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Paulo Ruffino, 04^{th} April 2010.

**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.
- To appear in the
**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).

**An apology for "A Mathematician's Apology" by G. H. Hardy.**- Preprint: 6 pages, (dvi) , (ps) or (pdf) .
**O problema da corda suspensa.***Matemática Universitária (SBM)*, no. 24/25 , pp. 1-9 , junho/dezembro 1998.**A Física: no mundo micro e macroscópico.***Revista Brasileira de Ensino de Física*, vol.**21**, no. 2, junho 1999.

Rua Sérgio Buarque de Holanda, 651

13083-859 Campinas, SP

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fone: 55-(0)19- 3521 6033 (office)

Email: ruffino@ime.unicamp.br

Last modified 19 August 2003.