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Explore the physical and mathematical foundations of spontaneous stochasticity in this comprehensive lecture from the Centre International de Rencontres Mathématiques. Delve into how stochastic classical equations with randomness in their initial data can converge to probability measures over non-unique weak solutions of limiting deterministic dynamics as randomness vanishes and dynamics become singular. Discover the universal nature of these limiting probability measures and their role as well-posed solutions to Cauchy problems for deterministic dynamics. Focus specifically on Lagrangian spontaneous stochasticity, tracing its origins to Lewis Fry Richardson's 1926 work on turbulent 2-particle dispersion and examining how Krzysztof Gawędzki and collaborators in 1997 demonstrated its necessity for anomalous dissipation of scalars advected by turbulent fluid flows. Gain insights into this fundamental phenomenon that bridges stochastic and deterministic approaches in fluid dynamics and turbulence theory. This lecture forms part of a three-part series recorded during the thematic meeting "Physics and Mathematics of hydrodynamic and wave turbulence" at CIRM in Marseille, France.
