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Infinite Dimensional Rough Dynamics

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Computation and Combinatorics in Dynamics, Stochastics and Control (Abelsymposium 2016)

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Abstract

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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Author information

Authors and Affiliations

  1. IAM and Hausdorff Center for Mathematics, Bonn, Germany

    Massimiliano Gubinelli

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  1. Massimiliano Gubinelli
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Correspondence to Massimiliano Gubinelli .

Editor information

Editors and Affiliations

  1. Department of Mathematical Sciences, Norwegian University of Science and Technology, Trondheim, Norway

    Elena Celledoni

  2. Department of Mathematics, University of Oslo, Oslo, Norway

    Giulia Di Nunno

  3. Department of Mathematical Sciences, Norwegian University of Science and Technology, Trondheim, Norway

    Kurusch Ebrahimi-Fard

  4. Department of Mathematics, University of Bergen, Bergen, Norway

    Hans Zanna Munthe-Kaas

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Gubinelli, M. (2018). Infinite Dimensional Rough Dynamics. In: Celledoni, E., Di Nunno, G., Ebrahimi-Fard, K., Munthe-Kaas, H. (eds) Computation and Combinatorics in Dynamics, Stochastics and Control. Abelsymposium 2016. Abel Symposia, vol 13. Springer, Cham. https://doi.org/10.1007/978-3-030-01593-0_14

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