On space-time regularity for the stochastic heat equation on Lie groups

We consider the stochastic heat equations on Lie groups, that is, equations of the form ∂tu = ∆xu + b(u) + F (u)Ẇ on R+ × G, where G is a compact Lie group, ∆ is the Laplace-Beltrami operator on G, b and F are Lipschitz coefficients, and where Ẇ is a Gaussian space-correlated noise, which is white-noise in time. We find necessary and sufficient conditions on the space correlation of Ẇ such that u is an L or Hölder-continuous function in the spatial variable x, using some basic tools of stochastic analysis and harmonic analysis on the Lie group G.