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=Δxu+b(u)+F(u) W on R+×G, where G is a compact Lie group, Δ is the Laplace–Beltrami operator on G, b and F are Lipschitz coefficients, and where W is a Gaussian space-correlated noise, which is white-noise in time. We find necessary and sufficient conditions on the space correlation of W such that u is an L2 or Hölder-continuous function in the spatial variable x, using some basic tools of stochastic analysis and harmonic analysis on the Lie group G.