Passage Problem Research Articles

On first passage times of sticky reflecting diffusion processes with double exponential jumps

AbstractWe explore the first passage problem for sticky reflecting diffusion processes with double exponential jumps. The joint Laplace transform of the first passage time to an upper level and the corresponding overshoot is studied. In particular, explicit solutions are presented when the diffusion part is driven by a drifted Brownian motion and by an Ornstein–Uhlenbeck process.

Optimized three-level quantum transfers based on frequency-modulated optical excitations

The difficulty in combining high fidelity with fast operation times and robustness against sources of noise is the central challenge of most quantum control problems, with immediate implications for the realization of quantum devices. We theoretically propose a protocol, based on the widespread stimulated Raman adiabatic passage technique, which achieves these objectives for quantum state transfers in generic three-level systems. Our protocol realizes accelerated adiabatic following through the application of additional control fields on the optical excitations. These act along frequency sidebands of the principal adiabatic pulses, dynamically counteracting undesired transitions. The scheme facilitates experimental control, not requiring new hardly-accessible resources. We show numerically that the method is efficient in a very wide set of control parameters, bringing the timescales closer to the quantum speed limit, also in the presence of environmental disturbance. These results hold for complete population transfers and for many applications, e.g., for realizing quantum gates, both for optical and microwave implementations. Furthermore, extensions to adiabatic passage problems in more-level systems are straightforward.

HPPRM: Hybrid Potential Based Probabilistic Roadmap Algorithm for Improved Dynamic Path Planning of Mobile Robots

Path planning and navigation is a very important problem in robotics, especially for mobile robots operating in complex environments. Sampling based planners such as the probabilistic roadmaps (PRM) have been widely used for different robot applications. However, due to the random sampling of nodes in PRM, it suffers from narrow passage problem that generates unconnected graph. The problem is addressed by increasing the number of nodes but at higher computation cost affecting real-time performance. To address this issue, in this paper, we propose an improved sampling-based path planning method for mobile robot navigation. The proposed method uses a layered hybrid Probabilistic Roadmap (PRM) and the Artificial Potential Field (APF) method for global planning. We used a decomposition method for node distribution that uses map segmentation to produce regions of high and low potential, and propose a method of reducing the dispersion of sample set during the roadmap construction. Our method produces better goal planning queries with a smaller graph and is computationally efficient than the traditional PRM. The proposed planner called the Hybrid Potential based Probabilistic Roadmap (HPPRM) is an improved sampling method with respect to success rate and calculation cost. Furthermore, we present a method for reactive local motion planning in the presence of static and dynamic obstacles in the environment. The advantage of the proposed method is that it can avoid local minima and successfully generate plans in complex maps such as narrow passages and bug trap scenarios that are otherwise difficult for the traditional sample-based methods. We show the validity of our method with experiments in simulation and real environments for both local and global planning. The results indicate that the proposed HPPRM is effective for autonomous mobile robot navigation in complex environments. The success rate of the proposed method is higher than 95% both for local and global planning.

TheW,Zscale functions kit for first passage problems of spectrally negative Lévy processes, and applications to control problems

In the last years there appeared a great variety of identities for first passage problems of spectrally negative Lévy processes, which can all be expressed in terms of two “q-harmonic functions” (or scale functions)WandZ. The reason behind that is that there are two ways of exiting an interval, and thus two fundamental “two-sided exit” problems from an interval (TSE). Since many other problems can be reduced to TSE, researchers developed in the last years a kit of formulas expressed in terms of the “W,Zalphabet”. It is important to note – as is currently being shown – that these identities apply equally to other spectrally negative Markov processes, where however theW,Zfunctions are typically much harder to compute. We collect below our favorite recipes from the “W,Zkit”, drawing from various applications in mathematical finance, risk, queueing, and inventory/storage theory. A small sample of applications concerning extensions of the classic de Finetti dividend problem is offered. An interesting use of the kit is for recognizing relationships between problems involving behaviors apparently unrelated at first sight (like reflection, absorption, etc.). Another is expressing results in a standardized form, improving thus the possibility to check when a formula is already known.

