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.
Dirr, N., & Souganidis, P. E. (2005). Large-time behavior for viscous and nonviscous Hamilton-Jacobi equations forced by additive noise. SIAM Journal on Mathematical Analysis (SIMA), 37(3), 777-796. https://doi.org/10.1137/040611896