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The two-dimensional parabolic Anderson model is the statistical mechanics model with Hamiltonian described by the two-dimensional random walk in random scenery on lattice. The particles gain energy whenever they visit the potential sites. The analogous continuum model, namely, the model with noise formally defined on $R^2$, is not well-defined. Instead, we consider that the particles only gain energy at their first visit. In the continuum and weak disorder regime, the partition function of our model as a random variable converges weakly to a Wiener Chaos expansion. This may solve the two-dimensional continuum parabolic Anderson equation.