@article{35644, author = {{Kolb, Martin and Klump, Alexander}}, journal = {{Theory of Probability and its Applications}}, number = {{4}}, pages = {{717--744}}, publisher = {{Society for Industrial and Applied Mathematics}}, title = {{{Uniqueness of the Inverse First Passage Time Problem and the Shape of the Shiryaev boundary}}}, volume = {{67}}, year = {{2022}}, } @article{35649, abstract = {{Motivated by the work [6] of Mariusz Bieniek, Krzysztof Burdzy and Soumik Pal we study a Fleming-Viot-type particle system consisting of independently moving particles each driven by generalized Bessel processes on the positive real line. Upon hitting the boundary {0} this particle is killed and an uniformly chosen different one branches into two particles. Using the symmetry of the model and the self similarity property of Bessel processes, we obtain a criterion to decide whether the particles converge to the origin at a finite time. This addresses open problem 1.4 in [6]. Specifically, inspired by [6, Open Problem 1.5], we investigate the case of three moving particles and refine the general result of [6, Theorem 1.1(ii)] extending the regime of drift parameters, where convergence does not occur – even to values, where it does occur when considering the case of only two particles.}}, author = {{Kolb, Martin and Liesenfeld, Matthias}}, journal = {{Electronic Journal of Probability}}, number = {{27}}, pages = {{1--28}}, publisher = {{Institute of Mathematical Statistics}}, title = {{{On non-extinction in a Fleming-Viot-type particle model with Bessel drift}}}, doi = {{https://doi.org/10.1214/22-EJP866}}, year = {{2022}}, } @article{35650, abstract = {{We consider autoregressive sequences Xn = aXn−1 + ξn and Mn = max{aMn−1 , ξn} with a constant a ∈ (0, 1) and with positive, in- dependent and identically distributed innovations {ξk }. It is known that if P(ξ1 > x) ∼ d log x with some d ∈ (0, − log a) then the chains {Xn} and {Mn} are null recurrent. We investigate the tail behaviour of recurrence times in this case of logarithmically decaying tails. More precisely, we show that the tails of recurrence times are regularly varying of index −1 − d/ log a. We also prove limit theorems for {Xn} and {Mn} conditioned to stay over a fixed level x0. Furthermore, we study tail asymptotics for recurrence times of {Xn} and {Mn} in the case when these chains are positive recurrent and the tail of log ξ1 is subexponential.}}, author = {{Denisov, Denis and Hinrichs, Günter and Kolb, Martin and Wachtel, Vitali}}, journal = {{Electronic Journal of Probability}}, pages = {{1--43}}, publisher = {{Institute of Mathematical Statistics}}, title = {{{Persistence of autoregressive sequences with logarithmic tails}}}, doi = {{https://doi.org/10.48550/arXiv.2203.14772}}, volume = {{27}}, year = {{2022}}, } @article{33278, abstract = {{The kinetic Brownian motion on the sphere bundle of a Riemannian manifold M is a stochastic process that models a random perturbation of the geodesic flow. If M is an orientable compact constantly curved surface, we show that in the limit of infinitely large perturbation the L2-spectrum of the infinitesimal generator of a time-rescaled version of the process converges to the Laplace spectrum of the base manifold.}}, author = {{Kolb, Martin and Weich, Tobias and Wolf, Lasse}}, journal = {{Annales Henri Poincaré }}, number = {{4}}, pages = {{1283--1296}}, publisher = {{Springer Science + Business Media}}, title = {{{Spectral Asymptotics for Kinetic Brownian Motion on Surfaces of Constant Curvature}}}, volume = {{23}}, year = {{2021}}, } @article{33481, abstract = {{While 2D Gibbsian particle systems might exhibit orientational order resulting in a lattice-like structure, these particle systems do not exhibit positional order if the interaction between particles satisfies some weak assumptions. Here we investigate to which extent particles within a box of size may fluctuate from their ideal lattice position. We show that particles near the center of the box typically show a displacement at least of order . Thus we extend recent results on the hard disk model to particle systems with fairly arbitrary particle spins and interaction. Our result applies to models such as rather general continuum Potts type models, e.g. with Widom–Rowlinson or Lenard-Jones-type interaction.