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Explore the third lecture in a comprehensive series examining the physical and mathematical foundations of spontaneous stochasticity, delivered at the Centre International de Rencontres Mathématiques. Delve into outstanding problems and recent developments in both Lagrangian and Eulerian spontaneous stochasticity, focusing on statistical-mechanical analogies and the chaotic dynamical properties necessary to achieve universality. Examine how renormalization group methods can be applied to calculate spontaneous statistics in dynamics with scale symmetries, and consider the significant challenge of observing spontaneous stochasticity in laboratory experiments. Learn about the phenomenon where stochastic classical equations with vanishing randomness and singular dynamics can have limits described by probability measures over non-unique weak solutions of deterministic dynamics, often resulting in universal limiting probability measures independent of the specific sequence considered. Understand how this framework connects to Edward Lorenz's 1969 work on predictability of turbulent flows and the convex integration studies that revealed infinitely many non-unique admissible weak solutions for Euler equations, providing new insights into the mathematical structure of turbulent phenomena and their probabilistic descriptions.
