Stochastic exponential in Lie groups and its applications.
- Preprint, February/2003. 10 pages
(ps)
.
Geometric aspects of stochastic delay differenctial equations on
manifolds.
- Preprint, August/2002. 12 pages
(ps)
.
Asymptotic angular stability in non-linear systems: rotation
numbers and winding numbers.
- Preprint, July/2002. 18 pages (ps)
.
-
(With Patrick
E. McSharry ).
Non-linear Iwasawa decomposition of stochastic flows: geometrical
characterization and examples.
- To appear in the Proceedings of
Semigroup Operators: Theory and Applications - SOTA2 , Rio de
Janeiro September 10-14 2001. 10 pages, (pdf)
.
-
Abstract:
-
Let $\varphi_t$ be the stochastic flow of a stochastic differential
equation on a Riemannian manifold $M$ of constant curvature. For a
given initial condition in the orthonormal frame bundle: $x_0\in M$
and $u$ an orthonormal frame in $T_{x_0}M$, there exists a unique
decomposition $\varphi_t=\xi_t \circ \Psi_t$ where $\xi_t$ is
isometry, $\Psi_t$ fixes $x_0$ and $D\Psi_t(u)=u\cdot s_t$ where
$s_t$ is an upper triangular matrix process. We present the results
and the main ideas by working in detailed examples.
Decomposition of stochastic flows and rotation matrix.
- Stochastics and Dynamics Vol.
2 (1), 2002.
-
Abstract:
-
We provide geometrical conditions on the manifold for the existence
of the Liao's factorization of stochastic flows (PTRF 25 (3), 2000).
If $M$ is simply connected and has constant curvature then this
decomposition holds for any stochastic flow, conversely, if every
flow on $M$ has this decomposition then $M$ has constant curvature.
Under certain conditions, it is possible to go further on the
factorization: $ \varphi_t = \xi_t \circ \Psi_t \circ \Theta_t$,
where $\xi_t$ and $\Psi_t$ have the same properties of Liao's
decomposition and $(\xi_t \circ \Psi_t)$ are affine transformations
on $M$. We study the asymptotic behaviour of the isometric component
$\xi_t$ via rotation matrix, providing a Furstenberg-Khasminskii
formula for this skew-symmetric matrix.
Regular Conditional Probability, Disintegration of Probability
and Radon Spaces.
- (With D. Leão Pinto Jr. and
Marcelo Dutra Fragoso). Preprint, 12 pages, (dvi)
or (ps).
-
Abstract:
-
We establish equivalence of several regular conditional probability
properties and Radon space. In addition, we introduce the
universally measurable disintegration concept and prove an existence
result.
Random Versions of Hartman-Grobman Theorem.
- Preprint IMECC, UNICAMP no. 27/01 (2001). 37 pages, (dvi)
.
-
(With Edson Alberto Coayla Teran ).
-
Abstract:
-
We present versions of Hartman-Grobman theorems for random dynamical
systems (RDS) in the discrete and continuous case. We apply the same
random norm used by Wanner (Dynamics Reported, Vol. 4, Springer,
1994), but instead of using difference equations, we perform an
apropriate generalization of the deterministic arguments in an
adequate space of measurable homeomorphisms to extend his result
with weaker hypotheses and simpler arguments.
Lyapunov Exponents for Stochastic Differential Equations in
Semi-simple Lie groups
- Archivum Mathematicum (Brno), Vol. 37 (3),
(2001).
-
(With Luiz
Antonio Barrera San Martin ).
-
Abstract:
-
We write an integral formula for the asymptotics of the A-part in
the Iwasawa decomposition of the solution of an invariant stochastic
equation in a semi-simple group. The integral is with respect to the
invariant measure on the maximal flag manifold, the Furstenberg
boundary. The integrand of the formula is related to the
Takeuchi-Kobayashi Riemannian metric in the flag manifold.
A Fourier analysis of white noise via canonical Wiener space.
- Proceedings of the 4th
Portuguese Conference on Automatic Control. 04-06 October 2000,
ISBN 972-98603-0-0 , pp. 144-148, 2000.
