It is doubtful that this document is apart of a pension package for Joseph Brown. I did not find a pension for Joseph Brown in the Old War Pension lists of 1818 for those soldiers who were wounded or to the heirs of the soldiers who were killed during the War of 1812.
Likewise, since the document is dated in 1855 then it is not apart of a service pension application which were first issued in 1872. Still, it maybe a special pension issued by Congress.
Most likely this affivadit is apart of the bounty land application for Mary Brown. Bounty land were issued to War of 1812 militia veterans or their widows beginning in 1855.
You will need to fill out NAFT Form 85 (Military Pension/Bounty Land Warrant Applications) from the National Archives and request the bounty land warrant application. There may be other documents in this package that will be of use to you.