Infinite Dimensional Rough Dynamics
We review recent results about the analysis of controlled or stochastic differential systems via local expansions in the time variable. This point of view has its origin in Lyons’ theory of rough paths and has been vastly generalised in Hairer’s theory of regularity structures. Here our concern is to understand this local expansions when they feature genuinely infinite dimensional objects like distributions in the space variable. Our analysis starts reviewing the simple situation of linear controlled rough equations in finite dimensions, then we introduce unbounded operators in such linear equations by looking at linear rough transport equations. Loss of derivatives in the estimates requires the introduction of new ideas, specific to this infinite dimensional setting. Subsequently we discuss how the analysis can be extended to systems which are not intrinsically rough but for which local expansion allows to highlight other phenomena: in our case, regularisation by noise in linear transport. Finally we comment about other application of these ideas to fully-nonlinear conservations laws and other PDEs.
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