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We study the two-dimensional wave equation with cubic nonlinearity posed on$\mathbb R^2$, with space-time white noise forcing. After a suitablerenormalisation of the nonlinearity, we prove global well-posedness for thisequation for initial data in $\mathcal H^s$, $s>\frac45$.

Global well-posedness of the two-dimensional stochastic nonlinear wave equation on an unbounded domain