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Xavier Bardina

Publications and source records attributed to Xavier Bardina.

At least 19 recordsLinked to original sources
arXiv2025

Weak approximation for Gaussian processes from renewal processes

In previous works, Bardina and Rovira (2023) constructed a family of processes that converge strongly towards Brownian motion, defined from renewal processes, are constructed. In this paper we prove that some of these processes can be utilized to build approximations of Gaussian processes such as fractional Brownian motion or multiple Stratonovich integrals and we provide sufficient conditions on renewal processes to ensure that the convergence holds. An illustrative example of such a Gaussian process is the fractional Brownian motion with any Hurst parameter.

math.PR↗
arXiv2025

Kac-Stroock type approximations for the Brownian motion

In the present paper we show that the processes $X_n = \{X_n(t) \colon t \in [0,1]\}$, $n \in \mathbb{N}$, defined by $X_n(t) = \sqrt{n}C\int_0^t (-1)^{L(nu)} du$, where $L = \{L(t) \colon t \geq 0\}$ is a renewal processes whose inter-arrival times satisfy some integrability conditions and $C > 0$ is some normalizing constant, weakly converge, in the space of continuous functions over $[0,1]$, $\mathcal{C}([0,1])$, to the Brownian motion as $n$ approaches infinity. Thus, generalizing the result of D. W. Stroock (1982), where $L$ is taken to be a standard Poisson process. In particular, we see that these results are a mere consequence of Donsker's invariance principle.

math.PR↗
arXiv2025

Weak convergence of stochastic integrals with applications to SPDEs

In this paper we provide sufficient conditions for sequences of random fields of the form $\int_{D} f(x,y) \theta_n(y) dy$ to weakly converge, in the space of continuous functions over $D$, to integrals with respect to the Brownian sheet, $\int_{D} f(x,y)W(dy)$, where $D \subset \mathbb{R}^d$ is a rectangular domain, $x \in D$, $f$ is a function satisfying some integrability conditions and $\{\theta_n\}_n$ is a sequence of stochastic processes whose integrals $\int_{[0,x]}\theta_n(y)dy$ converge in law to the Brownian sheet (in the sense of the finite dimensional distribution convergence). We then apply these results to stablish the weak convergence of solutions of the stochastic Poisson equation.

math.PR↗
arXiv2025

Weak convergence of stochastic integrals

In this paper we provide sufficient conditions for sequences of stochastic processes of the form $\int_{[0,t]} f_n(u) \theta_n(u) du$, to weakly converge, in the space of continuous functions over a closed interval, to integrals with respect to the Brownian motion, $\int_{[0,t]} f(u)W(du)$, where $\{f_n\}_n$ is a sequence satisfying some integrability conditions converging to $f$ and $\{\theta_n\}_n$ is a sequence of stochastic processes whose integrals $\int_{[0,t]}\theta_n(u)du$ converge in law to the Brownian motion (in the sense of the finite dimensional distribution convergence), in the multidimensional parameter set case.

math.PR↗
arXiv2022

Strong limit of processes constructed from a renewal proces

We construct a family of processes, from a renewal process, that have realizations that converge almost surely to the Brownian motion, uniformly on the unit time interval. Finally we compute the rate of convergence in a particular case.

math.PR↗
arXiv2020

A stochastic epidemic model of COVID-19 disease

To model the evolution of diseases with extended latency periods and the presence of asymptomatic patients like COVID-19, we define a simple discrete time stochastic SIR-type epidemic model. We include both latent periods as well as the presence of quarantine areas, to capture the evolutionary dynamics of such diseases.

q-bio.PE↗
arXiv2020

Weak approximation of the complex Brownian sheet from a Lévy sheet and applications to SPDEs

We consider a Lévy process in the plane and we use it to construct a family of complex-valued random fields that we show to converge in law, in the space of continuous functions, to a complex Brownian sheet. We apply this result to obtain weak approximations of the random field solution to a semilinear one-dimensional stochastic heat equation driven by the space-time white noise.

math.PR↗
arXiv2020

Weak convergence to the fractional Brownian sheet from a Lévy sheet

In this paper, we show an approximation in law, in the space of the continuous functions on $[0,1]^2$, of two-parameter Gaussian processes that can be represented as a Wiener type integral by processes constructed from processes that converge to the Brownian sheet. As an application, we obtain a sequence of processes constructed from a Lévy sheet that converges in law towards the fractional Brownian sheet.

math.PR↗
arXiv2020

On the strong convergence of multiple ordinary integrals to multiple Stratonovich integrals

