How do I implement cookieless subdomain?
All these cookies are set from/belong to the main domain www.example.com;


Check your $cookie_domain value in settings.php 

/**
 * Drupal automatically generates a unique session cookie name for each site
 * based on its full domain name. If you have multiple domains pointing at the
 * same Drupal site, you can either redirect them all to a single domain (see
 * comment in .htaccess), or uncomment the line below and specify their shared
 * base domain. Doing so assures that users remain logged in as they cross
 * between your various domains. Make sure to always start the $cookie_domain
 * with a leading dot, as per RFC 2109.
 */
# $cookie_domain = '.example.com';


the leading '.' means the cookies also apply to subdomains. 

You could try something more specific, to restrict the drupal cookie domain, like:

$cookie_domain = '.www.example.com';