Quantum Interferometry for Different Energy Landscapes in a Tuneable Josephson Junction Circuit

This paper presents a simple Josephson-junction circuit with two parameters (inductance and capacitance) which can be tuned to represent different energy landscapes with different physical properties. By tuning this quantum circuit through external accessible elements we can move from two to three and more energy levels depending on the parameter setting. The inductance, the capacitance as well as the external voltage (driving terms) condition the number of relevant energy levels as well as the model to be used. We show that the quantized circuit represents a multi-state system with tunneling induced by the Landau-Zener and Landau-Zener-Stückelberg transition. The special cases of single crossing and multi-crossing models are thoroughly studied and the transition probability is obtained in each case. It is proven that, the crossing time as well as the relaxation time affect drastically the transition probability; the system mimics a single passage for short relaxation and a multiple passage problem for large relaxation. The nonlinearity of energy levels modifies the transition probability and the derived adiabatic parameters help to redefine the Landau-Zener probability. The observed constructive and destructive interferences are parametrically conditioned by the initial condition set by the inductive branch. Moreover, the total population transfers as well as the complete blockage of the system are obtained in a permissible range of parameters only by changing the values of the inductance. Therefore, the system models a controllable level-crossing where the additional branches (inductive and capacitive) help in designing the number of states, the type of interferometry as well as the control of states occupation.

Limit Theorems for Random Walks with Absorption

We introduce a class of absorption mechanisms and study the behavior of real-valued centered random walks with finite variance that do not get absorbed. Our main results serve as a toolkit which allows obtaining persistence and scaling limit results for many different examples in this class. Further, our results reveal new connections between results in Kemperman (The passage problem for a stationary Markov chain. Statistical research monographs, The University of Chicago Press, Chicago, 1961) and Vysotsky (Stoch Processes Appl 125(5):1886–1910, 2015).

Random search on comb

Impact of geometry on the search optimization, as the first passage problem, is addressed for a comb model. This problem corresponds to the two dimensional search with a geometrical constraint. Various combinations of the Brownian and Lévy search strategies are considered and exact solutions for the first arrival time distributions are obtained. The obtained solutions are compared with corresponding solutions of the one dimensional counterparts.

The influences of correlated spatially random perturbations on first passage time in a linear-cubic potential.

The influences of correlated spatially random perturbations (SRPs) on the first passage problem are studied in a linear-cubic potential with a time-changing external force driven by a Gaussian white noise. First, the escape rate in the absence of SRPs is obtained by Kramers' theory. For the random potential case, we simplify the escape rate by multiplying the escape rate of smooth potentials with a specific coefficient, which is to evaluate the influences of randomness. Based on this assumption, the escape rates are derived in two scenarios, i.e., small/large correlation lengths. Consequently, the first passage time distributions (FPTDs) are generated for both smooth and random potential cases. We find that the position of the maximal FPTD has a very good agreement with that of numerical results, which verifies the validity of the proposed approximations. Besides, with increasing the correlation length, the FPTD shifts to the left gradually and tends to the smooth potential case. Second, we investigate the most probable passage time (MPPT) and mean first passage time (MFPT), which decrease with increasing the correlation length. We also find that the variation ranges of both MPPT and MFPT increase nonlinearly with increasing the intensity. Besides, we briefly give constraint conditions to guarantee the validity of our approximations. This work enables us to approximately evaluate the influences of the correlation length of SRPs in detail, which was always ignored previously.

The first passage problem for stable linear delay equations perturbed by power law Lévy noise.

This article studies a linear scalar delay differential equation subject to small multiplicative power tail Lévy noise. We solve the first passage (the Kramers) problem with probabilistic methods and discover an asymptotic loss of memory in this non-Markovian system. Furthermore, the mean exit time increases with the power of the small noise amplitude, whereas the prefactor accounts for memory effects. In particular, we discover a nonlinear delay-induced exit acceleration due to a non-normal growth phenomenon. Our results are illustrated by the example of a linear delay oscillator driven by α-stable Lévy flights.

First passage upwards for state-dependent-killed spectrally negative Lévy processes

AbstractFor a spectrally negative Lévy process X, killed according to a rate that is a function ω of its position, we complement the recent findings of [12] by analysing (in greater generality) the exit probability of the one-sided upwards passage problem. When ω is strictly positive, this problem is related to the determination of the Laplace transform of the first passage time upwards for X that has been time-changed by the inverse of the additive functional $$\int_0^ \cdot \omega ({X_u}){\kern 1pt} {\rm{d}}u$$. In particular, our findings thus shed extra light on related results concerning first passage times downwards (resp. upwards) of continuous-state branching processes (resp. spectrally negative positive self-similar Markov processes).

