arXiv subjects

Eldon Barros

Publications and source records attributed to Eldon Barros.

3 recordsLinked to original sources

arXiv2026

Second-order fields for stochastic partial differential equations

In this article, we study the second-order fluctuation of the solutions of one-dimensional polynomial stochastic partial differential equations (SPDEs) of the form \begin{equation*} (\partial_t - \Delta) \Phi_{\varepsilon} = -P(\Phi_{\varepsilon}) + \xi_{\varepsilon}, \end{equation*} where $P$ is a polynomial of degree greater than or equal to $2$, $\xi_\varepsilon$ is the white-noise after being convoluted (in space) by the heat kernel $K_\varepsilon = e^{\varepsilon \Delta}$. More precisely, taking advantage of the local solutions of pointwise well-posedness of limits $\Phi= \lim_{\varepsilon \to 0}\Phi_{\varepsilon}$, we characterise the limit of $\Phi^{err}=\lim_{\varepsilon \to 0} \varepsilon^{-1}(\Phi_{\varepsilon}-\Phi)$ as the solution of a more irregular stochastic partial differential equation. This is performed by applying the Da Prato--Debussche decomposition in the non-linear equation and characterising the limit of each of the terms. We also discuss possible characterisations of second-order fluctuations performed for higher-dimensional SPDEs in the weakly-coupled regime.

math.PR↗

arXiv2026

Delayed logistic equation as a limit of long memory Markov chains

We introduce and analyze a long-memory continuous-time Markov chain on $\mathbb{R}_{+}$ whose jump mechanism depends explicitly on a state in the past. From the present state $x_0$, the process jumps to $x_0\left(1+\frac{1}{N}\right)$ or $x_0\left(1-\frac{x_{-\lfloor \tau N \rfloor}}{N^2}\right)$, each at rate $\tfrac{1}{2}$, where $x_{-\lfloor \tau N \rfloor}$ denotes the state located $\lfloor \tau N \rfloor$ jumps backward in time. Here the delay $\tau > 0$ is fixed and $N$ is the scaling parameter. The initial condition is prescribed by a vector of length $\lfloor \tau N \rfloor + 1$, all of whose entries are equal to $\mu N$. Using a genuine space-time replacement lemma, we prove that, as $N \to \infty$, the rescaled process converges to a deterministic limit governed by the Delayed Logistic Equation (also known as the Hutchinson equation) with delay $\tau$ and initial condition $\rho(t) \equiv \mu$ for $t \in [-\tau, 0]$.

math.PR↗

arXiv2025

Scaling limit of boundary random walks: A martingale problem approach

We establish the scaling limit of a class of boundary random walks to the full spectrum of Brownian-type processes on the half-line. By solving the associated martingale problem and employing weak convergence techniques, we prove that under appropriate scaling, the process converges to the general Brownian motion in the $J_1$-Skorokhod topology. The main novelty of our approach lies in a result on the asymptotic behavior of the local time of the boundary random walks, allowing us to derive a CLT result for several Brownian-type limit processes on the half-line.

math.PR↗