For non-linear random walks on the 2-torus, we study the existence and regularity of stationary measures.  For random walks close to certain (Zariski dense) affine random walks, we show the only non-atomic (ergodic) stationary measure is absolutely continuous.  One can then study the top Lyapunov exponent relative to this stationary measure.  Entropy considerations given an inequality between the  top Lyapunov exponent for the non-linear and affine random walks; equality holds only when the non-linear random walk is smoothly conjugate to an affine random walk.  

This is joint with Homin Lee, Davi Obata, and Yuping Ruan and with Yi Shi.