arXiv subjects

Conrado da Costa

Publications and source records attributed to Conrado da Costa.

12 recordsLinked to original sources

arXiv2026

Superdiffusive limits for stochastic kinetics driven by self-similar drifts

We prove anomalous-diffusion scaling for a one-dimensional stochastic kinetic dynamics, in which the stochastic drift is driven by an exogenous self-similar noise, and also includes endogenous volatility which is permitted to have arbitrary dependence with the exogenous noise. We identify the superdiffusive scaling exponent for the model, and prove strong and weak convergence results on the corresponding scale. Our framework admits self-similar noise that is either a Bessel process, or, more generally, a self-similar continuous-state branching process with immigration, as well as more general processes satisfying certain asymptotic conditions.

math.PR↗

arXiv2026

A cohesive account on the ergodic behaviours and scaling limits of Random Walks in Cooling Random Environments

Transport in disordered media is a central theme in probability and statistical physics, where randomness in the underlying medium produces phenomena such as localization, anomalous scaling, and slow relaxation. A paradigmatic model for transport in disordered media is that of Random Walks in Random Environments (RWRE), which has been extensively studied since the 1970's and is by now well understood in one dimension. More recently, several works have explored perturbations of models of transport in disordered media aimed at interpolating between static disorder and fully homogenized dynamics. Random walks in cooling random environments (RWCRE), introduced in this context, constitute a key example: the environment is dynamically resampled at prescribed times and kept fixed in between, giving rise to a delicate ``quasi-ergodic'' structure in time allowing to interpolate between homogeneous random walks and classical RWRE. The purpose of this paper is twofold. A first goal is to offer an original survey on the main results for RWCRE in 1d: recurrence criteria, law of large numbers, large deviations, and fluctuation phenomena across different resampling regimes. Those results have been derived in a series of recent works and are complemented here with a number of new statements aiming at presenting a unified phenomenological picture. As a second goal, we try to extract a coherent conceptual picture that highlights structural mechanisms -- such as ergodic limits, persistence under perturbations and replacement principles -- that extend beyond the specific setting of RWCRE and are relevant for a broader class of disordered systems that can be perturbed by introducing independent resetting in the same fashion.

math.PR↗

arXiv2026

Long-range one-dimensional internal diffusion-limited aggregation

We study internal diffusion limited aggregation on $\mathbb{Z}$, where a cluster is grown incrementally by adding, for each random walk dispatched from the origin, the first site it reaches outside the cluster. We assume that the increment distribution $X$ of the driving random walks has $\mathbb{E} X =0$, but need neither be simple nor symmetric, and can have $\mathbb{E} (X^2) = \infty$, for example. For the case where $\mathbb{E} (X^2) < \infty$, we prove that after $m$ of the random walks have been dispatched, all but $o(m)$ sites in the cluster form an approximately symmetric contiguous block around the origin. This strengthens a result of Blachère, for centred random walks whose increments have finite $3$rd moments, to the optimal moments condition. On the other hand, if $X$ is in the domain of attraction of a symmetric $α$-stable law, $1 < α<2$, we prove that the cluster contains a contiguous block of $δm +o(m)$ sites, where $0 < δ< 1$, but, unlike the finite-variance case, one may not take $δ=1$.

math.PR↗

arXiv2025

Passage-times for partially-homogeneous reflected random walks on the quadrant

We consider a random walk on the first quadrant of the square lattice, whose increment law is, roughly speaking, homogeneous along a finite number of half-lines near each of the two boundaries, and hence essentially specified by finitely-many transition laws near each boundary, together with an interior transition law that applies at sufficient distance from both boundaries. Under mild assumptions, in the (most subtle) setting in which the mean drift in the interior is zero, we classify recurrence and transience and provide power-law bounds on tails of passage times; the classification depends on the interior covariance matrix, the (finitely many) drifts near the boundaries, and stationary distributions derived from two one-dimensional Markov chains associated to each of the two boundaries. As an application, we consider reflected random walks related to multidimensional variants of the Lindley process, for which the recurrence question was studied recently by Peigné and Woess (Ann. Appl. Probab., vol. 31, 2021) using different methods, but for which no previous quantitative results on passage-times appear to be known.

math.PR↗

arXiv2025

Limiting distributions for RWCRE in the sub-ballistic regime and in the critical Gaussian regime

Random Walks in Cooling Random Environments (RWCRE) is a model of random walks in dynamic random environments where the environment is frozen between a fixed sequence of times (called the cooling map) where it is resampled. Naturally the limiting distributions for this model depend both on the structure of the cooling sequence and on distribution $μ$ from which the environments are sampled. Previous results have considered the cases where $μ$ is such that the corresponding model of random walks in a fixed random environment (RWRE) is either (1) recurrent, (2) has a Gaussian limit with diffusive scaling (the $κ> 2$ case), or (3) has positive speed and a stable, non-Gaussian limit (the $κ\in (1,2)$ case). In this paper we examine the limiting distributions in two other transient regimes: the sub-ballistic, non-stable regime (i.e., $κ\in (0,1)$), and the Gaussian regime with non-diffusive scaling (i.e., $κ= 2$). In the first case we show that the limiting distributions are either Gaussian or a mixture of Gaussian and independent sums of Mittag-Leffler random variables, while in the second case the limiting distributions are always Gaussian but with a scaling that differs from the standard deviation by factor (which can oscillate, but which remains confined to some interval $[β,1]$) that depends very delicately on the properties of the cooling map.

