Stochastic logarithmic Schrodinger equations: energy regularized approach

In this paper, we prove the global existence and uniqueness of the solution of the stochastic logarithmic Schr\"odinger (SlogS) equation driven by additive noise or multiplicative noise. The key ingredient lies on the regularized stochastic logarithmic Schr\"odinger (RSlogS) equation with regularized energy and the strong convergence analysis of the solutions of (RSlogS) equations. In addition, temporal H\"older regularity estimates and uniform estimates in energy space $\mathbb H^1(\mathcal O)$ and weighted Sobolev space $L^2_{\alpha}(\mathcal O)$ of the solutions for both SlogS equation and RSlogS equation are also obtained.