First passage problems for upwards skip-free random walks via the scale functions paradigm

AbstractIn this paper we develop the theory of theWandZscale functions for right-continuous (upwards skip-free) discrete-time, discrete-space random walks, along the lines of the analogous theory for spectrally negative Lévy processes. Notably, we introduce for the first time in this context the one- and two-parameter scale functionsZ, which appear for example in the joint deficit at ruin and time of ruin problems of actuarial science. Comparisons are made between the various theories of scale functions as one makes time and/or space continuous.

Building a Better Bar Exam

In the wake of declining bar passage numbers and limited placement options for law grads, a new bar exam has emerged: the UBE. Drawn to an allusive promise of portability, thirty-six U.S. jurisdictions have adopted the UBE. I predict that in a few years’ time, the UBE will be administered in all states and U.S. territories. The UBE has snowballed from an idea into the primary gateway for entry into the practice of law. But the UBE is not a panacea that will solve the bar passage problems that U.S. law schools face. Whether or not to adopt a uniform exam is no longer the question. Now that the UBE has firmly taken root, the question to be answered is: “What can be done to make sure that the UBE does less harm than good?” This Article will, in four parts, examine the meteoric rise and spread of the UBE and the potential costs of its quick adoption. Part I will survey the gradual move away from state law exams to the jurisdictionally neutral UBE. Part II will identify correlations between recent changes to the multistate exams and a stark national decline in bar passage rates. Part III will address the limitations of the UBE, including the misleading promise of score portability and the consequences of forum shopping. Part IV will propose additional measures that can coexist with the UBE to counterbalance its limitations to make a better bar exam for law graduates and the clients they will serve.

Improved artificial potential field based lateral entry guidance for waypoints passage and no-fly zones avoidance

In this paper, a novel improved artificial potential field based lateral guidance algorithm for waypoints passage and no-fly zones avoidance is proposed. By introducing the improved attractive potential field and the improved repulsive potential field, the proposed algorithm converts the waypoints passage problem and the no-fly zones avoidance problem into the reference heading angle determination problem. Based on the heading error threshold and heading angle constraints, the reference heading corridor can be obtained in real time, which is adaptively updated during the entry flight according to the waypoint and no-fly zone constraints. Then the bank reversal logic is employed to regulate the sign of the bank angle, control the lateral motion of the vehicle, and steer the vehicle successfully to the desired terminal zone without any collision of the waypoint and no-fly zone constraints. The simulation results for verification and comparison cases show that the proposed algorithm has better adaptability, applicability and robustness than the alternative methods, and it can be successfully applied in lateral entry guidance of hypersonic glide vehicles with waypoint and no-fly zone constraints.

Return to the origin problem for a particle on a one-dimensional lattice with quasi-Zeno dynamics

In recent work, the so-called quasi-Zeno dynamics of a system has been investigated in the context of the quantum first passage problem. This dynamics considers the time evolution of a system subjected to a sequence of selective projective measurements made at small but finite intervals of time. This means that one has a sequence of steps, with each step consisting of a unitary transformation followed by a projection. The dynamics is non-unitary and, in recent work, it has been shown that it can be effectively described by two different non-Hermitian Hamiltonians. Here we explore this connection by considering the problem of detecting a free quantum particle moving on a one-dimensional lattice, where the detector is placed at the origin and the particle is initially located at some specified lattice point. We find that results for distribution times for the first detection probability, obtained from the non-Hermitian Hamiltonians, are in excellent agreement with known exact results as well as exact numerics. Interesting finite-size effects are discussed. We also study the first detection problem for the example of a particle moving in a quasi-periodic potential, an example where the unperturbed particle's motion can be ballistic, localized or diffusive.

The Problem of Time's Passage

While the B-theory of time seems to fit with the current physical theory, it also seems to require treating temporal passage as an illusion. The aim of this article is to show that by understanding cases of apparent motion in a particular way, one can maintain the B-theory while also retaining the privileged status that the phenomenon of temporal passage plays in human experience. However, to understand these cases correctly, one should turn to arguments in the history of philosophy. More specifically, arguments from Russell, Kant and Hume can be used to make the B-theory more plausible.

