@article{grigorescu_kang_2023, title={Asymptotics and criticality for a space-dependent branching process}, ISSN={["1744-2516"]}, DOI={10.1080/17442508.2023.2256922}, abstractNote={AbstractWe investigate a non-conservative semigroup (St)t≥0 determined by a branching process tracing the evolution of particles moving in a domain in Rd. When a particle is killed at the boundary, a new generation of particles with mean number K¯ is born at a random point in the domain. Between branching, the particles are driven by a diffusion process with Dirichlet boundary conditions. According to the sign of K¯−1, we distinguish super/sub-critical regimes and determine the exact exponential rate for the total number of particles n(t)∼exp(α∗t), with α∗ depending explicitly on K¯. We prove the Yaglom limit St/n(t)→ν, where the quasi-stationary distribution ν is determined by the resolvent of the Dirichlet kernel at the point α∗. The main application is in particle systems, where the normalization of the semigroup by its total mass gives the hydrodynamic limit of the Bak-Sneppen branching diffusions (BSBD). Since ν is the asymptotic profile under equilibrium, and the family of quasi-stationary distributions ν is indexed by K¯, the model provides an explicit example of self-organized criticality.Keywords: SemigroupYaglom limitbranching processessupercriticalqsdBak-SneppenFleming-ViotDirichlet kernelKey Words and Phrases: Primary: 60J3560J80Secondary: 47D0760K35 Disclosure statementNo potential conflict of interest was reported by the author(s).}, journal={STOCHASTICS-AN INTERNATIONAL JOURNAL OF PROBABILITY AND STOCHASTIC PROCESSES}, author={Grigorescu, Ilie and Kang, Min}, year={2023}, month={Sep} }
@article{ma_kang_silversteint_baron_2021, title={Local Convergence of an AMP Variant to the LASSO Solution in Finite Dimensions}, DOI={10.1109/ISIT45174.2021.9518079}, abstractNote={A common sparse linear regression formulation is $\ell_{1}$ regularized least squares, which is also known as least absolute shrinkage and selection operator (LASSO). Approximate message passing (AMP) has been proved to asymptotically achieve the LASSO solution when the regression matrix has independent and identically distributed (i.i.d.) Gaussian entries in the sense that the averaged per-coordinate $\ell_{2}$ distance between the AMP iterates and LASSO solution vanishes as the signal dimension goes to infinity before the iteration number. However, in finite dimensional settings, characterization of AMP iterates in the limit of large iteration number has not been established. In this work, we propose an AMP variant by including a parameter that depends on the largest singular value of the regression matrix. The proposed algorithm can also be considered as a primal dual hybrid gradient algorithm with adaptive stepsizes. We show that whenever the AMP variant converges, it converges to the LASSO solution for arbitrary finite dimensional regression matrices. Moreover, we show that our AMP variant is locally stable around the LASSO solution under the condition that the LASSO solution is unique and that the regression matrix is drawn from a continuous distribution. Our local stability result implies that when the regression matrix is large and has i.i.d. random entries, the original AMP, which is a special case of the proposed AMP variant, is locally stable around the LASSO solution.}, journal={2021 IEEE INTERNATIONAL SYMPOSIUM ON INFORMATION THEORY (ISIT)}, author={Ma, Yanting and Kang, Min and Silversteint, Jack W. and Baron, Dror}, year={2021}, pages={2256–2261} }
@article{chen_grigorescu_kang_2015, title={Optimal stopping for Shepp's urn with risk aversion}, volume={87}, DOI={10.1080/17442508.2014.995660}, abstractNote={An (m,p) urn contains m balls of value − 1 and p balls of value +1. A player starts with fortune k and in each game draws a ball without replacement with the fortune increasing by one unit if the ball is positive and decreasing by one unit if the ball is negative, having to stop when k = 0 (risk aversion). Let V(m,p,k) be the expected value of the game. We are studying the question of the minimum k such that the net gain function of the game V(m,p,k) − k is positive, in both the discrete and the continuous (Brownian bridge) settings. Monotonicity in various parameters m, p, k is established for both the value and the net gain functions of the game. For the cut-off value k, since the case m − p < 0 is trivial, for p → ∞, either , when the gain function cannot be positive, or , when it is sufficient to have , where α is a constant. We also determine an approximate optimal strategy with exponentially small probability of failure in terms of p. The problem goes back to Shepp [8], who determined the constant α in the unrestricted case when the net gain does not depend on k. A new proof of his result is given in the continuous setting.}, number={4}, journal={Stochastics-An International Journal of Probability and Stochastic Processes}, author={Chen, R. W. and Grigorescu, I. and Kang, M.}, year={2015}, pages={702–722} }
