Large-Time Behavior for Viscous and Nonviscous Hamilton-Jacobi Equations Forced by Additive Noise

We study the large-time behavior of the solutions to viscous and nonviscous Hamilton--Jacobi equations with additive noise and periodic spatial dependence. Under general structural conditions on the Hamiltonian, we show the existence of unique up to constants, global-in-time solutions, which attract any other solution.

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