arXiv subjects

Darío Mena

Publications and source records attributed to Darío Mena.

10 recordsLinked to original sources

arXiv2026

Itô's Formula for Itô processes defined with respect to a cylindrical-martingale valued measure

Using the authors' recently developed stochastic integration [Stoch PDE: Anal Comp, 2024], we prove an Itô formula for Hilbert space-valued Itô processes defined with respect to a cylindrical martingale-valued measure. We develop some tools from stochastic analysis, as are the predictable and optional quadratic variation of a stochastic integral, the continuous and purely discontinuous parts of an integral process, and a Riemann representation formula. As an application of our Itô formula, we prove a Burkholder inequality for the stochastic integral defined with respect to a cylindrical martingale-valued measure. Finally, we derive Itô formulas for Hilbert space-valued martingale-valued measures and for cylindrical square integrable martingales.

math.PR↗

arXiv2026

Markov property and path regularity for the solutions to SPDEs driven by cylindrical-martingale valued measures

In this paper we prove the Markov property for the solution to stochastic partial differential equations driven by a cylindrical orthogonal martingale-valued measure. We assume our coefficients are time-dependent and satisfy some growth and Lipschitz conditions. We also prove that for time-independent coefficients and under mild assumptions on the cylindrical orthogonal martingale-valued measure, the solutions to our stochastic partial differential equations are Feller. Finally, in the case that the $C_{0}$-semigroup is quasi-contraction, we show that the solution to our stochastic partial differential equation possesses a càdlàg version.

math.PR↗

arXiv2024

Riesz spaces of signed charges on semi-rings

A constructive definition of the supremum of a family of set functions is exploited in the context of Riesz spaces of signed measures and finitely additive functions (signed charges) on semi-rings. We explore applications, particularly to establish a Jordan decomposition for signed charges on semi-rings, whether the structure of Riesz space is present or not.

math.FA↗

arXiv2024

Cylindrical Martingale-Valued Measures, Stochastic Integration and SPDEs

We develop a theory of Hilbert-space valued stochastic integration with respect to cylindrical martingale-valued measures. As part of our construction, we expand the concept of quadratic variation, introduced by Veraar and Yaroslavtsev (2016), to the case of cylindrical martingale-valued measures that are allowed to have discontinuous paths (this is carried out within the context of separable Banach spaces). Our theory of stochastic integration is applied to address the existence and uniqueness of solutions to stochastic partial differential equations in Hilbert spaces.

math.PR↗

arXiv2023

A nonlinear relapse model with disaggregated contact rates: analysis of a forward-backward bifurcation

We developed a nonlinear differential equation model to explore the dynamics of relapse phenomena. Our incidence rate function is formulated, taking inspiration from recent adaptive algorithms. It incorporates contact behavior for individuals in each health class. We use constant contact rates at each health status for our analytical results and prove conditions for different forward-backward bifurcation scenarios. The relationship between the different contact rates influences these conditions. Numerical examples show the sensitivity of the model toward initial conditions. In particular, we highlight the effect of temporarily recovered individuals and infected initial conditions.

math.DS↗

arXiv2019

Sparse Bounds for the Discrete Spherical Maximal Function

We prove sparse bounds for the spherical maximal operator of Magyar, Stein and Wainger. The bounds are conjecturally sharp, and contain an endpoint estimate. The new method of proof is inspired by ones by Bourgain and Ionescu, is very efficient, and has not been used in the proof of sparse bounds before. The Hardy-Littlewood Circle method is used to decompose the multiplier into major and minor arc components. The efficiency arises as one only needs a single estimate on each element of the decomposition.

math.CA↗

arXiv2017

Characterization of two parameter matrix-valued BMO by commutator with the Hilbert transform

In this paper we prove that the space of two parameter, matrix-valued BMO functions can be characterized by considering iterated commutators with the Hilbert transform. Specifically, we prove that $$\| B \|_{BMO} \lesssim \| [[M_B, H_1],H_2] \|_{L^2(\mathbb{R}^2;\mathbb{C}^d) \rightarrow L^2(\mathbb{R}^2;\mathbb{C}^d)} \lesssim \| B \|_{BMO}.$$ The upper estimate relies on Petermichl's representation of the Hilbert transform as an average of dyadic shifts, and the boundedness of certain paraproduct operators, while the lower bound follows Ferguson and Lacey's proof for the scalar case.

math.CV↗

arXiv2017

Sparse Bounds for Bochner-Riesz Multipliers

The Bochner-Riesz multipliers $ B_{δ}$ on $ \mathbb R ^{n}$ are shown to satisfy a range of sparse bounds, for all $0< δ< \frac {n-1}2 $. The range of sparse bounds increases to the optimal range, as $ δ$ increases to the critical value, $ δ=\frac {n-1}2$, even assuming only partial information on the Bochner-Riesz conjecture in dimensions $ n \geq 3$. In dimension $n=2$, we prove a sharp range of sparse bounds. The method of proof is based upon a `single scale' analysis, and yields the sharpest known weighted estimates for the Bochner-Riesz multipliers in the category of Muckenhoupt weights.

math.CA↗

arXiv2017

Sparse Bounds for Discrete Quadratic Phase Hilbert Transform

Consider the discrete quadratic phase Hilbert Transform acting on $\ell^{2}$ finitely supported functions $$ H^α f(n) : = \sum_{m \neq 0} \frac{e^{2 πiαm^2} f(n - m)}{m}. $$ We prove that, uniformly in $α\in \mathbb{T}$, there is a sparse bound for the bilinear form $\langle H^α f , g \rangle$. The sparse bound implies several mapping properties such as weighted inequalities in an intersection of Muckenhoupt and reverse Hölder classes.

math.CA↗

arXiv2016

The Sparse T1 Theorem

We impose standard $ T1 $-type assumptions on a Calderón-Zygmund operator $ T $, and deduce that for bounded compactly supported functions $ f, g $ there is a sparse bilinear form $ Λ$ so that $$ \lvert \langle T f, g \rangle\rvert \lesssim Λ(f,g). $$ The proof is short and elementary. The sparse bound quickly implies all the standard mapping properties of a Calderón-Zygmund on a (weighted) $ L ^{p}$ space.

math.CA↗