Structure-Preserving Dynamical Low-Rank Approximations for Stochastic Vlasov--Poisson Equations

We develop structure-preserving dynamical low-rank methods for stochastic Vlasov–Poisson equations with transport noise. We first derive continuous low-rank evolution equations that inherit the mass, momentum, and energy balance laws through fixed velocity modes. We then construct two augmented basis-update Galerkin (BUG) integrators based on Euler–Maruyama discretization of the It^o formulation and Heun-type discretization of the Stratonovich formulation. Both integrators combine standard basis augmentation with conservative rank truncation to preserve the selected velocity moments. We prove pathwise mass conservation and conservation of total momentum in expectation. The energy balances contain distinct stochastic corrections with identical conditional expectations and an explicitly characterized time-discretization defect. Numerical experiments illustrate the conservation properties, rank adaptation, and temporal convergence behavior of the proposed methods.