764 research outputs found
G-Brownian Motion as Rough Paths and Differential Equations Driven by G-Brownian Motion
The present paper is devoted to the study of sample paths of G-Brownian
motion and stochastic differential equations (SDEs) driven by G-Brownian motion
from the view of rough path theory. As the starting point, we show that
quasi-surely, sample paths of G-Brownian motion can be enhanced to the second
level in a canonical way so that they become geometric rough paths of roughness
2 < p < 3. This result enables us to introduce the notion of rough differential
equations (RDEs) driven by G-Brownian motion in the pathwise sense under the
general framework of rough paths. Next we establish the fundamental relation
between SDEs and RDEs driven by G-Brownian motion. As an application, we
introduce the notion of SDEs on a differentiable manifold driven by GBrownian
motion and construct solutions from the RDE point of view by using pathwise
localization technique. This is the starting point of introducing G-Brownian
motion on a Riemannian manifold, based on the idea of Eells-Elworthy-Malliavin.
The last part of this paper is devoted to such construction for a wide and
interesting class of G-functions whose invariant group is the orthogonal group.
We also develop the Euler-Maruyama approximation for SDEs driven by G-Brownian
motion of independent interest
Quantum Brownian Motion on noncommutative manifolds: construction, deformation and exit times
We begin with a review and analytical construction of quantum Gaussian
process (and quantum Brownian motions) in the sense of [25],[10] and others,
and then formulate and study in details (with a number of interesting examples)
a definition of quantum Brownian motions on those noncommutative manifolds (a
la Connes) which are quantum homogeneous spaces of their quantum isometry
groups in the sense of [11]. We prove that bi-invariant quantum Brownian motion
can be 'deformed' in a suitable sense. Moreover, we propose a noncommutative
analogue of the well-known asymptotics of the exit time of classical Brownian
motion. We explicitly analyze such asymptotics for a specific example on
noncommutative two-torus A{\theta}, which seems to behave like a
one-dimensional manifold, perhaps reminiscent of the fact that A{\theta} is a
noncommutative model of the (locally one-dimensional) 'leaf-space' of the
Kronecker foliation
Inference on Riemannian Manifolds: Regression and Stochastic Differential Equations
Statistical inference for manifolds attracts much attention because of its power of working with more general forms of data or geometric objects. We study regression and stochastic differential equations on manifolds from the intrinsic point of view.
Firstly, we are able to provide alternative parametrizations for data that lie on Lie group in the problem of fitting a regression model, by mapping this space intrinsically onto its Lie algebra, while we explore the behaviour of fitted values when this base point is chosen differently. Due to the nature of our data in the application of soft tissue artefacts, we employ two correlation structures, namely Matern and quasi-periodic correlation functions when using the generalized least squares, and show that some patterns of the residuals are removed.
Secondly, we construct a generalization of the Ornstein-Uhlenbeck process on the cone of covariance matrices SP(n) endowed with two popular Riemannian metrics, namely Log-Euclidean (LE) and Affine-Invariant (AI) metrics. We show that the Riemannian Brownian motion on SP(n) has infinite explosion time as on the Euclidean space and establish the calculation for the horizontal lifts of smooth curves. Moreover, we provide Bayesian inference for discretely observed diffusion processes of covariance matrices associated with either the LE or the AI metrics, and present a novel diffusion bridge sampling method using guided proposals when equipping SP(n) with the AI metric. The estimation algorithms are illustrated with an application in finance, together with a goodness-of-fit test comparing models associated with different metrics. Furthermore, we explore the multivariate volatility models via simulation study, in which covariance matrices in the models are assumed to be unobservable
Developments in Random Matrix Theory
In this preface to the Journal of Physics A, Special Edition on Random Matrix
Theory, we give a review of the main historical developments of random matrix
theory. A short summary of the papers that appear in this special edition is
also given.Comment: 22 pages, Late
Recent advances in directional statistics
Mainstream statistical methodology is generally applicable to data observed
in Euclidean space. There are, however, numerous contexts of considerable
scientific interest in which the natural supports for the data under
consideration are Riemannian manifolds like the unit circle, torus, sphere and
their extensions. Typically, such data can be represented using one or more
directions, and directional statistics is the branch of statistics that deals
with their analysis. In this paper we provide a review of the many recent
developments in the field since the publication of Mardia and Jupp (1999),
still the most comprehensive text on directional statistics. Many of those
developments have been stimulated by interesting applications in fields as
diverse as astronomy, medicine, genetics, neurology, aeronautics, acoustics,
image analysis, text mining, environmetrics, and machine learning. We begin by
considering developments for the exploratory analysis of directional data
before progressing to distributional models, general approaches to inference,
hypothesis testing, regression, nonparametric curve estimation, methods for
dimension reduction, classification and clustering, and the modelling of time
series, spatial and spatio-temporal data. An overview of currently available
software for analysing directional data is also provided, and potential future
developments discussed.Comment: 61 page
Parabolic stable surfaces with constant mean curvature
We prove that if u is a bounded smooth function in the kernel of a
nonnegative Schrodinger operator on a parabolic Riemannian
manifold M, then u is either identically zero or it has no zeros on M, and the
linear space of such functions is 1-dimensional. We obtain consequences for
orientable, complete stable surfaces with constant mean curvature
in homogeneous spaces with four
dimensional isometry group. For instance, if M is an orientable, parabolic,
complete immersed surface with constant mean curvature H in
, then and if equality holds, then
M is either an entire graph or a vertical horocylinder.Comment: 15 pages, 1 figure. Minor changes have been incorporated (exchange
finite capacity by parabolicity, and simplify the proof of Theorem 1)
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