We consider the effect of random inhomogeneities on the focusing singularity of the nonlinear Schrödinger equation in three dimensions, in the high frequency limit. After giving a phase space formulation of the high frequency limit using the Wigner distribution, we derive a nonlinear diffusion equation for the evolution of the wave energy density when random inhomogeneities are present. We show that this equation is linearly stable even in the case of a focusing nonlinearity provided that it is

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... not too strong. The linear stability condition is related to the variance identity for the nonlinear Schrödinger equation in an unexpected way. We carry out extensive numerical computations to get a better understanding of the interaction between the focusing nonlinearity and the randomness. In the focusing case β < 0 and with a negative energy H < 0, the solution cannot remain bounded for all time. More precisely, it follows from the variance identity and the uncertainty inequality that the L 2 norm of the gradient of the solution blows up in finite time [6] . Many other properties of NLS can be found in [17] .