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In this paper, we construct unique, local-in-time strong solutions to theVlasov-Poisson (VP) and Vlasov-Poisson-Fokker-Planck (VPFP) systems subjectedto 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). Weadditionally show that solutions to the VPFP are instantly $C^\infty_{x,v}$ dueto hypoelliptic regularization if the external force fields are smooth. Theexternal forcing arises in the kinetic equation as a stochastic transport invelocity, which means, together with the anisotropy between $x$ and $v$ in thenonlinearity, that the local theory is a little more complicated thancomparable fluid mechanics equations subjected to either additive stochasticforcing or stochastic transport. Although stochastic electromagnetic fields areoften discussed in the plasma physics literature, to our knowledge, this is thefirst mathematical study of strong solutions to nonlinear stochastic kineticequations.

The Vlasov-Poisson and Vlasov-Poisson-Fokker-Planck systems in stochastic electromagnetic fields: local well-posedness