Paulo R. Ruffino

An averaging principle for diffusions in foliated spaces

Consider an SDE on a foliated manifold whose trajectories lay on compact leaves. We investigate the effective behavior of a small transversal perturbation of order e. An average principle is shown to hold such that the component transversal to the leaves converges to the solution of a deterministic ODE, according to the average of the perturbing vector field with respect to invariant measures on the leaves, as e goes to zero. An estimate of the rate of convergence is given. These results generalize the geometrical scope of previous approaches, including completely integrable stochastic Hamiltonian system.

STOCHASTIC VERSIONS OF HARTMAN–GROBMAN THEOREMS

We present versions of Hartman–Grobman theorems for random dynamical systems (RDS) in the discrete case. We use the same random norm like in Wanner [14], but instead of using difference equations, we perform an appropriate generalization of the deterministic arguments in an adequate space of measurable homeomorphisms to extend the results in [14] with weaker hypotheses (integrability instead of boundedness) and simpler arguments.

Rotation Numbers For Stochastic Dynamical Systems

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 St2M So, these numbers give an angular complementary information to that 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 on the dynamical system provided by the rotation number: rotation of the stable manifold (theorem 6.4)

Harmonic measures in embedded foliated manifolds

We study harmonic and totally invariant measures in a foliated compact Riemannian manifold. We construct explicitly a Stratonovich differential equation for the foliated Brownian motion. We present a characterization of totally invariant measures in terms of the flow of diffeomorphisms associated to this equation. We prove an ergodic formula for the sum of the Lyapunov exponents in terms of the geometry of the leaves.

DECOMPOSITION OF STOCHASTIC FLOWS IN MANIFOLDS WITH COMPLEMENTARY DISTRIBUTIONS

Let M be a differentiable manifold endowed locally with two complementary distributions, say horizontal and vertical. We consider the two subgroups of (local) diffeomorphisms of M generated by vector fields in each of of these distributions. Given a stochastic flow φt of diffeomorphisms of M, in a neighbourhood of an initial condition, up to a stopping time we decompose φt = ξt ◦ ψt where the first component is a diffusion in the group of horizontal diffeomorphisms and the second component is a process in the group of vertical diffeomorphisms. Further decomposition will include more than two components; it leads to a maximal cascade decomposition in local coordinates where each component acts only in the corresponding coordinate.

DECOMPOSITION OF STOCHASTIC FLOWS WITH AUTOMORPHISM OF SUBBUNDLES COMPONENT

We show that given a G-structure P on a differential manifold M, if the group G(M) of automorphisms of P is large enough, then there exists the quotient of a stochastic flows φt by G(M), in the sense that φt = ξt ◦ ρt where ξt ∈ G(M), the remainder ρt has derivative which is vertical, transversal to the fibres of P. This geometrical context generalises previous results where M is a Riemannian manifold and φt is decomposed with an isometric component, see [12] and [15], which in our context corresponds to the particular case of an SO(n)-structure on M.

Relative rotation number for stochastic systems: dynamical and topological applications

We introduce a concept of relative rotation number to unify many different approaches of rotation number in non-linear dynamical systems. We present an ergodic result of existence a.s. for stochastic systems. In higher dimension, we show that the natural idea of projecting into a plane does work well a.s. for any plane (different from deterministic systems where projections may be degenerate). A number of further properties (invariance by homotopy and by conjugacy) and applications are presented.

POISSON SPACES FOR PROPER SEMIGROUPS OF SEMI-SIMPLE LIE GROUPS

Let ν be a probability measure on a semi-simple Lie group G with finite center. Under the hypothesis that the semigroup S generated by ν has non-empty interior, we identify the Poisson space Π = G/MνAN, where bounded (l.u.c.) ν-harmonic functions in G have a one-to-one correspondence with measurable (continuous) functions in Π. This paper extends a classical result (see Furstenberg [7], Azencott [1] and others), where the semigroup generated by ν was assumed to be the whole (connected) group. We present two detailed examples.

Topology of Foliations and Decomposition of Stochastic Flows of Diffeomorphisms

Let M be a compact manifold equipped with a pair of complementary foliations, say horizontal and vertical. In Catuogno et al. (Stoch Dyn 13(4):1350009, 2013) it is shown that, up to a stopping time