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Abstract:
Random effects are ubiquitous in the natural sciences, from
quantum and fluid mechanics to biology and ecology. These fields provide a rich source of interesting and challenging problems which drive the development of theory at the intersection
of stochastic analysis and partial differential equations.
The heart of our session lies in the duality between `stochastic` and `deterministic` in the context of infinite-dimensional dynamical systems cast within a probabilistic framework. On
the one hand, stochasticity can have a regularizing effect. For instance, by randomizing the initial data, one can improve low regularity
well-posedness results for many dispersive equations, while in the context of dissipative systems such as the stochastically forced Navier-Stokes equations, one can establish the existence and uniqueness of ergodic, invariant measures through an emerging theory of hypo-ellipticity in infinite dimensions. On the other hand, as recent progress on the Karder-Parisi-Zhang equation has shown, stochastic versions of partial differential equations (PDEs)
can present unique, but arrestingly difficult challenges to their well-posedness or qualitative theory of solutions. Indeed, many analogues of classical questions are at the forefront of research in the stochastic setting.
This session promises to bring together a diverse group of leading researchers in probability, statistics, and the analysis of PDEs, particularly those working at their various interfaces.
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