  This article is concerned with the second order necessary conditions for the
stochastic optimal control problem of stochastic evolution equation with model
uncertainty when the traditional Pontryagin-type maximum principle holds
trivially and do not provide any information depicting the optimal control. The
diffusion term of the state equation is allowed to be control dependent with
convex control constraints. Transposition method is adopted to deal with the
adjoint operator-valued backward stochastic evolution equations, especially the
correction terms. Besides, weak convergence arguments are performed to obtain
the optimal uncertainty measure, among which the regularities of the state
processes, variational processes, and adjoint processes (in the transposition
sense) are carefully characterized. Malliavin calculus is applied to pave the
way for Lebesgue differentiation theorem to deduce the pointwise robust
optimality conditions.
