The solution is to create a rule that will delete the messages for you. Below are the steps to do this using Outlook 2010, 2013, or 2016. Note: this applies to people using Outlook to connect to an Exchange server account. Other types of distribution lists may behave differently.

Using Outlook 2010/2013/2016:

1. Click the File tab.
2. Click Manage Rules & Alerts.
3. In the Rules and Alerts dialog, select the E-mail Rules tab and click the New Rule... button.
4. In the Rules Wizard dialog, select Apply rule on messages I receive and then click Next.
5. Check the from people or public group condition. In the lower box, click the people or public group link and then select your own name in the Rule Address dialog. Click OK to close the Rule Address dialog, and then click Next.
6. Check the following actions:
   • delete it
   • stop processing more rules
7. Click Next.
8. (Optional) If you routinely send yourself mail for other reasons, you may want to check the except if my name is in the To or Cc box exception.
9. Click Next.
10. In the last step, specify the name for your rule.
11. (Optional) If you want to delete existing emails already in your inbox, check Run this rule now on messages already in "Inbox".
12. Click Finish to close the Rules Wizard.
13. Click OK to close the Rules and Alerts dialog.