Also JSession in the webbrowser doesn't gets chaged when the moving to another jvm but i could see some additional values getting appended along with JSession ID.
for eg:
one of the scenario this was my JSession ID
JSESSIONID=0001jjHtCtdqPCLrmNe_Y3RPP5H:-3003HJ
now if jvm goes down, my JSession ID is
JSESSIONID=0002jjHtCtdqPCLrmNe_Y3RPP5H:-3003HJ:1K4DR26OH3