Beschreibung
This book offers a rigorous yet accessible introduction to geometric methods for analyzing and predicting stochastic dynamics. Moving from individual trajectories to the infinite-dimensional space of probability distributions, it develops tools for understanding metastability, critical transitions, and tipping. Both Gaussian and non-Gaussian noise are considered, highlighting the role of jumps in shaping stochastic transitions.In deterministic systems, heteroclinic orbits (connecting orbits) form a geometric skeleton for global bifurcations and transitions between equilibria. Noise changes this picture: individual trajectories fluctuate, and connecting orbits must be understood probabilistically. The book develops a geometric perspective on stochastic transitions through most probable paths and evolving probability distributions, bringing together ideas from dynamical systems, variational analysis, and information geometry.Following a review of stochastic differential equations, the text develops three core frameworks:Most probable dynamics: transition paths between metastable states, obtained by minimizing the Onsager–Machlup action, with applications to rare-events and noise-induced transitions.Stochastic variational principles: stochastic Lagrangian mechanics, linking most probable transition paths to variational theories and clarifying their relationship to stochastic geometric mechanics.Schrödinger bridges and information geodesics: an introduction to local and nonlocal Otto calculus and information geometry, connecting optimal transport with stochastic dynamics through Schrödinger bridges.Throughout, geometric intuition accompanies mathematical rigor. The emphasis is on understanding how variational principles and probability-space geometry describe the structure and evolution of random systems.These ideas also connect to modern interdisciplinary science. Schrödinger bridges and transport geometry provide mathematical foundations for generative modeling, including diffusion models and flow matching. The same perspective supports the study of transitions in complex systems, from climate tipping to biological switching, and helps frame questions about prediction and intervention.Written as a textbook and reference, the book is intended for researchers and graduate students in applied mathematics, statistical mechanics, biophysics, systems biology, brain science, and engineering, as well as scientists in machine learning and artificial intelligence seeking a geometric foundation in stochastic dynamics.
