Spatial rough path lifts of stochastic convolutions

We present sufficient conditions for finite controlled rho-variation of the covariance of Gaussian processes with stationary increments, based on concavity or convexity of their variance function. The motivation for this type of conditions comes from recent work of Hairer [CPAM,2011]. Our results allow to construct rough paths lifts of solutions to a class of fractional stochastic heat equations with additive, possibly colored Wiener noise with respect to their space variable.