@article{grigorescu_kang_2014, title={CRITICAL SCALE FOR A CONTINUOUS AIMD MODEL}, volume={30}, ISSN={["1532-4214"]}, DOI={10.1080/15326349.2014.932697}, abstractNote={A scaled version of the general AIMD model of transmission control protocol (TCP) used in Internet traffic congestion management leads to a Markov process x(t) representing the time dependent data flow that moves forward with constant speed on the positive axis and jumps backward to γx(t), 0 < γ < 1 according to a Poisson clock whose rate α(x) depends on the interval swept in between jumps. We give sharp conditions for Harris recurrence and analyze the convergence to equilibrium on multiple scales (polynomial, fractional exponential, exponential) identifying the critical case xα(x) ∼ β. Criticality has different behavior according to whether it occurs at the origin or infinity. In each case, we determine the transient (possibly explosive), null—and positive—recurrent regimes by comparing β to ( − ln γ)− 1.}, number={3}, journal={STOCHASTIC MODELS}, author={Grigorescu, Ilie and Kang, Min}, year={2014}, pages={319–343} }
@article{grigorescu_kang_2013, title={Fixation Time for an Evolutionary Model}, volume={29}, ISSN={["1532-6349"]}, DOI={10.1080/15326349.2013.808908}, abstractNote={We study the asymptotic value as L → ∞ of the time for evolution τ, understood as the first time to reach a preferred word of length L using an alphabet with N letters. The word is updated at unit time intervals randomly, but configurations with letters matching with the preferred word are sticky, i.e., the probability to leave the configuration equals 0 ≤ γ ≤1, where γ may depend on the configuration. The model is introduced in Ref.[ 5 ] in the case γ = 0, where it was shown that E[τ] ∼ Nln (L). We first give an alternative proof of the logarithmic scale, by evaluating the mode of τ. We then answer positively a question posed by H. Wilf on whether τ is exponential when γ ≠ 0. The natural scaling γ =O(L −1) gives rise to several finite order limits, including the interacting model when γ depends linearly on the number of matches with the preferred word. The scaling limit of the number of non-matching letters follows a Galton-Watson process with immigration. In a related model, the empirical measure converges to the solution of a discrete logistic equation with possible nonzero steady state. In conclusion, the length of τ is a question of scaling.}, number={3}, journal={STOCHASTIC MODELS}, author={Grigorescu, Ilie and Kang, Min}, year={2013}, month={Jul}, pages={328–340} }
@article{grigorescu_kang_2013, title={Markov processes with redistribution}, volume={19}, number={3}, journal={Markov Processes and Related Fields}, author={Grigorescu, I. and Kang, M.}, year={2013}, pages={497–520} }
@article{grigorescu_kang_2012, title={Immortal particle for a catalytic branching process}, volume={153}, ISSN={["1432-2064"]}, DOI={10.1007/s00440-011-0347-6}, abstractNote={We study the existence and asymptotic properties of a conservative branching particle system driven by a diffusion with smooth coefficients for which birth and death are triggered by contact with a set. Sufficient conditions for the process to be non-explosive are given. In the Brownian motions case the domain of evolution can be non-smooth, including Lipschitz, with integrable Martin kernel. The results are valid for an arbitrary number of particles and non-uniform redistribution after branching. Additionally, with probability one, it is shown that only one ancestry line survives. In special cases, the evolution of the surviving particle is studied and for a two particle system on a half line we derive explicitly the transition function of a chain representing the position at successive branching times.}, number={1-2}, journal={PROBABILITY THEORY AND RELATED FIELDS}, author={Grigorescu, Ilie and Kang, Min}, year={2012}, month={Jun}, pages={333–361} }
@article{grigorescu_kang_2010, title={Steady state and scaling limit for a traffic congestion model}, volume={14}, ISSN={1292-8100 1262-3318}, url={http://dx.doi.org/10.1051/ps:2008029}, DOI={10.1051/ps:2008029}, abstractNote={In a general model (AIMD) of transmission control protocol (TCP) used in internet traffic congestion management, the time dependent data flow vector x ( t ) > 0 undergoes a biased random walk on two distinct scales. The amount of data of each component x i ( t ) goes up to x i ( t )+ a with probability 1- ζ i ( x ) on a unit scale or down to γ x i ( t ), 0 γ ζ i ( x ) on a logarithmic scale, where ζ i depends on the joint state of the system x . We investigate the long time behavior, mean field limit, and the one particle case. According to c = lim inf | x |→∞ | x | ζ i ( x ) , the process drifts to ∞ in the subcritical c c + ( n , γ ) case and has an invariant probability measure in the supercritical case c > c + ( n , γ ). Additionally, a scaling limit is proved when ζ i ( x ) and a are of order N –1 and t → N t , in the form of a continuum model with jump rate α ( x ).}, journal={ESAIM: Probability and Statistics}, publisher={EDP Sciences}, author={Grigorescu, Ilie and Kang, Min}, year={2010}, pages={271–285} }
