arXiv subjects

Martin Kolb

Publications and source records attributed to Martin Kolb.

At least 19 recordsLinked to original sources

arXiv2024

Restricted Monte Carlo wave function method and Lindblad equation for identifying entangling open-quantum-system dynamics

We develop an extension of the Monte Carlo wave function approach that unambiguously identifies dynamical entanglement in general composite, open systems. Our algorithm performs tangential projections onto the set of separable states, leading to classically correlated quantum trajectories. By comparing this restricted evolution with the unrestricted one, we can characterize the entangling capabilities of quantum channels without making use of input-output relations. Moreover, applying this method is equivalent to solving the nonlinear master equation in Lindblad form introduced in \cite{PAH24} for two-qubit systems. We here extend these equations to multipartite systems of qudits, describing non-entangling dynamics in terms of a stochastic differential equation. We identify the impact of dynamical entanglement in open systems by applying our approach to several correlated decay processes. Therefore, our methodology provides a complete and ready-to-use framework to characterize dynamical quantum correlations caused by arbitrary open-system processes.

quant-ph↗

arXiv2024

Separability Lindblad equation for dynamical open-system entanglement

Providing entanglement for the design of quantum technologies in the presence of noise constitutes today's main challenge in quantum information science. A framework is required that assesses the build-up of entanglement in realistic settings. In this work, we put forth a new class of nonlinear quantum master equations in Lindblad form that unambiguously identify dynamical entanglement in open quantum systems via deviations from a separable evolution. This separability Lindblad equation restricts quantum trajectories to classically correlated states only. Unlike many conventional approaches, here the entangling capabilities of a process are not characterized by input-output relations, but separability is imposed at each instant of time. We solve these equations for crucial examples, thereby quantifying the dynamical impact of entanglement in non-equilibrium scenarios. Our results allow to benchmark the engineering of entangled states through dissipation. The separability Lindblad equation provides a unique path to characterizing quantum correlations caused by arbitrary system-bath interactions, specifically tailored for the noisy intermediate-scale quantum era.

quant-ph↗

arXiv2023

Brownian motion conditioned to spend limited time outside a bounded interval -- an extreme example of entropic repulsion

We show that a Brownian motion on $\mathbb{R}_{\ge 0}$ which is allowed to spend a total of $s > 0$ time units outside a bounded interval does not leave the interval at all. This can be seen as an extreme example of entropic repulsion. Moreover, we explicitly determine the exact asymptotic behaviour of the probability that a Brownian motion on $[0,T]$ spends limited time outside a bounded interval, as $T \to \infty$.

math.PR↗

arXiv2022

Persistence of autoregressive sequences with logarithmic tails

We consider autoregressive sequences $X_n=aX_{n-1}+\xi_n$ and $M_n=\max\{aM_{n-1},\xi_n\}$ with a constant $a\in(0,1)$ and with positive, independent and identically distributed innovations $\{\xi_k\}$. It is known that if $\mathbf P(\xi_1>x)\sim\frac{d}{\log x}$ with some $d\in(0,-\log a)$ then the chains $\{X_n\}$ and $\{M_n\}$ 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 $\{X_n\}$ and $\{M_n\}$ conditioned to stay over a fixed level $x_0$. Furthermore, we study tail asymptotics for recurrence times of $\{X_n\}$ and $\{M_n\}$ in the case when these chains are positive recurrent and the tail of $\log\xi_1$ is subexponential.

math.PR↗

arXiv2020

Spectral Asymptotics for Kinetic Brownian Motion on Hyperbolic Surfaces

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 a orientable compact constant negatively curved surface, we show that in the limit of infinitely large perturbation the $L^2$-spectrum of the infinitesimal generator of a time rescaled version of the process converges to the Laplace spectrum of the base manifold. In addition, we give explicit error estimates for the convergence to equilibrium. The proofs are based on noncommutative harmonic analysis of $SL_2(\mathbb{R})$.

math.SP↗

arXiv2020

Spectral Asymptotics for Kinetic Brownian Motion on Surfaces of Constant Curvature

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 a orientable compact constantly curved surface, we show that in the limit of infinitely large perturbation the $L^2$-spectrum of the infinitesimal generator of a time rescaled version of the process converges to the Laplace spectrum of the base manifold.

math.SP↗

arXiv2020

Spectral analysis of the diffusion operator with random jumps from the boundary

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.

math.SP↗

arXiv2019

A Characterization of the Finiteness of Perpetual Integrals of Levy Processes

We derive a criterium for the almost sure finiteness of perpetual integrals of \LL processes for a class of real functions including all continuous functions and for general one-dimensional Lévy processes that drifts to plus infinity. This generalizes previous work of Döring and Kyprianou, who considered Lévy 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 criterium can not be reduced. This answers an open problem posed in \cite{doring}.

