1,343 research outputs found

### Quasilinear SPDEs via rough paths

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 $P$ is the projection on mean-zero functions, and $f$
is a distribution and only controlled in the low regularity norm of $C^{\alpha-2}$ for $\alpha > \frac{2}{3}$ on the parabolic H\"older scale.
The example we have in mind is a random forcing $f$ and our assumptions
allow, for example, for an $f$ which is white in the time variable $x_2$ and
only mildly coloured in the space variable $x_1$; any spatial covariance
operator $(1 + |\partial_1|)^{-\lambda_1 }$ with $\lambda_1 > \frac13$ is
admissible.
On the deterministic side we obtain a $C^\alpha$-estimate for $u$, assuming
that we control products of the form $v\partial_1^2v$ and $vf$ with $v$ solving
the constant-coefficient equation $\partial_2 v-a_0\partial_1^2v=f$. As a
consequence, we obtain existence, uniqueness and stability with respect to $(f,
vf, v \partial_1^2v)$ 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 $f$ using stochastic arguments.
For this we extend the treatment of the singular product $\sigma(u)f$ via a
space-time version of Gubinelli's notion of controlled rough paths to the
product $a(u)\partial_1^2u$, which has the same degree of singularity but is
more nonlinear since the solution $u$ 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

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
$\Phi^4_3$ 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

We study the Glauber dynamics of a two dimensional Blume-Capel model (or
dilute Ising model) with Kac potential parametrized by $(\beta,\theta)$ - the
"inverse temperature" and the "chemical potential". We prove that the locally
averaged spin field rescales to the solution of the dynamical $\Phi^4$ equation
near a curve in the $(\beta,\theta)$ plane and to the solution of the dynamical
$\Phi^6$ 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

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

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

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