3,825 research outputs found
Nonnegatively curved 5-manifolds with non-abelian symmetry
We classify compact simply-connected 5-dimensional manifolds which admit a
metric of nonnegative curvature with a connected non-abelian group acting by
isometries. We show that they are diffeomorphic to either S^5, S^3 x S^2, the
nontrivial S^3-bundle over S^2 or the Wu-manifold, SU(3)/SO(3). This result is
a consequence of our equivariant classification of all SO(3) and SU(2)-actions
on compact simply-connected 5-manifolds. In the case of positive curvature we
obtain a partial classification.Comment: 22 page
Principal Components Analysis for Semimartingales and Stochastic PDE
In this work, we develop a novel principal component analysis (PCA) for
semimartingales by introducing a suitable spectral analysis for the quadratic
variation operator. Motivated by high-dimensional complex systems typically
found in interest rate markets, we investigate correlation in high-dimensional
high-frequency data generated by continuous semimartingales. In contrast to the
traditional PCA methodology, the directions of large variations are not
deterministic, but rather they are bounded variation adapted processes which
maximize quadratic variation almost surely. This allows us to reduce
dimensionality from high-dimensional semimartingale systems in terms of
quadratic covariation rather than the usual covariance concept.
The proposed methodology allows us to investigate space-time data driven by
multi-dimensional latent semimartingale state processes. The theory is applied
to discretely-observed stochastic PDEs which admit finite-dimensional
realizations. In particular, we provide consistent estimators for
finite-dimensional invariant manifolds for Heath-Jarrow-Morton models. More
importantly, components of the invariant manifold associated to volatility and
drift dynamics are consistently estimated and identified. The proposed
methodology is illustrated with both simulated and real data sets.Comment: 54 page
A Maximal Inequality of the 2D Young Integral based on Bivariations
In this note, we establish a novel maximal inequality of the 2D Young
integral in terms of the -bivariation norms of
the section functions and where
is a controlled path satisfying
finite -variation conditions. The proof is reminiscent from the Young's
original ideas \cite{young1} in defining two-parameter integrals in terms of
-finite bivariations. Our result complements the standard maximal
inequality established by Towghi \cite{towghi1} in terms of joint variations.
We apply the maximal inequality to get novel strong approximations for 2D Young
integrals w.r.t the Brownian local time in terms of number of upcrossings of a
given approximating random walk.Comment: Some typos are correcte
Uniform approximation of the heat kernel on a manifold
We approximate the heat kernel on a compact connected Riemannian
manifold without boundary uniformly in ,
, by -fold integrals over of the densities of Brownian bridges.
Moreover, we provide an estimate for the uniform convergence rate. As an
immediate corollary, we get a uniform approximation of solutions of the Cauchy
problem for the heat equation on .Comment: Minor correction
Distance Closures on Complex Networks
To expand the toolbox available to network science, we study the isomorphism
between distance and Fuzzy (proximity or strength) graphs. Distinct transitive
closures in Fuzzy graphs lead to closures of their isomorphic distance graphs
with widely different structural properties. For instance, the All Pairs
Shortest Paths (APSP) problem, based on the Dijkstra algorithm, is equivalent
to a metric closure, which is only one of the possible ways to calculate
shortest paths. Understanding and mapping this isomorphism is necessary to
analyse models of complex networks based on weighted graphs. Any conclusions
derived from such models should take into account the distortions imposed on
graph topology when converting proximity/strength into distance graphs, to
subsequently compute path length and shortest path measures. We characterise
the isomorphism using the max-min and Dombi disjunction/conjunction pairs. This
allows us to: (1) study alternative distance closures, such as those based on
diffusion, metric, and ultra-metric distances; (2) identify the operators
closest to the metric closure of distance graphs (the APSP), but which are
logically consistent; and (3) propose a simple method to compute alternative
distance closures using existing algorithms for the APSP. In particular, we
show that a specific diffusion distance is promising for community detection in
complex networks, and is based on desirable axioms for logical inference or
approximate reasoning on networks; it also provides a simple algebraic means to
compute diffusion processes on networks. Based on these results, we argue that
choosing different distance closures can lead to different conclusions about
indirect associations on network data, as well as the structure of complex
networks, and are thus important to consider
A Note on the sharp -Covergence rate of Upcrossings to the Brownian local time
In this note, we prove a sharp -rate of convergence of the number of
upcrossings to the local time of the Brownian motion. In particular, it
provides novel -variation estimates () for the number of
upcrossings of the Brownian motion. Our result complements the fundamental work
of Koshnevisan \cite{kho} who obtains an almost sure exact rate of convergence
in the sup norm.Comment: Some typos are correcte
Homogenization of generalized second-order elliptic difference operators
Fix a function where each is a strictly increasing right continuous function
with left limits. For a diagonal matrix function , let be a generalized second-order
differential operator. We are interested in studying the homogenization of
generalized second-order difference operators, that is, we are interested in
the convergence of the solution of the equation to the solution of the equation where the superscript stands for some sort of
discretization. In the continuous case we study the problem in the context of
-Sobolev spaces, whereas in the discrete case the theory is developed here.
