arXiv subjects
Publications and source records attributed to Glauco Valle.
This work investigates the spectral and dynamical properties of transition operators associated with a class of stochastic adding machines. In particular, we characterize the spectra of these operators in terms of non-autonomous Julia sets. This description, in turn, provides new information about their dynamical behavior, including criteria for supercyclicity and hypercyclicity.
We introduce a generalized amnesic elephant random walk in which, at each time, the direction of an underlying amnesic elephant random walk selects one of two increment distributions, while the actual increment is sampled from the corresponding independent sequence. The selected distribution does not need to determine the sign of the increment. This construction combines memory reinforcement, amnesia, and randomness in jump sizes. Assuming finite second moments, we establish central limit theorems and functional limit theorems, identifying diffusive, critical, and superdiffusive regimes determined by the memory and amnesia parameters. We further prove a Gaussian fluctuation theorem around the random superdiffusive limit, with an explicit limiting variance that captures the contributions of both the underlying amnesic walk and the random jump magnitudes. For the special case without amnesia, we also establish stable limit theorems and their functional counterparts when the increments belong to the domain of attraction of a non-Gaussian stable law, in the regime where heavy-tailed jumps dominate the memory contribution.
Let $(S_n)_{n\geq 1}$ be the one-dimensional Elephant Random Walk (ERW) in the superdiffusive regime, with memory parameter $p\in(3/4,1)$ and first-step bias $\mathbb{P}(S_1=1)=q$. Set $a=2p-1\in(1/2,1)$. If $f_{q,a}$ denotes the density of the superdiffusive limit $L_{q,a}$, we prove the uniform local limit theorem $\lim_{n\to\infty}\sup_{j\in\mathbb{Z},\,j\equiv n\;(\mathrm{mod}\;2)}\left|\frac{n^a}{2}\mathbb{P}(S_n=j)-f_{q,a}(j/n^a)\right|=0$. The proof uses the exact recurrence for the probability mass function of $S_n$, yielding uniform $O(n^{-a})$ and $O(n^{-2a})$ bounds for the p.m.f. and its first discrete differences, respectively. We then disprove a conjecture of Guérin, Laulin, Raschel and Simon concerning eventual log-concavity. Writing $L_a=L_{1,a}$ and $f_a=f_{1,a}$, let $a_{\rm ev}$ be the eventual-log-concavity threshold and let $a_\star$ be the upper failure threshold for log-concavity of $f_a$. Their results imply $a_{\rm ev}\geq(\sqrt{5}-1)/2$, and we prove $(\sqrt{5}-1)/2\leq a_{\rm ev}\leq\min\{0.80399,a_\star\}\leq a_\star<0.918$. The bound $a_\star<0.918$ follows from the first three exact moments of $L_a$ and the sharp skewness inequality for centered log-concave distributions, while $a_{\rm ev}\leq0.80399$ follows from a certified finite-order analysis of the exact p.m.f. recurrence. Direct numerical iterations provide evidence that $a_{\rm ev}\approx0.80399$, equivalently $p_{\rm ev}\approx0.902$.
Sampling in score-based diffusion models can be performed by solving either a reverse-time stochastic differential equation (SDE) parameterized by an arbitrary stochasticity function or a probability flow ODE, corresponding to setting this stochasticity function to zero. In this work, we investigate the effect of this stochasticity on the generation process through the evolution of Kullback-Leibler (KL) divergences, obtaining general KL divergence bounds and a novel analysis of the impact of the time-profile of the score error on model performance. For exact score functions, stochasticity has a contractive effect, decreasing KL divergence along the sampling trajectory. For approximate scores, however, a trade-off arises between correcting accumulated errors and amplifying current score errors, meaning stochasticity can either improve or degrade generation performance. Theoretical considerations indicate that the gain from stochasticity depends on the time-localization of the trained model error. We test this in experiments on both toy and benchmark data sets, also comparing the KL divergence evolution with the obtained bounds. We also present a fully analytical example, where all the relevant quantities can be computed, and the optimal stochasticity function can be characterized via an optimal control analysis.
