Database mail uses a queue and Service Broker to handle the sending of mail, and it will activate when necessary.  It is normal to see the process that handles this stop when it has nothing to do.  Here's some queries to check:

USE msdb

-- see if enabled
SELECT * FROM sys.configurations
  WHERE name = 'Database Mail XPs'

-- all messages
SELECT * FROM dbo.sysmail_allitems
  WHERE send_request_date >= DATEADD(day, DATEDIFF(day, 0, GETDATE()), 0) --'2016-02-15'
  ORDER BY send_request_date DESC

-- messages not sent in past day
SELECT * FROM dbo.sysmail_allitems
  WHERE sent_date > GETDATE() - 1
    AND sent_status <> 'sent'
  ORDER BY send_request_date DESC

-- error log
SELECT i.mailitem_id, i.subject, i.send_request_date, l.log_date, i.body, i.file_attachments, l.description
  FROM dbo.sysmail_faileditems as i
    INNER JOIN dbo.sysmail_event_log AS l ON i.mailitem_id = l.mailitem_id
  WHERE i.last_mod_date > GETDATE() - 1
  ORDER BY i.send_request_date DESC, l.log_date DESC

-- unsent
SELECT * FROM dbo.sysmail_unsentitems

more detailed troubleshooting database mail:
https://technet.microsoft.com/en-us/library/ms189959(v=sql.105).aspx

to learn about database mail:
https://technet.microsoft.com/en-us/library/ms175887(v=sql.105).aspx