The main result is a homogenization result. Under minor assumptions regarding
weak convergence and ellipticity of these matrices , we show that every
such sequence admits a homogenization. We provide two examples of matrix
functions verifying these assumptions: The first one consists to fix a matrix
function with some minor regularity, and take to be a convenient
discretization. The second one consists on the case where represents a
random environment associated to an ergodic group, which we then show that the
homogenized matrix does not depend on the realization of the
environment. Finally, we apply this result in probability theory. More
precisely, we prove a hydrodynamic limit result for some gradient processes.Comment: arXiv admin note: text overlap with arXiv:0911.4177; text overlap
with arXiv:0806.3211 by other author
Asymptotic adjustments of Pearson residuals in exponential family nonlinear models
In this work we define a set of corrected Pearson residuals for continuous
exponential family nonlinear models that have the same distribution as the true
Pearson residuals up to order , where is the sample
size. Furthermore, we also introduce a new modification of the Pearson
residuals, which we call PCA Pearson residuals, that are approximately
uncorrelated. These PCA residuals are new even for the generalized linear
models. The numerical results show that the PCA residuals are approximately
normally distributed, thus improving previous results by Simas and Cordeiro
(2009). These numerical results also show that the corrected Pearson residuals
approximately follow the same distribution as the true residuals, which is a
considerable improvement with respect to the Pearson residuals and also extends
the previous work by Cordeiro and Simas (2009)
A weak version of path-dependent functional It\^o calculus
We introduce a variational theory for processes adapted to the
multi-dimensional Brownian motion filtration that provides a differential
structure allowing to describe infinitesimal evolution of Wiener functionals at
very small scales. The main novel idea is to compute the "sensitivities" of
processes, namely derivatives of martingale components and a weak notion of
infinitesimal generators, via a finite-dimensional approximation procedure
based on controlled inter-arrival times and approximating martingales. The
theory comes with convergence results that allow to interpret a large class of
Wiener functionals beyond semimartingales as limiting objects of differential
forms which can be computed path wisely over finite-dimensional spaces. The
theory reveals that solutions of BSDEs are minimizers of energy functionals
w.r.t Brownian motion driving noise.Comment: Version to appear in Annals of Probabilit
Weak differentiability of Wiener functionals and occupation times
In this paper, we establish a universal variational characterization of the
non-martingale components associated with weakly differentiable Wiener
functionals in the sense of Le\~ao, Ohashi and Simas. It is shown that any
Dirichlet process (in particular semimartingales) is a differential form w.r.t
Brownian motion driving noise. The drift components are characterized in terms
of limits of integral functionals of horizontal-type perturbations and
first-order variation driven by a two-parameter occupation time process.
Applications to a class of path-dependent rough transformations of Brownian
paths under finite -variation () regularity is also discussed. Under
stronger regularity conditions in the sense of finite -variation, the
connection between weak differentiability and two-parameter local time
integrals in the sense of Young is established.Comment: Revised version. To appear in Bulletin des Sciences Math\'ematiques.
arXiv admin note: text overlap with arXiv:1707.04972, arXiv:1408.142
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