We study a variant of the Generalized Excited Random Walk (GERW) on $\mathbb{Z}^d$ introduced by Menshikov, Popov, Ramírez and Vachkovskaia in [Ann. Probab. 40 (5), 2012]. It consists of a particular version of the model studied in [arXiv preprint arXiv:2211.05715, 2022] where excitation may or may not occur according to a time-dependent probability. Specifically, given $\{p_n\}_{n \ge 1}$, $p_n \in (0, 1]$ for all $n \ge 1$, whenever the process visits a site at time $n$ for the first time, with probability $p_n$ it gains a drift in a fixed direction. Otherwise, it behaves as a $d$-martingale with zero-mean vector. We refer to the model as $p_n$-GERW. Assuming bounded jumps and $p_n \approx n^{-β}$, we show a series of results for the $p_n$-\Name{} depending on the value of $β$ and on the dimension $d$. Specifically, for every $β\in(0,1]$ and $d=2$ or $d>h(β)$, with $h$ a decreasing function of $β$, we prove a SLLN for the range, while for $β<1/2$ we prove a sub-ballistic SLLN for the process whenever the SLLN for the range holds. We also study the $p_n$-\Name{} under diffusive scaling, and we obtain a Functional Central Limit Theorem for $β> 1/2$ and $d\geq 2$, or $β=1/2$ and $d=2$. Finally, for $β=1/2$ and $d \ge 11$ we show that the diffusively rescaled $p_n$-\Name{} converges in distribution to a Brownian Motion plus a multiple of the square root of time.
We study a multiscale stochastic optimal control problem subject to state constraints on the slow variable. To address this class of problems, we develop a rigorous theoretical framework based on singular perturbation analysis, tailored to settings with constrained dynamics. Our approach relies on the theory of viscosity solutions for degenerate Hamilton-Jacobi-Bellman equations with Neumann-type boundary conditions. We also establish the convergence of the multiscale value functions in the infinite-horizon regime. Finally, we present two illustrative examples that highlight the applicability and effectiveness of the proposed framework.
We study a stochastic optimal control problem with the state constrained to a smooth, compact domain. The control influences both the drift and a possibly degenerate, control-dependent dispersion matrix, leading to a fully nonlinear, degenerate elliptic Hamilton--Jacobi--Bellman (HJB) equation with a nontrivial Neumann boundary condition. Although these features have been studied separately, this work provides the first unified treatment combining them all. We establish that the optimal value function associated with the control problem is the unique viscosity solution of the HJB equation with a nontrivial Neumann boundary condition, and we present an illustrative example demonstrating the applicability of the framework.
We study the behavior of the solution of a stochastic Allen-Cahn equation $\frac{\partial u_\eps }{\partial t}=\frac 12 \frac{\partial^2 u_\eps }{\partial x^2}+ u_\eps -u_\eps^3+\sqrt\eps\, \dot W$, with Dirichlet boundary conditions on a suitably large space interval $[-L_\eps , L_\eps]$, starting from the identically zero function, and where $\dot W$ is a space-time white noise. Our main goal is the description, in the small noise limit, of the onset of the phase separation, with the emergence of spatial regions where $u_\eps$ becomes close $1$ or $-1$. The time scale and the spatial structure are determined by a suitable Gaussian process that appears as the solution of the corresponding linearized equation. This issue has been initially examined by De Masi et al. [Ann. Probab. {\bf 22}, (1994), 334-371] in the related context of a class of reaction-diffusion models obtained as a superposition of a speeded up stirring process and a spin flip dynamics on $\{-1,1\}^{\mathbb{Z}_\eps}$, where $\mathbb{Z}_\eps=\mathbb{Z}$ modulo $\lfloor\eps^{-1}L_\eps\rfloor$.
The Drainage Network is a system of coalescing random walks, exhibiting long-range dependence before coalescence, introduced by Gangopadhyay, Roy, and Sarkar. Coletti, Fontes, and Dias proved its convergence to the Brownian Web under diffusive scaling. In this work, we introduce a perturbation of the system allowing branching of the random walks with low probabilities varying with the scaling parameter. When the branching probability is inversely proportional to the scaling parameter, we show that this drainage network with branching consists of a tight family such that any weak limit point contains a Brownian Net. We conjecture that the limit is indeed the Brownian Net.
We consider a variation of the Generalized Excited Random Walk (GERW) in dimension $d\ge 2$ where the lower bound on the drift for excited jumps is time-dependent and decays to zero. We show that if the lower bound decays slower that $n^{-β}$ ($n$ is time), for $β$ depending on the transitions of the process, the GERW is transient in the direction of the drift.