math.PR↗

arXiv2018

Theoretical properties of quasi-stationary Monte Carlo methods

This paper gives foundational results for the application of quasi-stationarity to Monte Carlo inference problems. We prove natural sufficient conditions for the quasi-limiting distribution of a killed diffusion to coincide with a target density of interest. We also quantify the rate of convergence to quasi-stationarity by relating the killed diffusion to an appropriate Langevin diffusion. As an example, we consider in detail a killed Ornstein--Uhlenbeck process with Gaussian quasi-stationary distribution.

stat.ME↗

arXiv2018

Persistence of one-dimensional AR(1)-sequences

For a class of one-dimensional autoregressive processes $(X_n)$ we consider the tail behaviour of the stopping time $T_0=\min \lbrace n\geq 1: X_n\leq 0 \rbrace$. 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 $T_0$ and on the analytical Fredholm alternative. Using this method, we show that $\mathbb{P}_x(T_0=n)\sim V(x)R_0^n$ for some $0<R_0<1$ and a positive $R^{-1}_0$-harmonic function $V$. Further we prove that our conditions on the tail behaviour of the innovations are sharp in the sense that fatter tails produce non-exponential decay factors.

math.PR↗

arXiv2018

Stochastic spikes and Poisson Approximation of one-dimensional stochastic differential equations with applications to continuously measured Quantum Systems

Motivated by the recent contribution \cite{BB17} 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 \cite{BB17} and prove a general result concerning the convergence to a homogeneous Poisson process using only classical probabilistic tools.

math-ph↗

arXiv2018

Quasistationary distributions for one-dimensional diffusions with singular boundary points

In the present work we characterize the existence of quasistationary distributions for diffusions on $(0,\infty)$ allowing singular behavior at $0$ and $\infty$. 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. (Ann. Probab. 2009) and Kolb and Steinsaltz (Ann. Probab. 2012) for $0$ being a regular boundary point and extends results by Collet et al. (Ann. Probab. 2009) on singular diffusions.

math.PR↗

arXiv2017

On the Dipole Approximation with Error Estimates

The dipole approximation is employed to describe interactions between atoms and radiation. It essentially consists of neglecting the spatial variation of the external field over the atom. Heuristically, this is justified by arguing that the wavelength is considerably larger than the atomic length scale, which holds under usual experimental conditions. We prove the dipole approximation in the limit of infinite wavelengths compared to the atomic length scale and estimate the rate of convergence. Our results include N-body Coulomb potentials and experimentally relevant electromagnetic fields such as plane waves and laser pulses.

math-ph↗

arXiv2015

Transience and recurrence of a Brownian path with limited local time and its repulsion envelope

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 $f(t)/t^{3/2}$ 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 $f(t)/t^{3/2}$ 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.

math.PR↗

arXiv2015

Conditional survival distributions of Brownian trajectories in a one dimensional Poissonian environment in the critical case

In this work we consider a one-dimensional Brownian motion with constant drift moving among a Poissonian cloud of obstacles. Our main result proves convergence of the law of processes conditional on survival up to time $t$ as $t$ converges to infinity in the critical case where the drift coincides with the intensity of the Poisson process. The complements a previous result of T. Povel, who considered the same question in the case where the drift is strictly smaller than the intensity.

math.PR↗

arXiv2012

Quasilimiting behavior for one-dimensional diffusions with killing

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 infinity is larger than the principal eigenvalue, then the eigenfunction is always integrable. When the killing at infinity 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.

math.PR↗

arXiv2012

The rate of convergence to stationarity for M/G/1 models with admission controls via coupling

We study the workload processes of two restricted M/G/1 queueing systems: 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. However, we prove that for the class of all Model 1 workload processes there is no uniform bound. For Model 2 we prove that geometric ergodicity follows from the finiteness of the moment-generating function of the service time distribution and derive bounds for the convergence rates in special cases. The proofs use the coupling method.

math.PR↗

arXiv2011

Geometric Ergodicity of a Hypoelliptic Diffusion Modelling The Melt-Spinning Process of Nonwoven Materials

We analyze the large time behavior of a stochastic model for the lay-down of fibers on a conveyor belt in the production process of nonwovens. It is shown, that under weak conditions this degenerate diffusion process is strong mixing, confirming a conjecture of Grothaus and Klar. Moreover, under some additional assumptions even geometric ergodicity is established using probabilistic tools -- described in the book of Meyn and Tweedie -- in combination with methods from stochastic analysis.

math.PR↗