  We consider a linear parabolic problem with random elliptic operator in the
usual Gelfand triple setting. We do not assume uniform bounds on the coercivity
and boundedness constants, but allow them to be random variables. The parabolic
problem is studied in a weak space-time formulation, where we can derive
explicit formulas for the inf-sup constants. Under suitable assumptions we
prove existence of moments of the solution. We also prove quasi-optimal error
estimates for piecewise polynomial Petrov-Galerkin discretizations.
