Harmonic analysis, stochastics and PDEs
8-12 June 2020
- Zdzislaw Brzezniak, University of York
- Istvan Gyongy, University of Edinburgh
- Tadahiro Oh, University of Edinburgh
- Oana Pocovnicu, Heriot-Watt University
This five-day workshop is devoted to the areas of harmonic analysis, stochastic analysis and partial differential equations (PDEs). There will be a series of lectures from Franco Flandoli, Scuola Normale Superiore, Pisa, Massimiliano Gubinelli, Hausdorff Center for Mathematics, Bonn, Nicolai Krylov, University of Minnesota and Lutz Weis, Karlsruhe Institute for Technology. In addition, there will be 10 one-hour research presentations presented by a mixture of senior researchers and younger mathematicians in the relevant fields.
Harmonic analysis, stochastic analysis and PDEs are intimately related fields of great importance in mathematics. The relationship between harmonic analysis and PDE theory has a long and extremely successful history; harmonic analysis played a fundamental role in settling some of the central problems in the field of linear and nonlinear PDEs, including elliptic, parabolic, dispersive and hyperbolic equations and systems. On the other hand, recent advances in our theoretical understanding of PDEs has stimulated further development of analytical ideas and tools in harmonic analysis. For example, the interplay between the maximal inequalities in harmonic analysis and the martingale inequalities in stochastic analysis is well known and has played an important role in the developments of both disciplines. Moreover, methods and results in harmonic analysis have been playing an increasingly important role in the study of stochastic PDEs. For example, by the help of a generalisation of the Hardy-Littlewood inequality and by the Fefferman-Stein inequality, Krylov has recently built an theory of PDEs and SPDEs of parabolic type. Weis presented an alternative approach to the same question, based on infinite-dimensional harmonic analysis.
Harmonic analysis is important in the foundations of stochastic approach turbulence phenomena in fluids, understanding the very weak solutions and the phenomena of regularization by noise, in particular in the context of 3-d Navier-Stokes Equations.
Harmonic analysis has also played crucial roles in the recent spectacular progress in the theory of (stochastic) nonlinear dispersive and hyperbolic equations. We also mention a fundamental role of harmonic analysis in the recent theories of Hairer's regularity structures and Gubinelli's paracontrolled distributions. Over the last decade, there has been significant progress in the analysis of deterministic PDEs from non-deterministic points of view, in particular in dispersive and hyperbolic equations. This include probabilistic construction of solutions, invariant measures and transport properties of Gaussian measures under deterministic PDEs, where the latter extends a classical problem in probability theory to the PDE setting.
Details regarding participation will appear here in due course
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