arXiv subjects

Paulo Ruffino

Publications and source records attributed to Paulo Ruffino.

6 recordsLinked to original sources

arXiv2025

Minimal jointly uniform attractor for nonautonomous random dynamical systems

We introduce a notion of minimal uniform attractor for nonautonomous random dynamical systems, which depends jointly on time and on a random parameter. Several examples are provided to illustrate the concept and to compare it with existing notions of uniform attractors in the literature. We further apply the abstract theory to nonautonomous random differential equations with a non-compact symbol space. In particular, we develop a method to compactify the symbol space, by adapting techniques from the theory of deterministic nonautonomous differential equations. We also establish the stability of the minimal jointly uniform attractor by exploiting the relationship between deterministic and random dynamics. Finally, we show that such structures arise naturally in stochastic differential equations whose noise terms carry additional time dependence, by establishing a topological conjugacy between the resulting stochastic flows and suitable random differential equations.

math.DS↗

arXiv2025

Decomposition of discontinuous flows of diffeomorphisms: jumpings, geometrical and topological aspects

Let $M$ be a compact manifold equipped with a pair of complementary foliations, say horizontal $\mathcal{H}$ and vertical $\mathcal{V}$. In Melo, Morgado and Ruffino (Disc Cont Dyn Syst B, 2016, 21(9)) it is proved that if a semimartingale $X_t$ has a finite number of jumps in compact intervals then, up to a stopping time $τ$, a stochastic flow of local diffeomorphisms in $M$ driven by $X_t$ can be decomposed into a process in the Lie group of diffeomorphisms which fix the leaves of $\mathcal{H}$ composed with a process in the Lie group of diffeomorphisms which fix the leaves of $\mathcal{V}$. Dynamics at the discontinuities of $X_t$ here are interpreted in the Marcus sense as in Kurtz, Pardoux and Protter \cite{KPP}. Here we enlarge the scope of this geometric decomposition and consider flows driven by arbitrary semimartingales with jumps and show explicit equations for each component. Our technique is based in an extension of the Itô-Ventzel-Kunita formula for stochastic flows with jumps. Geometrical and others topological obstructions for the decomposition are also considered: e.g. an index of attainability is introduced to measure the complexity of the dynamics with respect to the pair of foliations.

math.DS↗

arXiv2022

Geometric decomposition of flows generated by rough path differential equations

Whenever an Itô-Wentsel type of formula holds for composition of flows of a certain differential dynamics, there exists locally a decomposition of the corresponding flow according to complementary distributions (or foliations, in the case of integrability of these distributions). Many examples have been proved in distinct context of dynamics: Stratonovich stochastic equations, Lévy driven noise, low regularity $α$-Hölder control functions ($ α\in (1/2,1]$), see e.g. [6], [7], [20], [21]. Here we present the proof of this categorical property: we illustrate with the $α$-Hölder rough path, $α\in (1/3, 1/2]$ using the Itô-Wentsel formula in this context proved in [5]. Different from the previous approaches, here however, instead of using an intrinsic rough path calculus on manifolds, the manifold has to be embedded in an Euclidean space. A cascade decomposition is also shown when we have multiple lower dimensional directions which span the whole space. As application, the linear case is treated in details: the cascade decomposition provides a row factorization of all matrices which allow real logarithm.

math.PR↗

arXiv2022

Geometric aspects of Young Integral: decomposition of flows

In this paper we study geometric aspects of dynamics generated by Young differential equations (YDE) driven by $α$-Hölder trajectories with $α\in (1/2, 1)$. We present a number of properties and geometrical constructions on this low regularity context: Young Itô geometrical formula, horizontal lift in principal fibre bundles, parallel transport, covariant derivative, development and anti-development, among others. Our main application here is a geometrical decomposition of flows generated by YDEs according to diffeomorphisms generated by complementary distributions (integrable or not). The proof of existence of this decomposition is based on an Young Itô-Kunita formula for $α$-H{ö}lder paths proved by Castrequini and Catuogno (Chaos Solitons Fractals, 2022).

math.DS↗

arXiv2020

Stochastic near-optimal control: additive, multiplicative, non-Markovian and applications

In this survey we present the near-optimal stochastic control problem according to some recent tools in the literature. In particular, we focus on the approach of a discretization of the noise values instead of the canonical time-discretization. This is the so called {\it skeleton} structure. This allows to obtain an $ε$-optimal control in non-Markovian systems (the main Theorem). A simple example illustrates the technique. The importance of the approach is emphasised in a final section on open problems related to more geometrical framework and discontinuous noise.

math.PR↗

arXiv2011

Decomposition of stochastic flows with automorphism of subbundles component

We show that given a $G$-structure $P$ on a differentiable manifold $M$, if the group $G(M)$ of automorphisms of $P$ is big enough, then there exists the quotient of an stochastic flows $phi_t$ by $G(M)$, in the sense that $ϕ_t = ξ_t \circ ρ_t$ where $ξ_t \in G(M)$, the remainder $ρ_t$ has derivative which is vertical but transversal to the fibre of $P$. This geometrical context generalizes previous results where $M$ is a Riemannian manifold and $ϕ_t$ is decomposed with an isometric component, see Liao \cite{Liao1} and Ruffino \cite{Ruffino}, which in our context corresponds to the particular case of an SO(n)-structure on $M$.

math.DS↗