}}, author = {{Richthammer, Thomas and Fiedler, Michael}}, journal = {{Stochastic Processes and their Applications}}, pages = {{1--32}}, publisher = {{Elsevier}}, title = {{{A lower bound on the displacement of particles in 2D Gibbsian particle systems}}}, doi = {{https://doi.org/10.1016/j.spa.2020.10.003}}, volume = {{132}}, year = {{2021}}, } @article{33282, abstract = {{We derive a criterium for the almost sure finiteness of perpetual integrals of L ́evy processes for a class of real functions including all continuous functions and for general one- dimensional L ́evy processes that drifts to plus infinity. This generalizes previous work of D ̈oring and Kyprianou, who considered L ́evy processes having a local time, leaving the general case as an open problem. It turns out, that the criterium in the general situation simplifies significantly in the situation, where the process has a local time, but we also demonstrate that in general our cri- terium can not be reduced. This answers an open problem posed in D ̈oring, L. and Kyprianou, A. (2015).}}, author = {{Kolb, Martin and Savov, Mladen}}, journal = {{Bernoulli}}, keywords = {{L ́evy processes, Perpetual integrals, Potential measures}}, number = {{2}}, pages = {{1453--1472}}, publisher = {{Bernoulli Society for Mathematical Statistics and Probability}}, title = {{{A Characterization of the Finiteness of Perpetual Integrals of Levy Processes}}}, doi = {{https://doi.org/10.48550/arXiv.1903.03792}}, volume = {{26}}, year = {{2020}}, } @article{33330, abstract = {{Reciprocal relations are binary relations Q with entries Q(i,j)∈[0,1], and such that Q(i,j)+Q(j,i)=1. Relations of this kind occur quite naturally in various domains, such as preference modeling and preference learning. For example, Q(i,j) could be the fraction of voters in a population who prefer candidate i to candidate j. In the literature, various attempts have been made at generalizing the notion of transitivity to reciprocal relations. In this paper, we compare three important frameworks of generalized transitivity: g-stochastic transitivity, T-transitivity, and cycle-transitivity. To this end, we introduce E-transitivity as an even more general notion. We also use this framework to extend an existing hierarchy of different types of transitivity. As an illustration, we study transitivity properties of probabilities of pairwise preferences, which are induced as marginals of an underlying probability distribution on rankings (strict total orders) of a set of alternatives. In particular, we analyze the interesting case of the so-called Babington Smith model, a parametric family of distributions of that kind.}}, author = {{Haddenhorst, Björn and Hüllermeier, Eyke and Kolb, Martin}}, journal = {{International Journal of Approximate Reasoning}}, number = {{2}}, pages = {{373--407}}, publisher = {{Elsevier}}, title = {{{Generalized transitivity: A systematic comparison of concepts with an application to preferences in the Babington Smith model}}}, doi = {{https://doi.org/10.1016/j.ijar.2020.01.007}}, volume = {{119}}, year = {{2020}}, } @article{33331, abstract = {{Motivated by the recent contribution (Bauer and Bernard in Annales Henri Poincaré 19:653–693, 2018), we study the scaling limit behavior of a class of one-dimensional stochastic differential equations which has a unique attracting point subject to a small additional repulsive perturbation. Problems of this type appear in the analysis of continuously monitored quantum systems. We extend the results of Bauer and Bernard (Annales Henri Poincaré 19:653–693, 2018) and prove a general result concerning the convergence to a homogeneous Poisson process using only classical probabilistic tools.}}, author = {{Kolb, Martin and Liesenfeld, Matthias}}, journal = {{Annales Henri Poincaré}}, number = {{6}}, pages = {{1753--1783}}, publisher = {{Institute Henri Poincaré}}, title = {{{Stochastic Spikes and Poisson Approximation of One-Dimensional Stochastic Differential Equations with Applications to Continuously Measured Quantum Systems}}}, doi = {{http://dx.doi.org/10.1007/s00023-019-00772-9}}, volume = {{20}}, year = {{2019}}, } @article{33333, author = {{Wang, Andi Q. and Kolb, Martin and Roberts, Gareth O. and Steinsaltz, David}}, journal = {{The Annals of Applied Probability}}, number = {{1}}, title = {{{Theoretical properties of quasi-stationary Monte Carlo methods}}}, doi = {{http://dx.doi.org/10.1214/18-AAP1422}}, volume = {{29}}, year = {{2019}}, } @article{33334, abstract = {{In the present work we characterize the existence of quasistationary distributions for diffusions on (0,∞) allowing singular behavior at 0 and ∞. If absorption at 0 is certain, we show that there exists a quasistationary distribution as soon as the spectrum of the generator is strictly positive. This complements results of Collet et al. and Kolb/Steinsaltz for 0 being a regular boundary point and extends results by Collet et al. on singular diffusions. We also study the existence and uniqueness of quasistationary distributions for a class of one-dimensional diffusions with killing that arise from a biological example and which have two inaccessible boundary points (more specifically 0 is natural and ∞ is entrance).