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We give necessary and sufficient conditions for the existence of weaksolutions of a parabolic problem corresponding to the Kolmogorov operatorsperturbed by a multipolar inverse square potential with respect to the Gaussianprobability measure which is the unique invariant measure forOrnstein-Uhlenbeck type operators. We state the optimality of the constant and,then, the nonexistence of positive exponentially bounded solutions to theparabolic problem.

Weighted Hardy inequalities and Ornstein-Uhlenbeck type operators perturbed by multipolar inverse square potentials