Merge of two oppositely biased Wiener processes

We introduce a technique to merge two biased Brownian motions into a single regular process. The outcome follows a stochastic differential equation with a constant diffusion coefficient and a non-linear drift. The emerging stochastic process has outstanding properties, such as spatial and temporal translational invariance of its mean squared displacement, and can be efficiently simulated via a random walk with site-dependent one-step transition probabilities.