Assuming you need to do this on linux, you may run any php script from the browser and from the CLI as well.
You may run a simple php script:
<? echo "Ana are mere"; ?>
like this:
php -f ./index.php
Be careful about file-permissions, and any bug that may creep inside your code, memory leaks or unallocated variables will become VERY visible now, as the process will run continuously.
If you dont want it running in the background all the time, take a look at crontab (http://unixgeeks.org/security/newbie/unix/cron-1.html) to be able to start jobs regularly.
-- edit--
take a look at http://stackoverflow.com/questions/45953/php-execute-a-background-process and http://stackoverflow.com/questions/122834/php-how-to-return-information-to-a-waiting-script-and-continue-processing
Basically you want to start a background process, and you may do this by either using exec() or fsockopen() or a file_get_contents() on your own script probably in this order, if don't have access to exec, or socket functions.
Also take a look at http://us2.php.net/manual/en/function.session-write-close.php so the "background script" won't "block" the request and http://us2.php.net/manual/en/function.ignore-user-abort.php