  We deal with an infinite horizon, infinite dimensional stochastic optimal
control problem arising in the study of economic growth in time-space. Such
problem has been the object of various papers in deterministic cases when the
possible presence of stochastic disturbances is ignored. Here we propose and
solve a stochastic generalization of such models where the stochastic term, in
line with the standard stochastic economic growth models, is a multiplicative
one, driven by a cylindrical Wiener process. The problem is studied using the
Dynamic Programming approach. We find an explicit solution of the associated
HJB equation and, using a verification type result, we prove that such solution
is the value function and we find the optimal feedback strategies. Finally we
use this result to study the asymptotic behavior of the optimal trajectories.
