Effects of random dispersion and driving noise on logarithmic Schrödinger equation via regularized approximation

In this paper, we study stochastic logarithmic Schrödinger (SLogS) equations subject to random dispersion and external driving noise. The singularity near zero density induced by the logarithmic nonlinearity is handled via an energy regularization method. Using the resulting regularized approximation, we establish well-posedness for the SLogS with white-noise dispersion driven by both additive and multiplicative noise. We further prove the stochastic translation invariance and gauge invariance for the SLogS equation and analyze the impact of random perturbations on standing waves. In addition, we develop a regularized Lie splitting approximation and derive strong error bounds that depend explicitly on the random dispersion coefficient, the deterministic dispersion coefficient, and the driving-noise amplitude. These results provide a unified analytical and numerical framework for SLogS with logarithmic nonlinearities under combined random effects.