  We study the persistence probability for processes with stationary
increments. Our results apply to a number of examples: sums of stationary
correlated random variables whose scaling limit is fractional Brownian motion,
random walks in random sceneries, random processes in Brownian scenery, and the
Matheron-de Marsily model in Z^2 with random orientations of the horizontal
layers. Using a new approach, strongly related to the study of the range, we
obtain an upper bound of optimal order in the general case and improved lower
bounds (compared to previous literature) for many processes.
