The Generic Chaining: Upper and Lower Bounds for Stochastic Processes

The fundamental question of characterizing continuity and boundedness of Gaussian processes goes back to Kolmogorov. After essential contributions by R. Dudley and X. Fernique, it was solved by the author in 1985. This advance was followed by a great improvement of our understanding of the boundedness of other fundamental classes of processes (empirical processes, infinitely divisible processes, etc.) This challenging body of work has now been considerably simplified through the notion of "generic chaining", a completely natural variation on the ideas of Kolmogorov. The entirely new presentation adopted here takes the reader from the first principles to the edge of current knowledge, and to the wonderful open problems that remain in this domain.