  We study the large time behaviour of mild solutions of HJB equations in
infinite dimension by a purely probabilistic approach. For that purpose, we
show that the solution of a BSDE in finite horizon $T$ taken at initial time
behaves like a linear term in $T$ shifted with the solution of the associated
EBSDE taken at initial time. Moreover we give an explicit speed of convergence,
which seems to appear very rarely in literature.
