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Pablo Blanc

Publications and source records attributed to Pablo Blanc.

18 recordsLinked to original sources
arXiv2026

Slot decomposition of continuous Box-Ball Systems

We study a piecewise constant function $\eta:\mathbb R\to\{-1,1\}$ with a finite number of discontinuities in any interval. We assume that the associated walk $\xi:\mathbb R\to\mathbb R$ satisfying $\xi'(x)=\eta(x)$, pinned by $\xi(0)=0$, has finite length excursions over past minima. This is the continuous generalization of an initial ball configuration in the discrete Box Ball System introduced by Takahashi and Satsuma, where solitons of integer sizes $k\ge1$ are identified. We extend the slot decomposition developed by Ferrari, Nguyen, Rolla and Wang in the discrete setting to the continuous case. Each soliton of $\xi$ is represented by a point in two dimensional space, one coordinate for position and the other for the soliton height, mapping $\xi$ to a point configuration. We consider a distribution on walks given by a product measure on the decomposition of the path into excursions over past minima. Excursions are distributed as products of their solitons weights, which are determined by the soliton heights. We show that when the weight function is in $L^1$ the slot decomposition of $\xi$ is a Poisson process. This extends to the continuous case an approach of Ferrari and Gabrielli. As an example, we compute the intensity measure of the Poisson process associated to the asymmetric telegraph process introduced by Kac. In a forthcoming paper we discuss the dynamic properties.

math.PR↗
arXiv2024

Krylov-Safonov theory for Pucci-type extremal inequalities on random data clouds

We establish Krylov-Safonov type H\"older regularity theory for solutions to quite general discrete dynamic programming equations or equivalently discrete stochastic processes on random geometric graphs. Such graphs arise for example from data clouds in graph-based machine learning. The results actually hold to functions satisfying Pucci-type extremal inequalities, and thus we cover many examples including tug-of-war games on random geometric graphs. As an application we show that under suitable assumptions when the number of data points increases, the graph functions converge to a solution of a partial differential equation.

math.AP↗
arXiv2023

Game-theoretic approach to Hölder regularity for PDEs involving eigenvalues of the Hessian

We prove a local Hölder estimate with an exponent $0<δ<\frac 12$ for solutions of the dynamic programming principle $$u^\varepsilon (x) =\sum_{j=1}^n α_j\inf_{\dim(S)=j}\sup_{\substack{v\in S\\ |v|=1}}\frac{ u^\varepsilon (x + \varepsilon v) + u^\varepsilon (x - \varepsilon v)}{2}.$$ The proof is based on a new coupling idea from game theory. As an application, we get the same regularity estimate for viscosity solutions of the PDE $$\sum_{i=1}^n α_iλ_i(D^2u)=0,$$ where $λ_1(D^2 u)\leq\cdots\leq λ_n(D^2 u)$ are the eigenvalues of the Hessian.

math.AP↗
arXiv2023

A bridge between convexity and quasiconvexity

We introduce a notion of convexity with respect to a one-dimensional operator and with this notion find a one-parameter family of different convexities that interpolates between classical convexity and quasiconvexity. We show that, for this interpolation family, the convex envelope of a continuous boundary datum in a strictly convex domain is continuous up to the boundary and is characterized as being the unique viscosity solution to the Dirichlet problem in the domain for a certain fully nonlinear partial differential equation that involves the associated operator. In addition we prove that the convex envelopes of a boundary datum constitute a one-parameter curve of functions that goes from the quasiconvex envelope to the convex envelope being continuous with respect to uniform convergence. Finally, we also show some regularity results for the convex envelopes proving that there is an analogous to a supporting hyperplane at every point and that convex envelopes are $C^1$ if the boundary data satisfies in particular $NV$-condition we introduce.

math.AP↗
arXiv2022

Hölder regularity for stochastic processes with bounded and measurable increments

We obtain an asymptotic Hölder estimate for expectations of a quite general class of discrete stochastic processes. Such expectations can also be described as solutions to a dynamic programming principle or as solutions to discretized PDEs. The result, which is also generalized to functions satisfying Pucci-type inequalities for discrete extremal operators, is a counterpart to the Krylov-Safonov regularity result in PDEs. However, the discrete step size $\varepsilon$ has some crucial effects compared to the PDE setting. The proof combines analytic and probabilistic arguments.

