Distribution Of Exit Times Research Articles

Monte Carlo treatment of resonance-radiation imprisonment in fluorescent lamps—revisited

We reported in 1985 a Monte Carlo treatment of the imprisonment of the 253.7 nm resonance radiation from mercury in the mercury–argon discharge of fluorescent lamps. The calculated spectra of the emitted radiation were found in good agreement with measured spectra. The addition of the isotope mercury-196 to natural mercury was found, also in agreement with experiments, to increase lamp efficiency. In this paper we report the extension of the earlier work with increased accuracy, analysis of photon exit-time distributions, recycling of energy released in quenching, analysis of dynamic similarity for different lamp sizes, variation of Mrozowski transfer rates, prediction and analysis of the hyperfine ultra-violet spectra, and optimization of tailored mercury isotope mixtures for increased lamp efficiency. The spectra were found insensitive to the extent of quenching and recycling. The optimized mixtures were found to increase efficiencies by as much as 5% for several lamp configurations. Optimization without increasing the mercury-196 fraction was found to increase efficiencies by nearly 1% for several configurations.

First exit time of a Lévy flight from a bounded region in ℝN

In this paper we characterize the mean and the distribution of the first exit time of a Lévy flight from a bounded region in N-dimensional spaces. We characterize tight upper and lower bounds on the tail distribution of the first exit time, and provide the exact asymptotics of the mean first exit time for a given range of step-length distribution parameters.

First exit time of a Lévy flight from a bounded region in ℝN

In this paper we characterize the mean and the distribution of the first exit time of a Lévy flight from a bounded region inN-dimensional spaces. We characterize tight upper and lower bounds on the tail distribution of the first exit time, and provide the exact asymptotics of the mean first exit time for a given range of step-length distribution parameters.

Thirty years of turnstiles and transport.

To characterize transport in a deterministic dynamical system is to compute exit time distributions from regions or transition time distributions between regions in phase space. This paper surveys the considerable progress on this problem over the past thirty years. Primary measures of transport for volume-preserving maps include the exiting and incoming fluxes to a region. For area-preserving maps, transport is impeded by curves formed from invariant manifolds that form partial barriers, e.g., stable and unstable manifolds bounding a resonance zone or cantori, the remnants of destroyed invariant tori. When the map is exact volume preserving, a Lagrangian differential form can be used to reduce the computation of fluxes to finding a difference between the actions of certain key orbits, such as homoclinic orbits to a saddle or to a cantorus. Given a partition of phase space into regions bounded by partial barriers, a Markov tree model of transport explains key observations, such as the algebraic decay of exit and recurrence distributions.

The exit-time problem for a Markov jump process

The purpose of this paper is to consider the exit-time problem for a finite-range Markov jump process, i.e, the distance the particle can jump is bounded independent of its location. Such jump diffusions are expedient models for anomalous transport exhibiting super-diffusion or nonstandard normal diffusion. We refer to the associated deterministic equation as a volume-constrained nonlocal diffusion equation. The volume constraint is the nonlocal analogue of a boundary condition necessary to demonstrate that the nonlocal diffusion equation is well-posed and is consistent with the jump process. A critical aspect of the analysis is a variational formulation and a recently developed nonlocal vector calculus. This calculus allows us to pose nonlocal backward and forward Kolmogorov equations, the former equation granting the various moments of the exit-time distribution.

Exit Time Distribution in Spherically Symmetric Two-Dimensional Domains

The distribution of exit times is computed for a Brownian particle in spherically symmetric two- dimensional domains (disks, angular sectors, annuli) and in rectangles that contain an exit on their boundary. The governing partial differential equation of Helmholtz type with mixed Dirichlet- Neumann boundary conditions is solved analytically. We propose both an exact solution relying on a matrix inversion, and an approximate explicit solution. The approximate solution is shown to be exact for an exit of vanishing size and to be accurate even for large exits. For angular sectors, we also derive exact explicit formulas for the moments of the exit time. For annuli and rectangles, the approximate expression of the mean exit time is shown to be very accurate even for large exits. The analysis is also extended to biased diffusion. Since the Helmholtz equation with mixed boundary conditions is encountered in microfluidics, heat propagation, quantum billiards, and acoustics, the developed method can find numerous applications beyond exit processes.

