An invariance principle for biased voter model interfaces

We consider one-dimensional biased voter models, where 1’s replace 0’s at a faster rate than the other way round, started in Heaviside initial state describing interface between two infinite populations of and 1’s. In limit weak bias, for diffusively rescaled process, we measure-valued process local fraction type 1 sites as function time. Under finite second moment condition on rates, show that diffusive scaling there is drifted Brownian path with property all but vanishingly small left (resp. right) this are 0 1). This extends known results unbiased models. Our proofs depend crucially recent about tightness