Stochastic Dynamics for AI: Continuous-Time Methods Learning, Sampling, Generation, and Control

Continuous-time mathematics has become a common language across modern artificial intelligence. Yet the same words-gradient flow, diffusion, entropy, sampling, control, and equilibrium-often refer to different mathematical objects, metrics, and regimes. This book is built around making those distinctions precise. Stochastic Dynamics for AI develops deterministic and stochastic dynamics from first principles and then uses them to connect Fokker-Planck equations and Wasserstein gradient flows; diffusion and flow-based generative models; Langevin, Hamiltonian, and non-reversible sampling; particle and variational inference; stochastic-gradient dynamics and the edge of stability; entropy-regularized control and continuous-time reinforcement learning; mean-field limits and games; and path-space relative entropy and Schrödinger bridges. Rather than presenting these subjects as isolated toolkits, the book asks a harder question: when does an idea transfer from one domain to another? Each major connection is separated into what is exact, what requires a limiting argument or a change of representation, and what fails outside its stated assumptions. Scope boxes, counterexamples, comparison tables, and four tiers of exercises make the hypotheses-and the boundaries of the conclusions-part of the mathematics. Designed for advanced undergraduates, graduate students, and researchers in mathematics, statistics, machine learning, and related fields, this volume offers a rigorous guide to the continuous-time structures that underlie learning, generation, sampling, optimization, control, and interacting AI systems.