Random walks in the quarter plane, discrete harmonic functions and conformal mappings

We propose a new approach for finding discrete harmonic functions in the quarter plane with Dirichlet conditions. It is based on solving functional equations that are satisfied by the generating functions of the values taken by the harmonic functions. As a first application of our results, we obtain a simple expression for the harmonic function that governs the asymptotic tail distribution of the first exit time for random walks from the quarter plane. As another corollary, we prove, in the zero drift case, the uniqueness of the discrete harmonic function.

Generalized similarity, renormalization groups, and nonlinear clocks for multiscaling.

Fixed points of the renormalization group operator Rp,rX(t)≡X(rt)/rp are said to be p-self-similar. Here X(t) is an arbitrary stochastic process. The concept of a p-self-similar process is generalized via the renormalization group operator RF,GX(t)=F[X(G(t))], where F and G are bijections on (-∞,∞) and [0,∞), respectively. If X(t) is a fixed point of RF,G, then X(t) is said to be (F,G)-self-similar. We say Y(t) is (F,G)-X(t)-similar if RF,GX(t)=Y(t) in distribution. Exit time distributions and finite-size Lyapunov exponents were obtained for these latter processes. A power law multiscaling process is defined with a multipower-law clock. This process is employed to statistically represent diffusion in a nanopore, a monolayer fluid confined between atomically structured surfaces. The tools presented provide a straightforward method to statistically represent any multiscaling process in time.

Reconstructing the Transition Rate Function of a Broadwell Random Walk from Exit Times

In this paper, we utilize the layer stripping method, a method used in seismology by geophysicists, to study a stochastic inverse problem arising from a one-dimensional Broadwell process. The Broadwell process can be described as a random walk of a particle that transitions, with a spatially dependent flip rate, between two states associated with the same speed but with different signs of velocities. Our goal is to reconstruct the spatially dependent flip rate function of a one-dimensional Broadwell process from exit time distributions out of a finite interval. In principle, we are able to reconstruct flip rate functions with error within $O(10^{-2})$ from $O(10^5)$ exit times when the particle speed is unity. For smaller particle speeds, the noise increases for a fixed number of exit times. This method is less time consuming compared with traditional projection methods which involve optimization.

On the exit time from a cone for brownian motion with drift

We investigate the tail distribution of the first exit time of Brownian motion with drift from a cone and find its exact asymptotics for a large class of cones. Our results show in particular that its exponential decreasing rate is a function of the distance between the drift and the cone, whereas the polynomial part in the asymptotics depends on the position of the drift with respect to the cone and its polar cone, and reflects the local geometry of the cone at the points that minimize the distance to the drift.

On the Distribution of First Exit Time for Brownian Motion with Double Linear Time-Dependent Barriers

This paper focuses on the first exit time for a Brownian motion with a double linear time-dependent barrier specified by y=a+bt, y=ct, (a>0, b<0, c>0). We are concerned in this paper with the distribution of the Brownian motion hitting the upper barrier before hitting the lower linear barrier. The main method we applied here is the Girsanov transform formula. As a result, we expressed the density of such exit time in terms of a finite series. This result principally provides us an analytical expression for the distribution of the aforementioned exit time and an easy way to compute the distribution of first exit time numerically.

Joint distribution of first exit times of a two dimensional Wiener process with jumps with application to a pair of coupled neurons

Motivated by a neuronal modeling problem, a bivariate Wiener process with two independent components is considered. Each component evolves independently until one of them reaches a threshold value. If the first component crosses the threshold value, it is reset while the dynamics of the other component remains unchanged. But, if this happens to the second component, the first one has a jump of constant amplitude; the second component is then reset to its starting value and its evolution restarts. Both processes evolve once again until one of them reaches again its boundary. In this work, the coupling of the first exit times of the two connected processes is studied.

