  In this paper we investigate mean-field backward doubly stochastic
differential equations (BDSDEs), i.e., BDSDEs whose driving coefficients also
depend on the joint law of the solution process as well as the solution of an
associated mean-field forward SDE. Unlike the pioneering paper on BDSDEs by
Pardoux-Peng (1994), we handle a driving coefficient in the backward integral
of the BDSDE for which the Lipschitz assumption w.r.t. the law of the solution
is sufficient, without assuming that this Lipschitz constant is small enough.
On the other hand, as the parameters $(x,P_\xi)$ and $(x,P_\xi,y)$ run an
infinite-dimensional space, unlike Pardoux and Peng, we cannot apply
Kolmogorov's continuity criterion to the value function
$V(t,x,P_{\xi}):=Y_t^{t,x,P_{\xi}}$, while in the classical case studied in
Pardoux-Peng the value function $V(t,x)=Y_t^{t,x}$ can be shown to be of class
$C^{1,2}([0,T]\times\mathbb{R}^d)$, we have for our value function
$V(t,x,P_{\xi})$ and its derivative $\partial_\mu V(t,x,P_{\xi},y)$ only the
$L^2$-differentiability with respect to $x$ and $y$, respectively. Using a new
method we prove the characterization of $V=(V(t,x,P_{\xi}))$ as the unique
solution of the associated mean-field backward stochastic PDE.
