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 partial derivative(t)u = Delta(x)u + b(u) + F(u) (W) over dot R+ x G, where G is a compact Lie group, Delta is the Laplace= Beltrami oper ator on G, b and F are Lipschitz coefficients, and where (W) over dot is a Gaussian space-correlated noise. which is white-noise in time. We find nece ssary and sufficient conditions on the space correlation of (W) over dot su ch that u is an L-2 or Holder-continuous function in the spatial variable x , using some basic tools of stochastic analysis and harmonic analysis on th e Lie group G. (C) 1999 Academic Press.