  In this paper, we construct unique, local-in-time strong solutions to the
Vlasov-Poisson (VP) and Vlasov-Poisson-Fokker-Planck (VPFP) systems subjected
to external, spatially regular, white-in-time electromagnetic fields in
$\mathbb T^d \times \mathbb R^d$. Initial conditions are taken $H^\sigma$ with
$\sigma > d/2 + 1$ (in addition to polynomial velocity weights). We
additionally show that solutions to the VPFP are instantly $C^\infty_{x,v}$ due
to hypoelliptic regularization if the external force fields are smooth. The
external forcing arises in the kinetic equation as a stochastic transport in
velocity, which means, together with the anisotropy between $x$ and $v$ in the
nonlinearity, that the local theory is a little more complicated than
comparable fluid mechanics equations subjected to either additive stochastic
forcing or stochastic transport. Although stochastic electromagnetic fields are
often discussed in the plasma physics literature, to our knowledge, this is the
first mathematical study of strong solutions to nonlinear stochastic kinetic
equations.