Given $\{W^{(m)}(t), t \in [0,T]\}_{m \ge 1}$ a sequence of approximations to a standard Brownian motion $W$ in $[0,T]$ such that $W^{(m)}(t)$ converges almost surely to $W(t)$ we show that, under regular conditions on the approximations, the multiple ordinary integrals with respect to $dW^{(m)}$ converge to the multiple Stratonovich integral. We are integrating functions of the type $$f(x_1,\ldots,x_n)=f_1(x_1)\ldots f_n(x_n) I_{\{x_1\le \ldots \le x_n\}},$$ where for each $i \in \{1,\ldots,n\}$, $f_i$ has continuous derivatives in $[0,T].$ We apply this result to approximations obtained from uniform transport processes.

math.PR↗
arXiv2019

Strong approximations of Brownian sheet by uniform transport processes

Many years ago, Griego, Heath and Ruiz-Moncayo proved that it is possible to define realizations of a sequence of uniform transform processes that converges almost surely to the standard Brownian motion, uniformly on the unit time interval. In this paper we extend their results to the multi parameter case. We begin constructing a family of processes, starting from a set of independent standard Poisson processes, that has realizations that converge almost surely to the Brownian sheet, uniformly on the unit square. At the end the extension to the $d$-parameter Wiener processes is presented.

math.PR↗
arXiv2017

Coinfection in a stochastic model for bacteriophage systems

A system modeling bacteriophage treatments with coinfections in a noisy context is analyzed. We prove that in a small noise regime, the system converges in the long term to a bacteria free equilibrium. Moreover, we compare the treatment with coinfection with the treatment without coinfection, showing how the coinfection affects the dose of bacteriophages that is needed to eliminate the bacteria and the velocity of convergence to the free bacteria equilibrium.

math.PR↗
arXiv2017

Stochastic epidemic SEIRS models with a constant latency period

In this paper we consider the stability of a class of deterministic and stochastic SEIRS epidemic models with delay. Indeed, we assume that the transmission rate could be stochastic and the presence of a latency period of $r$ consecutive days, where $r$ is a fixed positive integer, in the "exposed" individuals class E. Studying the eigenvalues of the linearized system, we obtain conditions for the stability of the free disease equilibrium, in both the cases of the deterministic model with and without delay. In this latter case, we also get conditions for the stability of the coexistence equilibrium. In the stochastic case we are able to derive a concentration result for the random fluctuations and then, using the Lyapunov method, that under suitable assumptions the free disease equilibrium is still stable.

math.PR↗
arXiv2013

Approximations of a complex Brownian motion with processes constructed from a process with independent increments

In this paper, we show an approximation in law of the complex Brownian motion by processes constructed from a stochastic process with independent increments. We give sufficient conditions for the characteristic function of the process with independent increments that ensure the existence of the approximation. We apply these results to Lévy processes. Finally we extend this results to the $m$-dimensional complex Brownian motion.

math.PR↗
arXiv2011

A stochastic model for bacteriophage therapies

In this article, we analyze a system modeling bacteriophage treatments for infections in a noisy context. In the small noise regime, we show that after a reasonable amount of time the system is close to a sane equilibrium (which is a relevant biologic information) with high probability. Mathematically speaking, our study hinges on concentration techniques for delayed stochastic differential equations.

math.PR↗
arXiv2011

An extension of bifractional Brownian motion

In this paper we introduce and study a self-similar Gaussian process that is the bifractional Brownian motion $B^{H,K}$ with parameters $H\in (0,1)$ and $K\in(1,2)$ such that $HK\in(0,1)$. A remarkable difference between the case $K\in(0,1)$ and our situation is that this process is a semimartingale when $2HK=1$.

math.PR↗
arXiv2009

Approximation of the finite dimensional distributions of multiple fractional integrals

We construct a family $I_{n_{\eps}}(f)_{t}$ of continuous stochastic processes that converges in the sense of finite dimensional distributions to a multiple Wiener-Itô integral $I_{n}^{H}(f1^{\otimes n}_{[0,t]})$ with respect to the fractional Brownian motion. We assume that $H>{1/2}$ and we prove our approximation result for the integrands $f$ in a rather general class.

math.PR↗
arXiv2009

Weak convergence towards two independent Gaussian processes from a unique Poisson process

We consider two independent Gaussian processes that admit a representation in terms of a stochastic integral of a deterministic kernel with respect to a standard Wiener process. In this paper we construct two families of processes, from a unique Poisson process, the finite dimensional distributions of which converge in law towards the finite dimensional distributions of the two independent Gaussian processes. As an application of this result we obtain families of processes that converge in law towards fractional Brownian motion and sub-fractional Brownian motion.

math.PR↗