Geordie Richards (University of Guelph)
Title: On the long-time statistical behavior of solutions to some stochastic dispersive PDE
Abstract: In the analysis of nonlinear stochastic evolutionary partial differential equations (PDEs), the roughness of noise terms can make questions of well-posedness technically challenging. At the same time, the presence of noise can simplify the long-time statistical behavior of solutions by inducing a smoothing at the level of the corresponding Markov semigroup. In particular, such a smoothing can make it tractable to prove that the system possesses a unique ergodic invariant measure. While this picture is now fairly well understood for stochastic parabolic PDEs, much less is known in the dispersive setting. In this talk we will review some recent progress and preliminary results on the ergodic theory of stochastic perturbations of the Korteweg-de Vries equation (KdV), a widely studied canonical dispersive PDE. Roughly speaking, the subtlety of regularization inherent to nonlinear dispersive PDEs such as KdV motivates our generalization of the ergodic theory framework for stochastic PDEs to e.g., rely on probabilistic (rather than pathwise) control arguments, and to incorporate nonlinear estimates in function spaces better suited to dispersive equations. This talk will discuss joint works with Cameron Bechthold (Guelph), Nathan Glatt-Holtz (Indiana), and Vincent Martinez (CUNY Hunter).
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