math.PR↗

arXiv2024

Gradual convergence for Langevin dynamics on a degenerate potential

In this paper, we study an ordinary differential equation with a degenerate global attractor at the origin, to which we add a white noise with a small parameter that regulates its intensity. Under general conditions, for any fixed intensity, as time tends to infinity, the solution of this stochastic dynamics converges exponentially fast in total variation distance to a unique equilibrium distribution. We suitably accelerate the random dynamics and show that the preceding convergence is gradual, that is, the function that associates to each fixed $t\geq 0$ the total variation distance between the accelerated random dynamics at time $t$ and its equilibrium distribution converges, as the noise intensity tends to zero, to a decreasing function with values in $(0,1)$. Moreover, we prove that this limit function for each fixed $t \geq 0$ corresponds to the total variation distance between the marginal, at time $t$, of a stochastic differential equation that comes down from infinity and its corresponding equilibrium distribution. This completes the classification of all possible behaviors of the total variation distance between the time marginal of the aforementioned stochastic dynamics and its invariant measure for one dimensional well-behaved convex potentials. In addition, there is no cut-off phenomenon for this one-parameter family of random processes and asymptotics of the mixing times are derived.

math.PR↗

arXiv2024

Superdiffusive planar random walks with polynomial space-time drifts

We quantify superdiffusive transience for a two-dimensional random walk in which the vertical coordinate is a martingale and the horizontal coordinate has a positive drift that is a polynomial function of the individual coordinates and of the present time. We describe how the model was motivated through an heuristic connection to a self-interacting, planar random walk which interacts with its own centre of mass via an excluded-volume mechanism, and is conjectured to be superdiffusive with a scale exponent $3/4$. The self-interacting process originated in discussions with Francis Comets.

math.PR↗

arXiv2023

Stochastic billiards with Markovian reflections in generalized parabolic domains

We study recurrence and transience for a particle that moves at constant velocity in the interior of an unbounded planar domain, with random reflections at the boundary governed by a Markov kernel producing outgoing angles from incoming angles. Our domains have a single unbounded direction and sublinear growth. We characterize recurrence in terms of the reflection kernel and growth rate of the domain. The results are obtained by transforming the stochastic billiards model to a Markov chain on a half-strip $\mathbb{R}_+ \!\times S$ where $S$ is a compact set. We develop the recurrence classification for such processes in the near-critical regime in which drifts of the $\mathbb{R}_+$ component are of generalized Lamperti type, and the $S$ component is asymptotically Markov; this extends earlier work that dealt with finite $S$.

math.PR↗

arXiv2021

Gaussian, stable, tempered stable and mixed limit laws for random walks in cooling random environments

Random Walks in Cooling Random Environments (RWCRE) is a model of random walks in dynamic random environments where the entire environment is resampled along a fixed sequence of times, called the "cooling sequence," and is kept fixed in between those times. This model interpolates between that of a homogenous random walk, where the environment is reset at every step, and Random Walks in (static) Random Environments (RWRE), where the environment is never resampled. In this work we focus on the limiting distributions of one-dimensional RWCRE in the regime where the fluctuations of the corresponding (static) RWRE is given by a $s$-stable random variable with $s\in(1,2)$. In this regime, due to the two extreme cases (resampling every step and never resampling, respectively), a crossover from Gaussian to stable limits for sufficiently regular cooling sequence was previously conjectured. Our first result answers affirmatively this conjecture by making clear critical exponent, norming sequences and limiting laws associated with the crossover which demonstrates a change from Gaussian to $s$-stable limits, passing at criticality through a certain generalized tempered stable distribution. We then explore the resulting RWCRE scaling limits for general cooling sequences. On the one hand, we offer sets of operative sufficient conditions that guarantee asymptotic emergence of either Gaussian, $s$-stable or generalized tempered distributions from a certain class. On the other hand, we give explicit examples and describe how to construct irregular cooling sequences for which the corresponding limit law is characterized by mixtures of the three above mentioned laws. To obtain these results, we need and derive a number of refined asymptotic results for the static RWRE with $s\in(1,2)$ which may be of independent interest.

math.PR↗

arXiv2021

Reaction-diffusion models for a class of infinite-dimensional non-linear stochastic differential equations

We establish the existence of solutions to a class of non-linear stochastic differential equation of reaction-diffusion type in an infinite-dimensional space, with diffusion corresponding to a given transition kernel. The solution obtained is the scaling limit of a sequence of interacting particle systems, and satisfies the martingale problem corresponding to the target differential equation.

math.PR↗

arXiv2021

Laws of large numbers for weighted sums of independent random variables: a game of mass

We consider weighted sums of independent random variables regulated by an increment sequence. We provide operative conditions that ensure strong law of large numbers for such sums to hold in both the centered and non-centered case. The existing criteria for the strong law are either implicit or assume some sufficient decay for the sequence of coefficients. In our set up we allow for arbitrary sequence of coefficients, possibly random, provided the random variables regulated by such increments satisfy some mild concentration conditions. In the non-centered case, convergence can be translated into the behavior of a deterministic sequence and it becomes a game of mass provided the expectation of the random variables is a function of the increments. We show how different limiting scenarios can emerge by identifying several classes of increments, for which concrete examples will be offered.

math.PR↗

arXiv2019

Random walk in cooling random environment: recurrence versus transience and mixed fluctuations

This is the third in a series of papers in which we consider one-dimensional Random Walk in Cooling Random Environment (RWCRE). The latter is obtained by starting from one-dimensional Random Walk in Random Environment (RWRE) and resampling the environment along a sequence of deterministic times, called refreshing times. In the present paper we explore two questions for general refreshing times. First, we investigate how the recurrence versus transience criterion known for RWRE changes for RWCRE. Second, we explore the fluctuations for RWCRE when RWRE is either recurrent or satisfies a classical central limit theorem. We show that the answer depends in a delicate way on the choice of the refreshing times. An overarching goal of our paper is to investigate how the behaviour of a random process with a rich correlation structure can be affected by resettings.

math.PR↗