Structure-Preserving Dynamical Low-Rank Approximations for Stochastic Vlasov--Poisson Equations
We propose structure-preserving dynamical low-rank methods for stochastic Vlasov--Poisson equations with transport noise. We first derive the continuous low-rank evolution equations and show that, by including fixed velocity modes associated with the conserved quantities, the low-rank dynamics inherits the mass, momentum, and energy balance laws of the original stochastic model. We then develop two augmented basis-update Galerkin (BUG) integrators based on two stochastic discretizations: an Euler--Maruyama scheme applied to the equivalent It\^o formulation and a Heun scheme applied directly to the Stratonovich formulation. These two choices allow us to study how the stochastic time discretization interacts with the conservative low-rank framework. In both cases, basis augmentation and conservative rank truncation retain the relevant moment spaces and provide robustness under rank reduction. Numerical experiments demonstrate the conservation properties of the proposed methods and compare the augmentation requirements, stochastic correction terms, and momentum and energy behavior induced by the two stochastic formulations.