arXiv subjects
Publications and source records attributed to Pablo Groisman.
Motivated by tumor growth models we establish conditions for the $R-$positivity of Markov processes and positive matrices. We then apply them to obtain the asymptotic behavior of the tumors sizes in the supercritical regime.
We study a particle system with the following diffusion-branching-selection mechanism. Particles perform independent one dimensional Brownian motions and on top of that, at a constant rate, a pair of particles is chosen uniformly at random and both particles adopt the position of the rightmost one among them. We show that the cumulative distribution function of the empirical measure converges to a solution of the Fisher-Kolmogorov-Petrovskii-Piskunov (F-KPP) equation and use this fact to prove that the system selects the minimal macroscopic speed as the number of particles goes to infinity.
We introduce a model for the spatial distribution of cell phone antennas in a urban environment. After showing that the complete spatial randomness (homogeneous Poisson distribution) hypothesis does not hold, we propose a model in which each point is distributed according to a bivariate Gaussian variable with mean given by the barycenter of its neighbors in the Delaunay triangulation. We show that this model is suitable, and can be used to generate a synthetic distribution of antennas. The generated distribution contains no sensitive or proprietary information, and can thus be freely shared with research groups, fostering further research on the subject.
We jointly investigate the existence of quasi-stationary distributions for one dimensional Lévy processes and the existence of traveling waves for the Fisher-Kolmogorov-Petrovskii-Piskunov (F-KPP) equation associated with the same motion. Using probabilistic ideas developed by S. Harris, we show that the existence of a traveling wave for the F-KPP equation associated with a centered Lévy processes that branches at rate $r$ and travels at velocity $c$ is equivalent to the existence of a quasi-stationary distribution for a Lévy process with the same movement but drifted by $-c$ and killed at zero, with mean absorption time $1/r$. This also extends the known existence conditions in both contexts. As it is discussed in a companion article, this is not just a coincidence but the consequence of a relation between these two phenomena.
The subcritical contact process seen from the rightmost infected site has no invariant measures. We prove that nevertheless it converges in distribution to a quasi-stationary measure supported on finite configurations.
For a general class of gas models ---which includes discrete and continuous Gibbsian models as well as contour or polymer ensembles--- we determine a \emph{diluteness condition} that implies: (1) Uniqueness of the infinite-volume equilibrium measure; (2) stability of this measure under perturbations of parameters and discretization schemes, and (3) existence of a coupled perfect-simulation scheme for the infinite-volume measure together with its perturbations and discretizations. Some of these results have previously been obtained through methods based on cluster expansions. In contrast, our treatment is purely probabilistic and its diluteness condition is weaker than existing convergence conditions for cluster expansions.
We study small random perturbations by additive space-time white noise of a reaction-diffusion equation with a unique stable equilibrium and solutions which blow up in finite time. We show that for initial data in the domain of attraction of the stable equilibrium the perturbed system exhibits metastable behavior: its time averages remain stable around this equilibrium until an abrupt and unpredictable transition occurs which leads to explosion in a finite (but exponentially large) time. On the other hand, for initial data in the domain of explosion we show that the explosion time of the perturbed system converges to the explosion time of the deterministic solution.
We consider Gibbs distributions on the set of permutations of $\mathbb Z^d$ associated to the Hamiltonian $H(σ):=\sum_{x} V(σ(x)-x)$, where $σ$ is a permutation and $V:\mathbb Z^d\to\mathbb R$ is a strictly convex potential. Call finite-cycle those permutations composed by finite cycles only. We give conditions on $V$ ensuring that for large enough temperature $α>0$ there exists a unique infinite volume ergodic Gibbs measure $μ^α$ concentrating mass on finite-cycle permutations; this measure is equal to the thermodynamic limit of the specifications with identity boundary conditions. We construct $μ^α$ as the unique invariant measure of a Markov process on the set of finite-cycle permutations that can be seen as a loss-network, a continuous-time birth and death process of cycles interacting by exclusion, an approach proposed by Fernández, Ferrari and Garcia. Define $τ_v$ as the shift permutation $τ_v(x)=x+v$. In the Gaussian case $V=\|\cdot\|^2$, we show that for each $v\in\mathbb Z^d$, $μ^α_v$ given by $μ^α_v(f)=μ^α[f(τ_v\cdot)]$ is an ergodic Gibbs measure equal to the thermodynamic limit of the specifications with $τ_v$ boundary conditions. For a general potential $V$, we prove the existence of Gibbs measures $μ^α_v$ when $α$ is bigger than some $v$-dependent value.