We study an extension of the generalized excited random walk (GERW) on $\mathbb{Z}^d$ introduced in [Ann. Probab. 40 (5), 2012, [7]] by Menshikov, Popov, Ramírez and Vachkovskaia. Our extension consists in studying a version of the GERW where excitation depends on a random environment. Given $p \in (0,1]$ (a parameter of the model) whenever the process visits a site for the first time, with probability $p$ it gains a drift in a given direction (could be any direction of the unit sphere). Otherwise, with probability $1-p$, it behaves as a $d$-martingale with zero-mean vector. Whenever the process visits an already-visited site, the process acts again as a $d$-martingale with zero-mean vector. We refer to the model as a GERW in Bernoulli environment, in short $p$-GERW. Under the same hypothesis of [7] (bounded jumps, uniform ellipticity), we show that the $p$-GERW is ballistic for all $p\in (0,1]$. Under the stronger assumptions that the increments of the regeneration times associated to the $p$-GERW are i.i.d. (condition which is satisfied, for example, for the excited random walk in a Bernoulli i.i.d. environment), we also obtain a Law of Large Numbers and a Central Limit Theorem.
Several authors have studied convergence in distribution to the Brownian web under diffusive scaling of Markovian random walks. In a paper by R. Roy, K. Saha and A. Sarkar, convergence to the Brownian web is proved for a system of coalescing random paths -- the Random Directed Forest -- which are not Markovian. Paths in the Random Directed Forest do not cross each other before coalescence. Here we study a generalization of the non-Markovian Random Directed Forest where paths can cross each other and prove convergence to the Brownian web. This provides an example of how the techniques to prove convergence to the Brownian web for systems allowing crossings can be applied to non-Markovian systems.
We prove ballistic behaviour as well as an annealed functional central limit theorem for random walks in mixing random environments (RWRE). The ballistic hypothesis will be an effective polynomial condition as the one introduced by Berger, Drewitz, and Ram\'ırez (\emph{Comm. Pure Appl. Math,} {\bf 67}, (2014) 1947--1973). The novel idea therein was the construction of several simultaneous renormalization steps, providing more flexibility for seed estimates. For our proof, we indeed follow a similar path, and introduce a new mixing effective criterion which will be implied by the polynomial condition. This allows us to prove, in a mixing framework, the RWRE conjecture concerning the equivalence between each condition $(T^γ)|\ell$, for $γ\in (0,1)$ and $\ell \in \mathbb S^{d-1}$. This work complements the previous work of Guerra (\emph{Ann. Probab.} {\bf 47} (2019) 3003--3054) and completes the answer about the meaning of condition $(T')|\ell$ in a mixing setting, an open question posed by Comets and Zeitouni (\emph{Ann. Probab.} {\bf 32} (2004) 880--914).
The Tree Builder Random Walk is a special random walk that evolves on trees whose size increases with time, randomly and depending upon the walker. After every s steps of the walker, a random number of vertices are added to the tree and attached to the current position of the walker. These processes share similarities with other important classes of markovian and non-markovian random walks presenting a large variety of behaviors according to parameters specifications. We show that for a large and most significant class of tree builder random walks, the process is either null recurrent or transient. If s is odd, the walker is ballistic and thus transient. If s is even, the walker's behavior can be explained from local properties of the growing tree and it can be either null recurrent or it gets trapped on some limited part of the growing tree.
In this paper, we define some Markov Chains associated to Vershik maps on Bratteli diagrams. We study probabilistic and spectral properties of their transition operators and we prove that the spectra of these operators are connected to Julia sets in higher dimensions. We also study topological properties of these spectra.
We provide a process on the space of coalescing cadlag stable paths and show convergence in the appropriate topology for coalescing stable random walks on the integer lattice.
In this paper, we study dynamical properties as hypercyclicity, supercyclicity, frequent hypercyclicity and chaoticity for transition operators associated to countable irreductible Markov chains. As particular cases, we consider simple random walks on Z and Z+.
We prove exponential convergence to equilibrium for the Fredrikson-Andersen one spin facilitated model on bounded degree graphs satisfying a subexponential, but larger than polynomial, growth condition. This was a classical conjecture related to non-attractive spin systems. Our proof rely on coupling techniques based on Harris graphical construction for interacting particle systems.