Dependence of V in ergodic infinite time optimal control problem on cost function

Question

Consider the ergodic, infinite time optimal control problem: dx = [F(x) + G1 u]dt + G2 dW J = lim T->infinity E{ 1/T\int_0^T [Q(x) + u'Ru]dt}, F(0) = 0, Q(0) =0, Q(x) >= 0 Now suppose that Q(x) is replaced by Q1(x); Q1(0) = 0, Q1(x) >= Q(x). Then one would expect that V, and V1, the solutions of the HJB equation for the two problems would satisfy V1(x) >= V(x). Although in its LQG version the proof of this fact is trivial, I have been unable to prove it for the present more general case. Any help would be very much appreciated