@article{grigorescu_kang_2007, title={Ergodic properties of multidimensional Brownian motion with rebirth}, volume={12}, ISSN={["1083-6489"]}, DOI={10.1214/ejp.v12-450}, abstractNote={In a bounded open region of the $d$ dimensional space we consider a Brownian motion which is reborn at a fixed interior point as soon as it reaches the boundary. The evolution is invariant with respect to a density equal, modulo a constant, to the Green function of the Dirichlet Laplacian centered at the point of return. We calculate the resolvent in closed form, study its spectral properties and determine explicitly the spectrum in dimension one. Two proofs of the exponential ergodicity are given, one using the inverse Laplace transform and properties of analytic semigroups, and the other based on Doeblin's condition. Both methods admit generalizations to a wide class of processes.}, journal={ELECTRONIC JOURNAL OF PROBABILITY}, author={Grigorescu, Ilie and Kang, Min}, year={2007}, month={Oct}, pages={1299–1322} }
@article{grigorescu_kang_2006, title={Tagged particle limit for a Fleming-Viot type system}, volume={11}, ISSN={["1083-6489"]}, DOI={10.1214/ejp.v11-316}, abstractNote={We consider a branching system of $N$ Brownian particles evolving independently in a domain $D$ during any time interval between boundary hits. As soon as one particle reaches the boundary it is killed and one of the other particles splits into two independent particles, the complement of the set $D$ acting as a catalyst or hard obstacle. Identifying the newly born particle with the one killed upon contact with the catalyst, we determine the exact law of the tagged particle as $N$ approaches infinity. In addition, we show that any finite number of labelled particles become independent in the limit. Both results can be seen as scaling limits of a genome population undergoing redistribution present in the Fleming-Viot dynamics.}, journal={ELECTRONIC JOURNAL OF PROBABILITY}, author={Grigorescu, I and Kang, M}, year={2006}, month={Apr}, pages={311–331} }
@article{grigorescu_kang_seppalainen_2004, title={Behavior dominated by slow particles in a disordered asymmetric exclusion process}, volume={14}, ISSN={["1050-5164"]}, DOI={10.1214/105051604000000387}, abstractNote={We study the large space and time scale behavior of a totally asymmetric, nearest-neighbor exclusion process in one dimension with random jump rates attached to the particles. When slow particles are sufficiently rare the system has a phase transition. At low densities there are no equilibrium distributions, and on the hydrodynamic scale the initial profile is transported rigidly. We elaborate this situation further by finding the correct order of the correction from the hydrodynamic limit, together with distributional bounds averaged over the disorder. We consider two settings, a macroscopically constant low density profile and the outflow from a large jam.}, number={3}, journal={ANNALS OF APPLIED PROBABILITY}, author={Grigorescu, I and Kang, M and Seppalainen, T}, year={2004}, month={Aug}, pages={1577–1602} }
@article{grigorescu_kang_2004, title={Hydrodynamic limit for a Fleming-Viot type system}, volume={110}, ISSN={["1879-209X"]}, DOI={10.1016/j.spa.2003.10.010}, abstractNote={We consider a system of N Brownian particles evolving independently in a domain D. As soon as one particle reaches the boundary it is killed and one of the other particles is chosen uniformly and splits into two independent particles resuming a new cycle of independent motion until the next boundary hit. We prove the hydrodynamic limit for the joint law of the empirical measure process and the average number of visits to the boundary as N approaches infinity.}, number={1}, journal={STOCHASTIC PROCESSES AND THEIR APPLICATIONS}, author={Grigorescu, I and Kang, M}, year={2004}, month={Mar}, pages={111–143} }
@article{grigorescu_kang_2004, title={Path collapse for multidimensional Brownian motion with rebirth}, volume={70}, ISSN={["1879-2103"]}, DOI={10.1016/j.spl.2004.10.006}, abstractNote={In a bounded open region of the d-dimensional Euclidean space we consider a Brownian motion which is reborn at a fixed interior point as soon as it reaches the boundary. It was shown that in dimension one coupled paths starting at different points but driven by the same Brownian motion either collapse with probability one or never meet. In higher dimensions, for convex or polyhedral regions, the paths with positive probability of collapse differ at start by a vector from a set of codimension one. The problem can be interpreted in terms of the long term mixing properties of the payoff of a portfolio of knock-out barrier options in derivatives markets.}, number={3}, journal={STATISTICS & PROBABILITY LETTERS}, author={Grigorescu, I and Kang, M}, year={2004}, month={Dec}, pages={199–209} }