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 -