@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}}, } @article{33362, abstract = {{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.}}, author = {{Kolb, Martin and Krejčiřík, David}}, journal = {{Journal of Spectral Theory}}, number = {{2}}, pages = {{235--281}}, publisher = {{EMS Press}}, title = {{{The Brownian traveller on manifolds}}}, doi = {{https://doi.org/10.4171/jst/69}}, volume = {{4}}, year = {{2014}}, } @article{37503, abstract = {{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. }}, author = {{Kolb, Martin and Steinsaltz, David}}, journal = {{Annals of Probability}}, number = {{1}}, pages = {{162--212}}, publisher = {{Institute of Mathematical Statistics}}, title = {{{Quasilimiting behavior for one-dimensional diffusions with killing}}}, doi = {{https://doi.org/10.1214/10-AOP623}}, volume = {{40}}, year = {{2012}}, } @article{45765, author = {{Grummt, Robert and Kolb, Martin}}, journal = {{ Journal of Mathematical Analysis and Applications}}, pages = {{480--489}}, title = {{{Essential selfadjointness of singular magnetic Schrödinger operators on Riemannian manifolds}}}, doi = {{10.1016/j.jmaa.2011.09.060}}, volume = {{388}}, year = {{2012}}, } @article{37500, abstract = {{We discuss the time evolution of the wave function which is the solution of a stochastic Schrödinger equation describing the dynamics of a free quantum particle subject to spontaneous localizations in space. We prove global existence and uniqueness of solutions. We observe that there exist three time regimes: the collapse regime, the classical regime and the diffusive regime. Concerning the latter, we assert that the general solution converges almost surely to a diffusing Gaussian wave function having a finite spread both in position as well as in momentum. This paper corrects and completes earlier works on this issue.}}, author = {{Bassi, Angelo and Dürr, Detlef and Kolb, Martin}}, journal = {{Reviews in Mathematical Physics}}, number = {{1}}, pages = {{55--89}}, title = {{{ON THE LONG TIME BEHAVIOR OF FREE STOCHASTIC SCHRÖDINGER EVOLUTIONS}}}, doi = {{https://doi.org/10.1142/S0129055X10003886}}, volume = {{22}}, year = {{2009}}, }