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DDC: 510 × Subject: Regularity structures ×

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    A rough path perspective on renormalization
    (Amsterdam [u.a.] : Elsevier, 2019) Bruned, Y.; Chevyrev, I.; Friz, P.K.; Preiß, R.
    We develop the algebraic theory of rough path translation. Particular attention is given to the case of branched rough paths, whose underlying algebraic structure (Connes-Kreimer, Grossman-Larson) makes it a useful model case of a regularity structure in the sense of Hairer. Pre-Lie structures are seen to play a fundamental rule which allow a direct understanding of the translated (i.e. renormalized) equation under consideration. This construction is also novel with regard to the algebraic renormalization theory for regularity structures due to Bruned–Hairer–Zambotti (2016), the links with which are discussed in detail. © 2019 The Author(s)
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    Precise Laplace asymptotics for singular stochastic PDEs: The case of 2D gPAM
    (Amsterdam [u.a.] : Elsevier, 2022) Friz, Peter K.; Klose, Tom
    We implement a Laplace method for the renormalised solution to the generalised 2D Parabolic Anderson Model (gPAM) driven by a small spatial white noise. Our work rests upon Hairer's theory of regularity structures which allows to generalise classical ideas of Azencott and Ben Arous on path space as well as Aida and Inahama and Kawabi on rough path space to the space of models. The technical cornerstone of our argument is a Taylor expansion of the solution in the noise intensity parameter: We prove precise bounds for its terms and the remainder and use them to estimate asymptotically irrevelant terms to arbitrary order. While most of our arguments are not specific to gPAM, we also outline how to adapt those that are.