | January 2023 |
| Creators | Yeung, Lane Chun Lanston |
| Source Sets | Columbia University |
| Language | English |
| Detected Language | English |
| Type | Theses |
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