arXiv subjects

Paulo R. Ruffino

Publications and source records attributed to Paulo R. Ruffino.

13 recordsLinked to original sources

arXiv2022

Stochastic n-point D-bifurcations of stochastic Lévy flows and their complexity on finite spaces

This article refines the classical notion of a stochastic D-bifurcation to the respective family of n-point motions for homogeneous Markovian stochastic semiflows, such as stochastic Brownian flows of homeomorphisms, and their generalizations. This notion essentially detects at which level $k\leq n$ the support of the invariant measure of the k-point bifurcation has more than one connected component. Stochastic Brownian flows and their invariant measures which were shown by Kunita (1990) to be rigid, in the sense of being uniquely determined by the $1$-and $2$-point motions, and hence only stochastic n-point bifurcation of level $n=1$ or $n=2$ can occur. For general homogeneous stochastic Markov semiflows this turns out to be false. This article constructs minimal examples of where this rigidity is false in general on finite space and studies the complexity of the resulting n-point bifurcations.

math.PR↗

arXiv2018

A strong averaging principle for Lévy diffusions in foliated spaces with unbounded leaves

This article extends a strong averaging principle for Lévy diffusions which live on the leaves of a foliated manifold subject to small transversal Lévy type perturbation to the case of non-compact leaves. The main result states that the existence of $p$-th moments of the foliated Lévy diffusion for $p\geq 2$ and an ergodic convergence of its coefficients in $L^p$ implies the strong $L^p$ convergence of the fast perturbed motion on the time scale $t/ε$ to the system driven by the averaged coefficients. In order to compensate the non-compactness of the leaves we use an estimate of the dynamical system for each of the increments of the canonical Marcus equation derived in da Costa and Hoegele (2017), the boundedness of the coefficients in $L^p$ and a nonlinear Gronwall-Bihari type estimate. The price for the non-compactness are slower rates of convergence, given as $p$-dependent powers of $ε$ strictly smaller than $1/4$.

math.DS↗

arXiv2016

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 $\varepsilon$. 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 $\varepsilon$ 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.

math.DS↗

arXiv2015

Decomposition of stochastic flows generated by Stratonovich SDEs with jumps

Consider a manifold $M$ endowed locally with a pair of complementary distributions $Δ^H \oplus Δ^V=TM$ and let $\text{Diff}(Δ^H, M)$ and $\text{Diff}(Δ^V, M)$ be the corresponding Lie subgroups generated by vector fields in the corresponding distributions. We decompose a stochastic flow with jumps, up to a stopping time, as $φ_t = ξ_t \circ ψ_t$, where $ξ_t \in \text{Diff}(Δ^H, M)$ and $ψ_t \in \text{Diff}(Δ^V, M)$. Our main result provides Stratonovich stochastic differential equations with jumps for each of these two components in the corresponding infinite dimensional Lie groups. We present an extension of the Itô-Ventzel-Kunita formula for stochastic flows with jumps generated by classical Marcus equation (as in Kurtz, Pardoux and Protter, Annales de L'I.H.P. section B, 1995, among others). The results here correspond to an extension of Catuogno, da Silva and Ruffino, Stoch. Dyn. 2013, where this decomposition was studied for the continuous case.

math.DS↗

arXiv2015

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, Silva and Ruffino ($Stoch$. $Dyn$., 2013) it is shown that, up to a stopping time $τ$, a stochastic flow of local diffeomorphisms $φ_t$ in $M$ can be written as a Markovian process in the subgroup of diffeomorphisms which preserve the horizontal foliation composed with a process in the subgroup of diffeomorphisms which preserve the vertical foliation. Here, we discuss topological aspects of this decomposition. The main result guarantees the global decomposition of a flow if it preserves the orientation of a transversely orientable foliation. In the last section, we present an Itô-Liouville formula for subdeterminants of linearised flows. We use this formula to obtain sufficient conditions for the existence of the decomposition for all $t\geq 0$.

math.DS↗

arXiv2015

Stochastic delay differential equations with jumps in differentiable manifolds

In this article we propose a model for stochastic delay differential equation with jumps (SDDEJ) in a differentiable manifold $M$ endowed with a connection $\nabla$. In our model, the continuous part is driven by vector fields with a fixed delay and the jumps are assumed to come from a distinct source of (càdlàg) noise, without delay. The jumps occur along adopted differentiable curves with some dynamical relevance (with fictitious time) which allow to take parallel transport along them. Using a geometrical approach, in the last section, we show that the horizontal lift of the solution of an SDDEJ is again a solution of an SDDEJ in the linear frame bundle $BM$ with respect to a connection $\nabla^H$ in $BM$.

