  Mean-field backward doubly stochastic differential equations (MF-BDSDEs, for
short) are introduced and studied. The existence and uniqueness of solutions
for MF-BDSDEs is established. One probabilistic interpretation for the
solutions to a class of nonlocal stochastic partial differential equations
(SPDEs, for short) is given. A Pontryagin's type maximum principle is
established for optimal control problem of MF-BDSDEs. Finally, one backward
linear quadratic problem of mean-field type is discussed to illustrate the
direct application of above maximum principle.
