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

A probabilistic approach to large time behaviour of mild solutions of Hamilton-Jacobi-Bellman equations in infinite dimension