math.DS↗

arXiv2014

Extension of time for decomposition of stochastic flows in spaces with complementary foliations

Let $M$ be a manifold equipped (locally) with a pair of complementary foliations. In Catuogno, da Silva and Ruffino (Stoch. Dyn. 2013), it is shown that, up to a stopping time $τ$, a stochastic flow of local diffeomorphisms $φ_t$ in $M$ can be decomposed in diffeomorphisms that preserves this foliations. In this article we present techniques which allows us to extend the time of this decomposition. For this extension, we use two techniques: In the first one, assuming that the vector fields of the system commute with each other, we apply Marcus equation to jump nondecomposable diffeomorphisms. The second approach deals with the general case: we introduce a `stop and go' technique that allows us to construct a process that follows the original flow in the `good zones' for the decomposition, and remains paused in `bad zones'. Among other applications, our results open the possibility of studying the asymptotic behaviour of each component. \end{abstract}

math.DS↗

arXiv2014

Application of an averaging principle on foliated diffusions: topology of the leaves

We consider an $εK$ transversal perturbing vector field in a foliated Brownian motion defined in a foliated tubular neighbourhood of an embedded compact submanifold in $\R^3$. We study the effective behaviour of the system under this $ε$ perturbation. If the perturbing vector field $K$ is proportional to the Gaussian curvature at the corresponding leaf, we have that the transversal component, after rescaling the time by $t/ε$, approaches a linear increasing behaviour proportional to the Euler characteristic of $M$, as $ε$ goes to zero. An estimate of the rate of convergence is presented.

math.PR↗

arXiv2014

Foliated stochastic calculus: Harmonic measures

In this article we present an intrinsec construction of foliated Brownian motion via stochastic calculus adapted to foliation. The stochastic approach together with a proposed foliated vector calculus provide a natural method to work on harmonic measures. Other results include a decomposition of the Laplacian in terms of the foliated and basic Laplacians, a characterization of totally invariant measures and a differential equation for the density of harmonic measures.

math.DG↗

arXiv2013

Degenerate semigroups and stochastic flows of mappings in foliated manifolds

Let $(M, \mathcal{F})$ be a compact Riemannian foliated manifold. We consider a family of compatible Feller semigroups in $C(M^n)$ associated to laws of the $n$-point motion. Under some assumptions (Le Jan and Raimond, \cite{Le Jan-Raimond}) there exists a stochastic flow of measurable mappings in $M$. We study the degeneracy of these semigroups such that the flow of mappings is foliated, i.e. each trajectory lays in a single leaf of the foliation a.s, hence creating a geometrical obstruction for coalescence of trajectories in different leaves. As an application, an averaging principle is proved for a first order perturbation transversal to the leaves. Estimates for the rate of convergence are calculated.

math.PR↗

arXiv2012

Harmonic measures in embedded foliated manifolds

We study harmonic and totally invariant measures in a foliated compact Riemannian manifold isometrically embedded in an Euclidean space. We introduce geometrical techniques for stochastic calculus in this space. In particular, using these techniques we can construct explicitely an Stratonovich equation for the foliated Brownian motion (cf. L. Garnett \cite{LG} and others). We present a characterization of totally invariant measures in terms of the flow of diffeomorphisms of associated to this equation. We prove an ergodic formula for the sum of the Lyapunov exponents in terms of the geometry of the leaves.

math.DG↗

arXiv2012

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 $φ_t$ of diffeomorphisms of $M$, in a neighbourhood of initial condition, up to a stopping time we decompose $φ_t = ξ_t \circ ψ_t$ 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.

math.DS↗

arXiv2011

A note on stochastic calculus in vector bundles

The aim of these notes is to relate covariant stochastic integration in a vector bundle $E$ (as in Norris \cite{Norris}) with the usual Stratonovich calculus via the connector $\K:TE \rightarrow E$ (cf. e.g. Paterson \cite{Paterson} or Poor \cite{Poor}) which carries the connection dependence.

math.DG↗