Abstract: The KdV equation is known to be an infinitely dimensional completely integrable system. In finite dimension, if a system is completely integrable and is confined in a bounded domain, then it is known that the motion is quasi-periodic, and as a consequence the motion is recurrent. If the space dimension is infinite, this statement is not trivial. However, people believe that it would be true, namely solutions to the KdV equation starting from a certain class of ergodic processes have a property of recurrence. This conjecture has not been solved yet, but it seems to contain a new aspect of the KdV equation as well as of ergode theory. I will explain several problems behind.