In this talk, we will discuss three aspects of SPDEs:
• A Framework for Nonlinear Singular SPDEs: As the Gelfand-triple argument for local monotone SPDEs turns out to be inadequate in the singular cases, we propose a new framework based on proper regularization. Some recent results in the literature are
extended in a cohesive and unified manner.
• Pseudo-differential Noise: To study/model intricate non-local random interactions, we extend the transport type noise by replacing the classical gradient operator with pseudodifferential operators. We establish two new cancellation properties, which play key roles
in proving existence/uniqueness of solutions.
• Stability of Exiting Time: For certain stochastic fluid models, we demonstrate that (small) multiplicative noise does not simultaneously enhance the stability of the exiting time and the continuity of initial-data-dependence.