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Pedro Catuogno

Publications and source records attributed to Pedro Catuogno.

9 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↗
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

Moderate averaged deviations for a multi-scale system with jumps and memory

This work studies a two-time-scale functional system given by two jump-diffusions under the scale separation by a small parameter $\varepsilon \rightarrow 0$. The coefficients of the equations that govern the dynamics of the system depend on the segment process of the slow variable (responsible for capturing delay effects on the slow component) and on the state of the fast variable. We derive a moderate deviations principle for the slow component of the system in the small noise limit using the weak convergence approach. The rate function is written in terms of the averaged dynamics associated to the multi-scale system. The core of the proof of the moderate deviations principle is the establishment of an averaging principle for the controlled processes associated to the slow variable in the framework of the weak convergence approach. The controlled version of the averaging principle for jump multi-scale diffusions relies on the classical Khasminkii's technique.

math.PR↗
arXiv2022

Large Deviations for Lévy Diffusions in small regime

This article concerns the large deviations regime and the consequent solution of the Kramers problem for a two-time scale stochastic system driven by a common jump noise signal perturbed in small intensity $\varepsilon>0$ and with accelerated jumps by intensity $\frac{1}{\varepsilon}$. We establish Freidlin-Wentzell estimates for the slow process of the multiscale system in the small noise limit $\varepsilon \rightarrow 0$ using the weak convergence approach to large deviations theory. The core of our proof is the reduction of the large deviations principle to the establishment of a stochastic averaging principle for auxiliary controlled processes. As consequence we solve the first exit time/ exit locus problem from a bounded domain containing the stable state of the averaged dynamics for the family of the slow processes in the small noise limit.

math.PR↗
arXiv2022

Martingales on Principal Fiber Bundles

Let $P(M,G)$ be a principal fiber bundle, let $ω$ be a connection form on $P(M,G)$, and consider a projectable connection $\nabla^{P}$ on $P(M,G)$. The aim of this work is to determine the $\nabla^{P}$-martingales in $P(M,G)$. Our results allow establishing new characterizations of harmonic maps from Riemannian manifolds to principal fiber bundles.

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↗
arXiv2011

$L^{p}-$solutions of the stochastic transport equation

We consider the stochastic transport linear equation and we prove existence and uniqueness of weak $L^{p}-$solutions. Moreover, we obtain a representation of the general solution and a Wong-Zakai principle for this equation. We make only minimal assumptions, similar to the deterministic problem. The proof is supported on the generalized Itô-Ventzel-Kunita formula (see 15) and the theory of Lions-DiPerna on transport linear equation (see 9).

math.FA↗
arXiv2010

On Stochastic generalized functions

We introduced a new algebra of stochastic generalized functions which contains to the space of stochastic distributions G, [25]. As an application, we prove existence and uniqueness of the solution of a stochastic Cauchy problem involving singularities.

math.FA↗