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End date: 2019 × Start date: 2010 × Type: Text × Publisher: Cambridge : Cambridge Univ. Press × WGL Institute: WIAS ×

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    Varieties of Signature Tensors
    (Cambridge : Cambridge Univ. Press, 2019) Améndola, Carlos; Friz, Peter; Sturmfels, Bernd
    The signature of a parametric curve is a sequence of tensors whose entries are iterated integrals. This construction is central to the theory of rough paths in stochastic analysis. It is examined here through the lens of algebraic geometry. We introduce varieties of signature tensors for both deterministic paths and random paths. For the former, we focus on piecewise linear paths, on polynomial paths, and on varieties derived from free nilpotent Lie groups. For the latter, we focus on Brownian motion and its mixtures.
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    On the regularity of SLE trace
    (Cambridge : Cambridge Univ. Press, 2017) Friz, Peter K.; Tran, Huy
    We revisit regularity of SLE trace, for all κ≠8, and establish Besov regularity under the usual half-space capacity parametrization. With an embedding theorem of Garsia–Rodemich–Rumsey type, we obtain finite moments (and hence almost surely) optimal variation regularity with index min(1+κ/8,2), improving on previous works of Werness, and also (optimal) Hölder regularity à la Johansson Viklund and Lawler.