Historical Passages and Scenes of Transport in Uwe Johnson's Jahrestage

This contribution explores the problem of a historical passage between the German past and the American present in Uwe Johnson's Jahrestage: Aus dem Leben von Gesine Cresspahl through an analysis of its scenes of transit and technologized movement. Reading Johnson's work in terms of the Freudian theory of trauma, it argues that Jahrestage registers traumatic experience not only in the challenge of remembering the past but also in the repetition of trauma in the present. Scenes of transportation in the New York level of the novel provide a surface on which the psychic processes of Gesine's response to trauma—from repression to transference and repetition—become visible. Yet by encoding trauma in scenes rather than in narrative terms, Johnson does not resolve the disconnect between the two historical levels of Jahrestage.

On fluctuation theory for spectrally negative Lévy processes with Parisian reflection below, and applications

As well known, all functionals of a Markov process may be expressed in terms of the generator operator, modulo some analytic work. In the case of spectrally negative Markov processes however, it is conjectured that everything can be expressed in a more direct way using the $W$ scale function which intervenes in the two-sided first passage problem, modulo performing various integrals. This conjecture arises from work on Levy processes \cite{AKP,Pispot,APP,Iva,IP, ivanovs2013potential,AIZ,APY}, where the $W$ scale function has explicit Laplace transform, and is therefore easily computable; furthermore it was found in the papers above that a second scale function $Z$ introduced in \cite{AKP} greatly simplifies first passage laws, especially for reflected processes. This paper gathers a collection of first passage formulas for spectrally negative Parisian L\'evy processes, expressed in terms of $W,Z$ which may serve as an "instruction kit" for computing quantities of interest in applications, for example in risk theory and mathematical finance. To illustrate the usefulness of our list, we construct a new index for the valuation of financial companies modeled by spectrally negative L\'evy processes, based on a Dickson-Waters modifications of the de Finetti optimal expected discounted dividends objective. We offer as well an index for the valuation of conglomerates of financial companies. An implicit question arising is to investigate analog results for other classes of spectrally negative Markovian processes.

Navigation-oriented design for in-pipe robot in recursively divided sampling space with rapidly exploring random tree

Gas pipelines are subject to periodic inspection and maintenance for safety and longevity. Many robotic inspection systems have been developed for in-pipe applications, but systematic geometric design methodology that is suitable for in-pipe navigation has not been well studied so far due to difficulties in predicting the capability of maneuvering through the obstacles inside of pipelines such as bend, miter, and T-branch joint. The geometric design of the robot is critical to the performance of such in-pipe robots because the actuation and the measurement are constrained by the shape and the size of the robot. In this paper, we propose a design methodology that finds the maximum value of geometric design parameters of the robot with recursive evaluation of the parameter values in the design parameter space. The role of the design space division is to reduce the search region and to increase the number of parametric samples to near optimal values. As a parameter evaluation method, we adapt Rapidly exploring random tree (RRT) because it is known to be suitable for solving narrow passage problems for high-dimensional systems. Our design method makes it possible to find an optimal parameter set without computing complex cost functions. The design result of the in-pipe robot is 8 % larger than that of a heuristic geometry-based approach in three-parameter design problem.

Sample path large deviations for Lévy processes and random walks with Weibull increments

We study sample path large deviations for Lévy processes and random walks with heavy-tailed jump-size distributions that are of Weibull type. The sharpness and applicability of these results are illustrated by a counterexample proving the nonexistence of a full LDP in the $J_{1}$ topology, and by an application to a first passage problem.

First passage times for multiple particles with reversible target-binding kinetics.

We investigate the first passage problem for multiple particles that diffuse towards a target, partially adsorb there, and then desorb after a finite exponentially distributed residence time. We search for the first time when m particles undergoing such reversible target-binding kinetics are found simultaneously on the target that may trigger an irreversible chemical reaction or a biophysical event. Even if the particles are independent, the finite residence time on the target yields an intricate temporal coupling between particles. We compute analytically the mean first passage time (MFPT) for two independent particles by mapping the original problem to higher-dimensional surface-mediated diffusion and solving the coupled partial differential equations. The respective effects of the adsorption and desorption rates on the MFPT are revealed and discussed.