According to the homogenization principle, solutions to Hamilton-Jacobi equations with certain ergodic stationary Hamiltonian functions can be approximated by solutions to deterministic homogenized Hamilton-Jacobi equations after a linear scaling of the space of time.
Central Limit Theorem (CLT) addresses the nature of the fluctuations of such an approximation and Large Deviation Principle (LDP) provides some kind of Gibbsian description for the probability of a large deviation from solutions to the homogenized problems. Both CLT and LDP have been understood for a limited number of examples for a discrete variation of stochastic Hamilton-Jacobi equations. I give an overview of some of the known results and discuss some conjectures in this talk.
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