"An Auxiliary facility under orders and on Patrol observes a boat speeding in a Slow or No Wake Zone. Have we got the authority to speed up, and chase him down to tell him he is in violation? Do we have the authority to speed after and chase down any person?"

Speeding up, chasing down or any other action that is contrary to the law is a violation of both the law in question and the Operations Policy Manual.  Simply put; no you may not.  You should observe and report said infractions to your radio guard for further prosecution.

Per the Auxiliary Operations Policy Manual, Chapter 1, A.3.e. Law Enforcement Support:  Auxiliarists do not have and cannot be given general police powers or direct law enforcement authority. Coast Guard unit commanders may, however, assign Auxiliarists to missions that support Coast Guard law enforcement efforts (see Section E of Chapter 4).

Also see the Auxiliary Manual, COMDINST M16790.1 (series), Ch. 1, A.3.a.: In contrast to Coast Guard active duty personnel and Reserve military components, the Auxiliary is specifically declared by statute to be non-military. This definition puts the Auxiliary‘s role entirely within the Coast Guard‘s civil function responsibilities. The Auxiliary‘s role does not extend to any Coast Guard military or direct law enforcement missions assigned to active or Reserve forces.