}}, author = {{Hening, Alexandru and Kolb, Martin}}, journal = {{Stochastic Processes and their Applications}}, number = {{5}}, pages = {{1659--1696}}, publisher = {{Bernoulli Society for Mathematical Statistics and Probability}}, title = {{{Quasistationary distributions for one-dimensional diffusions with two singular boundary points}}}, doi = {{http://dx.doi.org/10.1016/j.spa.2018.05.012}}, volume = {{129}}, year = {{2019}}, } @article{33335, abstract = {{For a class of one-dimensional autoregressive sequences (Xn), we consider the tail behaviour of the stopping time T0=min{n≥1:Xn≤0}. We discuss existing general analytical approaches to this and related problems and propose a new one, which is based on a renewal-type decomposition for the moment generating function of T0 and on the analytical Fredholm alternative. Using this method, we show that Px(T0=n)∼V(x)Rn0 for some 00. We thus prove that it is sub-ballistic and estimate the speed of escape. The latter is in a sharp contrast with discrete models of dimension larger or equal to 2 when the behaviour at criticality is ballistic, see [7], and even to many one dimensional models which exhibit ballistic behaviour at criticality, see [8].}}, author = {{Savov, Mladen and Kolb, Martin}}, journal = {{Electronic Journal of Probability}}, publisher = {{ Institute of Mathematical Statistics & Bernoulli Society}}, title = {{{Conditional survival distributions of Brownian trajectories in a one dimensional Poissonian environment in the critical case}}}, doi = {{https://doi.org/10.1214/17-EJP4468}}, volume = {{22}}, year = {{2017}}, } @article{33343, abstract = {{Using an operator-theoretic framework in a Hilbert-space setting, we perform a detailed spectral analysis of the one-dimensional Laplacian in a bounded interval, subject to specific non-self-adjoint connected boundary conditions modelling a random jump from the boundary to a point inside the interval. In accordance with previous works, we find that all the eigenvalues are real. As the new results, we derive and analyse the adjoint operator, determine the geometric and algebraic multiplicities of the eigenvalues, write down formulae for the eigenfunctions together with the generalised eigenfunctions and study their basis properties. It turns out that the latter heavily depend on whether the distance of the interior point to the centre of the interval divided by the length of the interval is rational or irrational. Finally, we find a closed formula for the metric operator that provides a similarity transform of the problem to a self-adjoint operator.}}, author = {{Kolb, Martin and Krejčiřík, David}}, journal = {{Mathematische Zeitschrift}}, pages = {{877--900}}, publisher = {{Springer}}, title = {{{Spectral analysis of the diffusion operator with random jumps from the boundary}}}, doi = {{https://link.springer.com/content/pdf/10.1007/s00209-016-1677-y.pdf}}, volume = {{284}}, year = {{2016}}, } @article{33344, abstract = {{The hard disk model is a 2D Gibbsian process of particles interacting via pure hard core repulsion. At high particle density the model is believed to show orientational order, however, it is known not to exhibit positional order. Here we investigate to what extent particle positions may fluctuate. We consider a finite volume version of the model in a box of dimensions 2n × 2n with arbitrary boundary configuration, and we show that the mean square displacement of particles near the center of the box is bounded from below by c log n. The result generalizes to a large class of models with fairly arbitrary interaction.}}, author = {{Richthammer, Thomas}}, journal = {{Communications in Mathematical Physics }}, pages = {{1077--1099}}, title = {{{Lower Bound on the Mean Square Displacement of Particles in the Hard Disk Model}}}, doi = {{https://link.springer.com/article/10.1007/s00220-016-2584-0}}, volume = {{345}}, year = {{2016}}, } @article{33357, abstract = {{In this note we investigate the behaviour of Brownian motion conditioned on a growth constraint of its local time which has been previously investigated by Berestycki and Benjamini. For a class of non-decreasing positive functions f(t);t>0, we consider the Wiener measure under the condition that the Brownian local time is dominated by the function f up to time T. In the case where f(t)/t3/2 is integrable we describe the limiting process as T goes to infinity. Moreover, we prove two conjectures in [BB10] in the case for a class of functions f, for which f(t)/t3/2 just fails to be integrable. Our methodology