Universal non stationary dynamics at the depinning transition

We study the non-stationary dynamics of an elastic interface in a disordered medium at the depinning transition. We compute the two-time response and correlation functions, found to be universal and characterized by two independent critical exponents. We find a good agreement between two-loop Functional Renormalization Group calculations and molecular dynamics simulations for the scaling forms, and for the response aging exponent $\theta_R$. We also describe a dynamical dimensional crossover, observed at long times in the relaxation of a finite system. Our results are relevant for the non-steady driven dynamics of domain walls in ferromagnetic films and contact lines in wetting.