ncts

The field of well-posedness by noise investigates how the analytic properties of a deterministic equation can change after the introduction of a slight stochastic perturbation. More specifically, this field focuses on whether stochastic noise can render ill-posed equations well-posed. While the transformative impact of noise addition is a classical result for ordinary differential equations (ODEs), its exploration within the realm of partial differential equations (PDEs) is relatively recent and evolving rapidly. Some even argue that it could potentially have an impact on the Millennium Problem concerning the well-posedness of the Navier-Stokes equations. The first and most important contribution to the field was made by Flandolli, Gubinelli, and Priola in 2011. They showcased a dramatic transformation in the behaviour of solutions to the linear transport equation following the addition of simple Brownian motion, marking the first documented instance of stochastic noise rendering a PDE well-posed. This pioneering work spurred a wave of subsequent research, proving new results and extensions to various equations and relaxing regularity in underlying spaces. In this talk, we introduce the field of well-posedness by noise, highlight seminal results, and show that the well-posedness mechanism identified by Flandolli, Gubinelli, and Priola extends to a broader geometric framework applicable to a range of equations derived from conservation laws, including the transport and continuity equations, as well as fundamental equations in Magnetohydrodynamics.