SPDEs with rough noise in space: Hölder continuity of the solution

We consider the stochastic wave and heat equations with affine multiplicative Gaussian noise which is white in time and behaves in space like the fractional Brownian motion with index $H \in (\frac14,\frac12)$. The existence and uniqueness of the solution to these equations has been proved recently by the authors. In the present note we show that these solutions have modifications which are Hölder continuous in space of order smaller than $H$, and Hölder continuous in time of order smaller than $γ$, where $γ=H$ for the wave equation and $γ=H/2$ for the heat equation.

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