No you can't connect USB3 ports to and external eSATA drive. Your best bet will be to go with a USB 3.0 external.
Also, make sure your external drive or array has good cooling. Often USB 3.0 ext. drives are designed for backups not video editing, and the later does push the hardware harder (and hotter!).
Thanks everyone for their replies. Will look into the sites you listed.
Please send the model number of your "new SONY L-SERIES DESKTOP".
Everything points to that series being obsolete
Does it have USB 3 port?
How much RAM
Check the date of the original post. That post is three-and-a-half years old now.