TY - JOUR AB - AbstractThe kinetic Brownian motion on the sphere bundle of a Riemannian manifold $$\mathbb {M}$$ M is a stochastic process that models a random perturbation of the geodesic flow. If $$\mathbb {M}$$ M is an orientable compact constantly curved surface, we show that in the limit of infinitely large perturbation the $$L^2$$ L 2 -spectrum of the infinitesimal generator of a time-rescaled version of the process converges to the Laplace spectrum of the base manifold. AU - Kolb, Martin AU - Weich, Tobias AU - Wolf, Lasse Lennart ID - 31193 IS - 4 JF - Annales Henri Poincaré KW - Mathematical Physics KW - Nuclear and High Energy Physics KW - Statistical and Nonlinear Physics SN - 1424-0637 TI - Spectral Asymptotics for Kinetic Brownian Motion on Surfaces of Constant Curvature VL - 23 ER - TY - JOUR AU - Kolb, Martin AU - Klump, Alexander ID - 35644 IS - 4 JF - Theory of Probability and its Applications TI - Uniqueness of the Inverse First Passage Time Problem and the Shape of the Shiryaev boundary VL - 67 ER - TY - JOUR AB - 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. AU - Kolb, Martin AU - Liesenfeld, Matthias ID - 35649 IS - 27 JF - Electronic Journal of Probability TI - On non-extinction in a Fleming-Viot-type particle model with Bessel drift ER - TY - JOUR AB - 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. AU - Denisov, Denis AU - Hinrichs, Günter AU - Kolb, Martin AU - Wachtel, Vitali ID - 35650 JF - Electronic Journal of Probability TI - Persistence of autoregressive sequences with logarithmic tails VL - 27 ER - TY - JOUR AB - 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. AU - Kolb, Martin AU - Weich, Tobias AU - Wolf, Lasse ID - 33278 IS - 4 JF - Annales Henri Poincaré TI - Spectral Asymptotics for Kinetic Brownian Motion on Surfaces of Constant Curvature VL - 23 ER - TY - JOUR AB - 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). AU - Kolb, Martin AU - Savov, Mladen ID - 33282 IS - 2 JF - Bernoulli KW - L ́evy processes KW - Perpetual integrals KW - Potential measures TI - A Characterization of the Finiteness of Perpetual Integrals of Levy Processes VL - 26 ER - TY - JOUR AB - 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. AU - Haddenhorst, Björn AU - Hüllermeier, Eyke AU - Kolb, Martin ID - 33330 IS - 2 JF - International Journal of Approximate Reasoning TI - Generalized transitivity: A systematic comparison of concepts with an application to preferences in the Babington Smith model VL - 119 ER - TY - JOUR AB - 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. AU - Kolb, Martin AU - Liesenfeld, Matthias ID - 33331 IS - 6 JF - Annales Henri Poincaré TI - Stochastic Spikes and Poisson Approximation of One-Dimensional Stochastic Differential Equations with Applications to Continuously Measured Quantum Systems VL - 20 ER - TY - JOUR AU - Wang, Andi Q. AU - Kolb, Martin AU - Roberts, Gareth O. AU - Steinsaltz, David ID - 33333 IS - 1 JF - The Annals of Applied Probability TI - Theoretical properties of quasi-stationary Monte Carlo methods VL - 29 ER - TY - JOUR AB - 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]. AU - Savov, Mladen AU - Kolb, Martin ID - 33342 JF - Electronic Journal of Probability TI - Conditional survival distributions of Brownian trajectories in a one dimensional Poissonian environment in the critical case VL - 22 ER - TY - JOUR AB - 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. AU - Kolb, Martin AU - Krejčiřík, David ID - 33343 JF - Mathematische Zeitschrift TI - Spectral analysis of the diffusion operator with random jumps from the boundary VL - 284 ER - TY - JOUR AB - 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. AU - Kolb, Martin AU - Savov, Mladen ID - 33357 IS - 6 JF - The Annals of Probability TI - Transience and recurrence of a Brownian path with limited local time VL - 44 ER - TY - JOUR AB - 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. AU - Kolb, Martin AU - Stadje, Wolfgang AU - Wübker, Achim ID - 33358 IS - 1 JF - Stochastic Models TI - The rate of convergence to stationarity for M/G/1 models with admission controls via coupling VL - 32 ER - TY - JOUR AB - 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. AU - Kolb, Martin AU - Denisov, Denis AU - Wachtel, Vitali ID - 33360 IS - 2 JF - Journal of the London Mathematical Society TI - Local asymptotics for the area of random walk excursions VL - 91 ER - TY - JOUR AB - 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.
AU - Kolb, Martin AU - Savov, Mladen ID - 33361 IS - 31 JF - Electronic Communications in Probability TI - Exponential ergodicity of killed Lévy processes in a finite interval VL - 19 ER - TY - JOUR AB - We study the influence of the intrinsic curvature on the large time behaviour of the heat equation in a tubular neighbourhood of an unbounded geodesic in a two-dimensional Riemannian manifold. Since we consider killing boundary conditions, there is always an exponential-type decay for the heat semigroup. We show that this exponential-type decay is slower for positively curved manifolds comparing to the flat case. As the main result, we establish a sharp extra polynomial-type decay for the heat semigroup on negatively curved manifolds comparing to the flat case. The proof employs the existence of Hardy-type inequalities for the Dirichlet Laplacian in the tubular neighbourhoods on negatively curved manifolds and the method of self-similar variables and weighted Sobolev spaces for the heat equation. AU - Kolb, Martin AU - Krejčiřík, David ID - 33362 IS - 2 JF - Journal of Spectral Theory TI - The Brownian traveller on manifolds VL - 4 ER - TY - JOUR AB - This paper extends and clarifies results of Steinsaltz and Evans [Trans. Amer. Math. Soc. 359 (2007) 1285–1234], which found conditions for convergence of a killed one-dimensional diffusion conditioned on survival, to a quasistationary distribution whose density is given by the principal eigenfunction of the generator. Under the assumption that the limit of the killing at infinity differs from the principal eigenvalue we prove that convergence to quasistationarity occurs if and only if the principal eigenfunction is integrable. When the killing at ∞ is larger than the principal eigenvalue, then the eigenfunction is always integrable. When the killing at ∞ is smaller, the eigenfunction is integrable only when the unkilled process is recurrent; otherwise, the process conditioned on survival converges to 0 density on any bounded interval. AU - Kolb, Martin AU - Steinsaltz, David ID - 37503 IS - 1 JF - Annals of Probability TI - Quasilimiting behavior for one-dimensional diffusions with killing VL - 40 ER - TY - JOUR AU - Grummt, Robert AU - Kolb, Martin ID - 45765 JF - Journal of Mathematical Analysis and Applications TI - Essential selfadjointness of singular magnetic Schrödinger operators on Riemannian manifolds VL - 388 ER -