On the maximal operator of a general Ornstein--Uhlenbeck semigroup

If Q is a real, symmetric and positive definite nxn matrix, and B a real nxn matrix whose eigenvalues have negative real parts, we consider the Ornstein--Uhlenbeck semigroup on R^n with covariance Q and drift matrix B. Our main result is that the associated maximal operator is of weak type (1,1) with respect to the invariant measure. The proof has a geometric gist and hinges on the ``forbidden zones method'' previously introduced by the third author. For large values of the time parameter, we also prove a refinement of this result, in the spirit of a conjecture due to Talagrand.