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Title:
Spontaneous stochasticity and renormalization group in discrete multi-scale dynamics
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Abstract:
We study an initial value problem for a class of ideal scale-invariant systems with discrete time, by introducing a regularization and noise at small scales. Such systems are motivated by scale-invariant physical models, like equations for an ideal fluid regularized by viscosity and microscopic fluctuations. We propose a qualitative theory describing the vanishing regularization (inviscid) limit in this problem as an attractor of the renormalization group (RG) operator. The RG operator reflects symmetries and interactions of the ideal scale-invariant system. It acts on evolution maps of regularized systems, therefore, different regularizations provide different initial conditions for the RG dynamics. A fixed-point RG attractor defines a unique dynamics of the ideal system, akin to shock solutions in viscous conservation laws. Otherwise, a chaotic attractor defines a Markov kernel of spontaneously stochastic solutions. These are intrinsically probabilistic solutions solving deterministic equations of the ideal system with deterministic initial conditions. The results are illustrated with solvable models. This is a joint work with Artem Raibekas.
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