1,343 research outputs found

    Quasilinear SPDEs via rough paths

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    We are interested in (uniformly) parabolic PDEs with a nonlinear dependance of the leading-order coefficients, driven by a rough right hand side. For simplicity, we consider a space-time periodic setting with a single spatial variable: \begin{equation*} \partial_2u -P( a(u)\partial_1^2u - \sigma(u)f ) =0 \end{equation*} where PP is the projection on mean-zero functions, and ff is a distribution and only controlled in the low regularity norm of Cα2 C^{\alpha-2} for α>23\alpha > \frac{2}{3} on the parabolic H\"older scale. The example we have in mind is a random forcing ff and our assumptions allow, for example, for an ff which is white in the time variable x2x_2 and only mildly coloured in the space variable x1x_1; any spatial covariance operator (1+1)λ1(1 + |\partial_1|)^{-\lambda_1 } with λ1>13\lambda_1 > \frac13 is admissible. On the deterministic side we obtain a CαC^\alpha-estimate for uu, assuming that we control products of the form v12vv\partial_1^2v and vfvf with vv solving the constant-coefficient equation 2va012v=f\partial_2 v-a_0\partial_1^2v=f. As a consequence, we obtain existence, uniqueness and stability with respect to (f,vf,v12v)(f, vf, v \partial_1^2v) of small space-time periodic solutions for small data. We then demonstrate how the required products can be bounded in the case of a random forcing ff using stochastic arguments. For this we extend the treatment of the singular product σ(u)f\sigma(u)f via a space-time version of Gubinelli's notion of controlled rough paths to the product a(u)12ua(u)\partial_1^2u, which has the same degree of singularity but is more nonlinear since the solution uu appears in both factors. The PDE ingredient mimics the (kernel-free) Krylov-Safanov approach to ordinary Schauder theory.Comment: 65 page

    Stochastic PDEs, Regularity Structures, and Interacting Particle Systems

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    These lecture notes grew out of a series of lectures given by the second named author in short courses in Toulouse, Matsumoto, and Darmstadt. The main aim is to explain some aspects of the theory of "Regularity structures" developed recently by Hairer in arXiv:1303.5113 . This theory gives a way to study well-posedness for a class of stochastic PDEs that could not be treated previously. Prominent examples include the KPZ equation as well as the dynamic Φ34\Phi^4_3 model. Such equations can be expanded into formal perturbative expansions. Roughly speaking the theory of regularity structures provides a way to truncate this expansion after finitely many terms and to solve a fixed point problem for the "remainder". The key ingredient is a new notion of "regularity" which is based on the terms of this expansion.Comment: Fixed typo

    Glauber dynamics of 2D Kac-Blume-Capel model and their stochastic PDE limits

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    We study the Glauber dynamics of a two dimensional Blume-Capel model (or dilute Ising model) with Kac potential parametrized by (β,θ)(\beta,\theta) - the "inverse temperature" and the "chemical potential". We prove that the locally averaged spin field rescales to the solution of the dynamical Φ4\Phi^4 equation near a curve in the (β,θ)(\beta,\theta) plane and to the solution of the dynamical Φ6\Phi^6 equation near one point on this curve. Our proof relies on a discrete implementation of Da Prato-Debussche method as in a result by Mourrat-Weber but an additional coupling argument is needed to show convergence of the linearized dynamics.Comment: 42 pages, 1 figur

    On the Use of Underspecified Data-Type Semantics for Type Safety in Low-Level Code

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    In recent projects on operating-system verification, C and C++ data types are often formalized using a semantics that does not fully specify the precise byte encoding of objects. It is well-known that such an underspecified data-type semantics can be used to detect certain kinds of type errors. In general, however, underspecified data-type semantics are unsound: they assign well-defined meaning to programs that have undefined behavior according to the C and C++ language standards. A precise characterization of the type-correctness properties that can be enforced with underspecified data-type semantics is still missing. In this paper, we identify strengths and weaknesses of underspecified data-type semantics for ensuring type safety of low-level systems code. We prove sufficient conditions to detect certain classes of type errors and, finally, identify a trade-off between the complexity of underspecified data-type semantics and their type-checking capabilities.Comment: In Proceedings SSV 2012, arXiv:1211.587

    Nuclear Masses in Astrophysics

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    Among all nuclear ground-state properties, atomic masses are highly specific for each particular combination of N and Z and the data obtained apply to a variety of physics topics. One of the most crucial questions to be addressed in mass spectrometry of unstable radionuclides is the one of understanding the processes of element formation in the Universe. To this end, accurate atomic mass values of a large number of exotic nuclei participating in nucleosynthesis are among the key input data in large-scale reaction network calculations. In this paper, a review on the latest achievements in mass spectrometry for nuclear astrophysics is given.Comment: Proceedings of the 10th Symposium on Nuclei in the Cosmos, NIC X - Mackinac Island, Michigan, USA (10 pages, 4 figures