We analyze the connection between selection principles in front propagation and quasi-stationary distributions. We describe the missing link through the microscopic models known as Branching Brownian Motion with selection and Fleming-Viot.
Consider a system of independent random walks in the discrete torus with creation-annihilation of particles and possible explosion of the total number of particles in finite time. Rescaling space and rates for diffusion/creation/annihilation of particles, we obtain a stong law of large numbers for the density of particles in the supremum norm. The limiting object is a classical solution to the semilinear heat equation u_t =u_{xx} + f(u). If f(u)=u^p, 1<p \le 3, we also obtain a law of large numbers for the explosion time.
Quasi-stationary distributions (QSD) have been widely studied since the pioneering work of Kolmogorov (1938), Yaglom (1947) and Sevastyanov (1951). They appear as a natural object when considering Markov processes that are certainly absorbed since they are invariant for the evolution of the distribution of the process conditioned on not being absorbed. They hence appropriately describe the state of the process at large times for non absorbed paths. Unlike invariant distributions for Markov processes, QSD are solutions of a non-linear equation and there can be 0, 1 or an infinity of them. Also, they cannot be obtained as Cesàro limits of Markovian dynamics. These facts make the computation of QSDs a nontrivial matter. We review different approximation methods for QSD that are useful for simulation purposes, mainly focused on Fleming-Viot dynamics. We also give some alternative proofs and extensions of known results.
Consider N particles moving independently, each one according to a subcritical continuous-time Galton-Watson process unless it hits 0, at which time it jumps instantaneously to the position of one of the other particles chosen uniformly at random. The resulting dynamics is called Fleming-Viot process. We show that for each N there exists a unique invariant measure for the Fleming-Viot process, and that its stationary empirical distribution converges, as N goes to infinity, to the minimal quasi-stationary distribution of the Galton-Watson process conditioned on non-extinction.
We construct harmonic functions on random graphs given by Delaunay triangulations of ergodic point processes as the limit of the zero-temperature harness process.
We study small random perturbations by additive white-noise of a spatial discretization of a reaction-diffusion equation with a stable equilibrium and solutions that blow up in finite time. We prove that the perturbed system blows up with total probability and establish its order of magnitude and asymptotic distribution. For initial data in the domain of explosion we prove that the explosion time converges to the deterministic one while for initial data in the domain of attraction of the stable equilibrium we show that the system exhibits metastable behavior.
Consider a continuous time Markov chain with rates Q in the state space Λ\cup\{0\} with 0 as an absorbing state. In the associated Fleming-Viot process N particles evolve independently in Λwith rates Q until one of them attempts to jump to the absorbing state 0. At this moment the particle comes back to Λinstantaneously, by jumping to one of the positions of the other particles, chosen uniformly at random. When Λis finite, we show that the empirical distribution of the particles at a fixed time converges as N\to\infty to the distribution of a single particle at the same time conditioned on non absorption. Furthermore, the empirical profile of the unique invariant measure for the Fleming-Viot process with N particles converges as N\to\infty to the unique quasi-stationary distribution of the one-particle motion. A key element of the approach is to show that the two-particle correlations is of order 1/N.
We prove that perturbing the reaction--diffusion equation $u_t=u_{xx} + (u_+)^p$ ($p>1$), with time--space white noise produces that solutions explodes with probability one for every initial datum, opposite to the deterministic model where a positive stationary solution exists.
The equation $u_t = Δu + u^p$ with homegeneous Dirichlet boundary conditions has solutions with blow-up if $p > 1$. An adaptive time-step procedure is given to reproduce the asymptotic behvior of the solutions in the numerical approximations. We prove that the numerical method reproduces the blow-up cases, the blow-up rate and the blow-up time. We also localize the numerical blow-up set.