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In this article, we consider the quasi-linear stochastic wave and heatequations on the real line and with an additive Gaussian noise which is whitein time and behaves in space like a fractional Brownian motion with Hurst index$H\in (0,1)$. The drift term is assumed to be globally Lipschitz. We prove thatthe solution of each of the above equations is continuous in terms of the index$H$, with respect to the convergence in law in the space of continuousfunctions.

SPDEs with fractional noise in space: continuity in law with respect to the Hurst index