math.AP↗
arXiv2022

Local regularity estimates for general discrete dynamic programming equations

We obtain an analytic proof for asymptotic Hölder estimate and Harnack's inequality for solutions to a discrete dynamic programming equation. The results also generalize to functions satisfying Pucci-type inequalities for discrete extremal operators. Thus the results cover a quite general class of equations.

math.AP↗
arXiv2021

Asymptotic Mean-Value Formulas for Solutions of General Second-Order Elliptic Equations

We obtain asymptotic mean-value formulas for solutions of second-order elliptic equations. Our approach is very flexible and allows us to consider several families of operators obtained as an infimum, a supremum, or a combination of both infimum and supremum, of linear operators. The families of equations that we consider include well-known operators such as Pucci, Issacs, and $k$-Hessian operators.

math.AP↗
arXiv2021

Asymptotic mean value formulas for parabolic nonlinear equations

In this paper we characterize viscosity solutions to nonlinear parabolic equations (including parabolic Monge-Ampère equations) by asymptotic mean value formulas. Our asymptotic mean value formulas can be interpreted from a probabilistic point of view in terms of Dynamic Programming Principles for certain two-player, zero-sum games.

math.AP↗
arXiv2020

A Nonlinear Mean Value Property for Monge-Ampère

In recent years there has been an increasing interest in whether a mean value property, known to characterize harmonic functions, can be extended in some weak form to solutions of nonlinear equations. This question has been partially motivated by the surprising connection between Random Tug-of-War games and the normalized $p-$Laplacian discovered some years ago, where a nonlinear asymptotic mean value property for solutions of a PDE is related to a dynamic programming principle for an appropriate game. Currently, asymptotic nonlinear mean value formulas are rare in the literature and our goal is to show that an asymptotic nonlinear mean value formula holds for the classical Monge-Ampère equation.

math.AP↗
arXiv2019

A lower bound for the principal eigenvalue of fully nonlinear elliptic operators

In this article we present a new technique to obtain a lower bound for the principal Dirichlet eigenvalue of a fully nonlinear elliptic operator. We ilustrate the construction of an appropriate radial function required to obtain the bound in several examples. In particular we use our results to prove that $\lim_{p\to \infty}λ_{1,p}=λ_{1,\infty}=\left(\fracπ{2R}\right)^2$ where $λ_{1,p}$ and $λ_{1,\infty}$ are the principal eigenvalue for the homogeneous $p$-laplacian and the homogeneous infinity laplacian respectively.

math.AP↗
arXiv2019

Inference of Demographic Attributes based on Mobile Phone Usage Patterns and Social Network Topology

Mobile phone usage provides a wealth of information, which can be used to better understand the demographic structure of a population. In this paper, we focus on the population of Mexican mobile phone users. We first present an observational study of mobile phone usage according to gender and age groups. We are able to detect significant differences in phone usage among different subgroups of the population. We then study the performance of different machine learning (ML) methods to predict demographic features (namely, age and gender) of unlabeled users by leveraging individual calling patterns, as well as the structure of the communication graph. We show how a specific implementation of a diffusion model, harnessing the graph structure, has significantly better performance over other node-based standard ML methods. We provide details of the methodology together with an analysis of the robustness of our results to changes in the model parameters. Furthermore, by carefully examining the topological relations of the training nodes (seed nodes) to the rest of the nodes in the network, we find topological metrics which have a direct influence on the performance of the algorithm.

