Space regularity of stochastic heat equations driven by irregular Gaussian processes

We study linear stochastic evolution equations driven by various innite-dimension al Gaussian processes, some of which are more irregular in time than fractional Brownian motion (fBm) with any Hurst parameter H, while others are comparable to fBm with H < 1 . Sharp necessary and su- cient conditions for the existence and uniqueness of solutions are presented. Specializing to stochastic heat equations on compact manifolds, especially on the unit circle, sharp Gaussian regularity results are used to determine suf- cien t conditions for a given xed function to be an almost-sure modulus of continuity for the solution in space; these sucien t conditions are also proved necessary in highly irregular cases, and are nearly necessary (logarithmic cor- rections are given) in other cases, including the Holder scale.