It's possible to create the appearance of that kind of behavior, although you can't implement it by having the sketch continuously running on a server.  Processing.js sketches run on the client.  To make this happen, you would need to retrieve a file from the server that indicates the start time, and any other useful state information.  If you are making something simple like a clock that shows elapsed time from a fixed date, you don't need anything this elaborate.

Thanks jbum for your answer. Actually I was thinking in having like brushes in two different colors with random motion painting in the canvas... say fighting to complete the canvas with it's color. I thought it would never happen and the fight should goes for ever... But if two people got to the page at the same time they should see the same thing, like a server, with the brushes always there fighting... And if one got there and a short time later goes again he shoud see the continuation of the same drawing.  Not simple i guess... Thinking loud,  if i have a machine hooked to the internet and never dad I could run one applet indefinitely and make it feed a website with short pieces of the animation to be reproduced for who hits the page. Like server/client. But what I was looking for, is a way to "create the appearance of that kind of behavior" as you suggested. And not running it forever, unless is was on the server that is suposed to be 24/7 running. Any idea?