-
Abstract:
-
We present a Fourier analysis of the white noise, where this process
is considered as the formal derivative of the Brownian motion in the
time interval [0,T] with T \geq 0. By a convenient construction of
an isomorphism of abstract Wiener space we identify each trajectory
of the white noise with a sequence of complex numbers whose modulus
and argument represent respectively the amplitude and the phase of
each harmonic component exp {i(pi/T)nt} of this (formal) stochastic
trajectory.
Wiener Integral in the space of sequences of real numbers.
- Archivum Mathematicum (BRNO) , Vol. 36 (2), pp.
95-101, (2000).
-
(With Alexandre de Andrade).
-
Abstract:
-
Let i:H --> W be the classical Wiener space , where H is the
Cameron-Martin space and W={\sigma :[0,1] --> R continuous with
\sigma(0) =0}. We extend the canonical isometry H --> l_{2} to a
linear isomorphism \Phi :W --> V \subset R^{\infty} which pushes
forward the Wiener structure into the abstract Wiener space i:l_{2}
--> V . The Wiener integration assumes a new interesting face
when it is taken in this space.
A sampling theorem for rotation numbers of linear processes in
R^2.
- Random Operators and Stochastic
Equations, , Vol. 8 (2), pp. 175-188, (2000).
-
Abstract:
-
We prove an ergodic theorem for the rotation number of the
composition of a sequence os stationary random homeomorphisms in
$S^{1}$. In particular, the concept of rotation number of a matrix
$g\in Gl^{+}(2,{\Bbb R})$ can be generalized to a product of a
sequence of stationary random matrices in $% Gl^{+}(2,{\Bbb R})$. In
this particular case this result provides a counter-part of the
Osseledec's multiplicative ergodic theorem which guarantees the
existence of Lyapunov exponents. A random sampling theorem is then
proved to show that the concept we propose is consistent by
discretization in time with the rotation number of continuous linear
processes on ${\Bbb R}^{2}.$
Characterizations of Radon Spaces.
- Statistics and Probability Letters, , Vol. 48
(4), pp. 409-413, 1999.
-
(With D. Leão Pinto Jr. and Marcelo Dutra Fragoso).
-
Abstract:
-
Assuming hypothesis only on the $\sigma $-algebra ${\cal F},$ we
characterize (via Radon spaces) the class of measurable spaces
($\Omega ,{\cal F})$ that admits regular conditional probability for
all probabilities on ${\cal F}$.
Matrix of rotation for stochastic dynamical systems.
- Computacional and Applied
Mathematics,, Vol. 18 (2), pp. 213-226, 1999.
-
Abstract:
-
Matrix of rotation generalizes the concept of rotation number for
stochastic dynamical systems given in Ruffino (Stoch. Stoch.
Reports, 1997). This matrix is the asymptotic time average of the
Maurer--Cartan form composed with the Riemannian connection along
the induced trajectory in the orthonormal frame bundle $OM$ over an
$n$-dimensional Riemannian manifold $M$. It provides the asymptotic
behaviour of an orthonormal $n$-frame under the action of the
derivative flow and the Gram--Schmidt orthonormalization. We lift
the stochastic differential equation of the system on $M$ to a
stochastic differential equation in $OM$ and we use
Furstenberg-Khasminskii argument to prove that the matrix of
rotation exists almost surely with respect to invariant measures on
this bundle.
Rotation number for stochastic dynamical systems.
- Stochastics and Stochastics
Reports , Vol. 60, pp. 289-318, 1997.
-
Abstract:
-
Rotation number is the asymptotic time average of the angular
rotation of a given tangent vector under the action of the
derivative flow in the tangent bundle over a Riemannian manifold
$M$. This angle in higher dimension is taken with respect to a
reference given by the stochastic parallel transport along the
trajectories and the canonical connection in the Stiefel bundle
$St_2 M$. So, these numbers give an angular complementary
information of that one given by the Lyapunov exponents. We lift the
stochastic differential equation on $M$ to a stochastic equation in
the Stiefel bundle and we use Furstenberg-Khasminskii argument to
prove the existence almost surely of the rotation numbers with
respect to any invariant measure on this bundle. Finally we present
some information of the dynamical system provided by the rotation
number: rotation of the stable manifold (Theorem 6.4).
Last modified 19 August 2003.