cs.SI↗
arXiv2019

The evolution problem associated with eigenvalues of the Hessian

In this paper we study the evolution problem \[ \left\lbrace\begin{array}{ll} u_t (x,t)- λ_j(D^2 u(x,t)) = 0, & \text{in } Ω\times (0,+\infty), \\ u(x,t) = g(x,t), & \text{on } \partial Ω\times (0,+\infty), \\ u(x,0) = u_0(x), & \text{in } Ω, \end{array}\right. \] where $Ω$ is a bounded domain in $\mathbb{R}^N$ (that verifies a suitable geometric condition on its boundary) and $λ_j(D^2 u)$ stands for the $j-$st eigenvalue of the Hessian matrix $D^2u$. We assume that $u_0 $ and $g$ are continuous functions with the compatibility condition $u_0(x) = g(x,0)$, $x\in \partial Ω$. We show that the (unique) solution to this problem exists in the viscosity sense and can be approximated by the value function of a two-player zero-sum game as the parameter measuring the size of the step that we move in each round of the game goes to zero. In addition, when the boundary datum is independent of time, $g(x,t) =g(x)$, we show that viscosity solutions to this evolution problem stabilize and converge exponentially fast to the unique stationary solution as $t\to \infty$. For $j=1$ the limit profile is just the convex envelope inside $Ω$ of the boundary datum $g$, while for $j=N$ it is the concave envelope. We obtain this result with two different techniques: with PDE tools and and with game theoretical arguments. Moreover, in some special cases (for affine boundary data) we can show that solutions coincide with the stationary solution in finite time (that depends only on $Ω$ and not on the initial condition $u_0$).

math.AP↗
arXiv2018

Games for Pucci's maximal operators

In this paper we introduce a game whose value functions converge (as a parameter that measures the size of the steps goes to zero) uniformly to solutions to the second order Pucci maximal operators.

math.AP↗
arXiv2018

Games for eigenvalues of the Hessian and concave/convex envelopes

We study the PDE $λ_j(D^2 u) = 0$, in $Ω$, with $u=g$, on $\partial Ω$. Here $λ_1(D^2 u) \leq ... \leq λ_N (D^2 u)$ are the ordered eigenvalues of the Hessian $D^2 u$. First, we show a geometric interpretation of the viscosity solutions to the problem in terms of convex/concave envelopes over affine spaces of dimension $j$. In one of our main results, we give necessary and sufficient conditions on the domain so that the problem has a continuous solution for every continuous datum $g$. Next, we introduce a two-player zero-sum game whose values approximate solutions to this PDE problem. In addition, we show an asymptotic mean value characterization for the solution the the PDE.

math.AP↗
arXiv2017

A limiting free boundary problem with gradient constraint and Tug-of-War games

In this manuscript we deal with regularity issues and the asymptotic behaviour (as $p \to \infty$) of solutions for elliptic free boundary problems of $p-$Laplacian type ($2 \leq p< \infty$): \begin{equation*} -Δ_p u(x) + λ_0(x)χ_{\{u>0\}}(x) = 0 \quad \mbox{in} \quad Ω\subset \mathbb{R}^N, \end{equation*} with a prescribed Dirichlet boundary data, where $λ_0>0$ is a bounded function and $Ω$ is a regular domain. First, we prove the convergence as $p\to \infty$ of any family of solutions $(u_p)_{p\geq 2}$, as well as we obtain the corresponding limit operator (in non-divergence form) ruling the limit equation, $$ \left\{ \begin{array}{rcrcl} \max\left\{-Δ_{\infty} u_{\infty}, \,\, -|\nabla u_{\infty}| + χ_{\{u_{\infty}>0\}}\right\} & = & 0 & \text{in} & Ω\cap \{u_{\infty} \geq 0\} \\ u_{\infty} & = & g & \text{on} & \partial Ω. \end{array} \right. $$ Next, we obtain uniqueness for solutions to this limit problem together with a number of weak geometric and measure theoretical properties as non-degeneracy, uniform positive density, porosity and convergence of the free boundaries. Finally, we show that any solution to the limit operator is a limit of value functions for a specific Tug-of-War game.

math.AP↗
arXiv2017

Secretary Problem with quality-based payoff

We consider a variant of the classical Secretary Problem. In this setting, the candidates are ranked according to some exchangeable random variable and the quest is to maximize the expected quality of the chosen aspirant. We find an upper bound for the optimal hiring rule, present examples showing it is sharp, and recover the classical case, among other results.

math.PR↗
arXiv2015

A Study of Age and Gender seen through Mobile Phone Usage Patterns in Mexico

Mobile phone usage provides a wealth of information, which can be used to better understand the demographic structure of a population. In this paper we focus on the population of Mexican mobile phone users. Our first contribution is an observational study of mobile phone usage according to gender and age groups. We were able to detect significant differences in phone usage among different subgroups of the population. Our second contribution is to provide a novel methodology to predict demographic features (namely age and gender) of unlabeled users by leveraging individual calling patterns, as well as the structure of the communication graph. We provide details of the methodology and show experimental results on a real world dataset that involves millions of users.

cs.SI↗