is more general as it relies on the study of the asymptotic of the probability of subordinators to stay above a given curve. Immediately or with adaptations one can study questions like the Brownian motioned conditioned on a growth constraint of its local time at the maximum or more generally a Levy process conditioned on a growth constraint of its local time at the maximum or at zero. We discuss briefly the former. }}, author = {{Kolb, Martin and Savov, Mladen}}, journal = {{The Annals of Probability}}, number = {{6}}, publisher = {{Institute of Mathematical Statistics}}, title = {{{Transience and recurrence of a Brownian path with limited local time}}}, doi = {{http://dx.doi.org/10.1214/15-AOP1069}}, volume = {{44}}, year = {{2016}}, } @article{33359, abstract = {{We consider Gibbs distributions on permutations of a locally finite infinite set X⊂R, where a permutation σ of X is assigned (formal) energy ∑x∈XV(σ(x)−x). This is motivated by Feynman’s path representation of the quantum Bose gas; the choice X:=Z and V(x):=αx2 is of principal interest. Under suitable regularity conditions on the set X and the potential V, we establish existence and a full classification of the infinite-volume Gibbs measures for this problem, including a result on the number of infinite cycles of typical permutations. Unlike earlier results, our conclusions are not limited to small densities and/or high temperatures. }}, author = {{Richthammer, Thomas and Biskup, Marek}}, journal = {{Communications in Mathematical Physics}}, number = {{2}}, pages = {{898--929}}, publisher = {{Springer Science+Business Media}}, title = {{{Gibbs measures on permutations over one-dimensional discrete point sets}}}, doi = {{https://doi.org/10.48550/arXiv.1310.0248}}, volume = {{25}}, year = {{2015}}, } @article{33358, abstract = {{We study the workload processes of two M/G/1 queueing systems with restricted capacity: in Model 1 any service requirement that would exceed a certain capacity threshold is truncated; in Model 2 new arrivals do not enter the system if they have to wait more than a fixed threshold time in line. For Model 1 we obtain several results concerning the rate of convergence to equilibrium. In particular, we derive uniform bounds for geometric ergodicity with respect to certain subclasses. For Model 2 geometric ergodicity follows from the finiteness of the moment-generating function of the service time distribution. We derive bounds for the convergence rates in special cases. The proofs use the coupling method.}}, author = {{Kolb, Martin and Stadje, Wolfgang and Wübker, Achim}}, journal = {{Stochastic Models}}, number = {{1}}, pages = {{121--135}}, publisher = {{INFORMS}}, title = {{{The rate of convergence to stationarity for M/G/1 models with admission controls via coupling}}}, doi = {{http://dx.doi.org/10.1080/15326349.2015.1090322}}, volume = {{32}}, year = {{2015}}, } @article{33360, abstract = {{We prove a local limit theorem for the area of the positive excursion of random walks with zero mean and finite variance. Our main result complements previous work of Caravenna and Chaumont; Sohier, as well as Kim and Pittel.}}, author = {{Kolb, Martin and Denisov, Denis and Wachtel, Vitali}}, journal = {{Journal of the London Mathematical Society}}, number = {{2}}, pages = {{495--513}}, publisher = {{London Mathematical Society}}, title = {{{Local asymptotics for the area of random walk excursions}}}, volume = {{91}}, year = {{2015}}, } @article{33361, abstract = {{Following Bertoin who considered the ergodicity and exponential decay of Lévy processes in a finite domain, we consider general Lévy processes and their ergodicity and exponential decay in a finite interval. More precisely, given Ta=inf{t>0:Xt∉. Under general conditions, e.g. absolute continuity of the transition semigroup of the unkilled Lévy process, we prove that the killed semigroup is a compact operator. Thus, we prove stronger results in view of the exponential ergodicity and estimates of the speed of convergence. Our results are presented in a Lévy processes setting but are well applicable for Markov processes in a finite interval under information about Lebesgue irreducibility of the killed semigroup and that the killed process is a double Feller process. For example, this scheme is applicable to a work of Pistorius.
}}, author = {{Kolb, Martin and Savov, Mladen}}, journal = {{Electronic Communications in Probability}}, number = {{31}}, pages = {{1--9}}, publisher = {{Institute of Mathematical Statistics (IMS)}}, title = {{{Exponential ergodicity of killed Lévy processes in a finite interval}}}, doi = {{http://dx.doi.org/10.1214/ECP.v19-3006}}, volume = {{19}}, year = {{2014}}, }