Diffusion limited aggregation (DLA) in $\mathbb{Z}^2$ is a
stochastic process first defined by Witten and Sander in order to study
aggregation systems governed by diffusive laws. In this talk, I will
construct an infinite stationary DLA on the upper half planar lattice.
The stationary DLA is ergodic with respect to integer left-right
translations. This is a joint work with Eviatar Procaccia and Yuan
Zhang.