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This paper is aimed at proving homogenization for viscous Hamilton-Jacobiequations in dimension one in stationary ergodic environments with aquasiconvex and superlinear Hamiltonian of fairly general type. The effectiveHamiltonian is furthermore shown to be quasiconvex. This latter result is neweven in the periodic setting, despite homogenization has been known for quitesome time. Unfortunately, the proof of one of our key theorems contains a flaw,so this contribution can be regarded as an attempt. The homogenization resultsdescribed above hold when the diffusion coefficient vanishes at some points butis not identically zero, as we have recently proved in [12].

Stochastic homogenization of quasiconvex viscous HJ equations in 1d