Stretched exponential relaxation in the biased random voter model

We study the relaxation properties of the voter model with i.i.d, random bias. We prove under mild conditions that the disorder-averaged relaxation of this biased random voter model is faster than a stretched exponential with exponent d/(d + alpha), where 0 < alpha less than or equal to 2 depends on the transition rates of the non-biased Voter model. Under an additional assumption, we show that the above upper bound is optimal. The main ingredient of our proof is a result of Donsker and Varadhan.