I wrote a processing.js script directly into my website.
Say what? Processing does not run javascripts and it does not run "in a website".
Is there like a javascript library for the Arduino that I can import in my website for this to work?
No.
If your javascript can write to the serial port that the Arduino is connected to (very unlikely unless your desktop/laptop is what is hosting the web site), then it could send information to the Arduino, which could do something based on the information.
PaulS:
Say what? Processing does not run javascripts and it does not run "in a website".
Processing.js does work when wrote directly into the website. Mine is working and it is playing music on my website, but the problem is that I want it to connect my Arduino at the same time.
PS: proxy permit to not acces directly device from internet but can exchanghe results for queries as relay,broker,intermediate,dispatcher...
The port must be forwarding or can use ngrok