Familysearch.org has his name as Jessram Walker born 1757 died Townshend, Vermont on 17 June 1813 at age 56.
FamilySearch also has Jesse Walker, Military Service 1 Jan 1777 Townshend, Windham, Vermont. 3 years, Private.
One of Daniel Hemenway's sons, Asa, was in the Battle of Bunker Hill and he is listed on page 700 of the MASSACHUSETTS SOLDIERS AND SAILORS. I wonder if your Mr. Marsh who died in the battle, would be on a similar list, but I don't know where to look for it