Reconstruction of a persistent random walk from exit time distributions

In this paper, we study the inverse problem of reconstructing the spatially dependent transition rate F (x) of a one-dimensional Broadwell process from exit time distributions. In such a process, an advecting particle is assumed to undergo transitions between states with constant positive (+v) and negative (−v) velocities. The goal is to reconstruct the transition rate function F (x) from the exit time distributions out of a finite interval. Using the associated backward equation, we compute the distribution of exit times and its Laplace transform, given a fixed starting position and velocity. We propose two methods (called ‘t’ and ‘s’) for finding F (x). In both methods, we represent F (x) as a linear combination of polynomials and repeatedly solve the backward equation to minimize the difference between its solution and given first exit time data. In the t-method we work in the time domain, using exit times directly and leveraging a novel series solution for the exit time distribution. In the s-method, we work with the Laplace-transformed equation and Laplace-transformed exit times. Noisy data is generated using a custom-designed algorithm to simulate the trajectories of a Broadwell process. In most cases we can find 4 coefficients to within O(10−1) accuracy from O(104) exit times, with the t-method method slightly out-performing the s-method. We also explore the effectiveness of our algorithms for a fixed number of exit times under different advection speeds and find that optimal reconstruction occurs when v = O(1).

The Poincaré Map of Randomly Perturbed Periodic Motion

A system of autonomous differential equations with a stable limit cycle and perturbed by small white noise is analyzed in this work. In the vicinity of the limit cycle of the unperturbed deterministic system, we define, construct, and analyze the Poincare map of the randomly perturbed periodic motion. We show that the time of the first exit from a small neighborhood of the fixed point of the map, which corresponds to the unperturbed periodic orbit, is well approximated by the geometric distribution. The parameter of the geometric distribution tends zero together with the noise intensity. Therefore, our result can be interpreted as an estimate of stability of periodic motion to random perturbations. In addition, we show that the geometric distribution of the first exit times translates into statistical properties of solutions of important differential equation models in applications. To this end, we demonstrate three examples from mathematical neuroscience featuring complex oscillatory patterns characterized by the geometric distribution. We show that in each of these models the statistical properties of emerging oscillations are fully explained by the general properties of randomly perturbed periodic motions identified in this paper.

On the Convergence to Equilibrium for Degenerate Transport Problems

We give a counterexample which shows that the asymptotic rate of convergence to the equilibrium state for the transport equation, with a degenerate cross section and in the periodic setting, cannot be better than t −1/2 in the general case. We suggest, moreover, that the geometrical properties of the cross section are the key feature of the problem and impose, through the distribution of the forward exit time, the speed of convergence to the stationary state.

On the distribution of the first exit time and overshoot in a two-sided boundary crossing problem

We consider a random walk generated by a sequence of independent identically distributed random variables. We assume that the distribution function of a jump of the random walk equals an exponential polynomial on the negative half-axis. For double transforms of the joint distribution of the first exit time from an interval and overshoot, we obtain explicit expressions depending on finitely many parameters that, in turn, we can derive from the system of linear equations. The principal difference of the present article from similar results in this direction is the rejection of using factorization components and projection operators connected with them.

Two-Sided First Exit Problem for Jump Diffusion Distribution Processes Having Jumps with a Mixture of Erlang

In this paper, we consider the two-sided first exit problem for jump diffusion processes having jumps with rational Laplace transforms. We investigate the probabilistic property of conditional memorylessness, and drive the joint distribution of the first exit time from an interval and the overshoot over the boundary at the exit time.

Nonparametric regression of state occupation, entry, exit, and waiting times with multistate right‐censored data

We construct nonparametric regression estimators of a number of temporal functions in a multistate system based on a continuous univariate baseline covariate. These estimators include state occupation probabilities, state entry, exit, and waiting (sojourn) time distribution functions of a general progressive (e.g., acyclic) multistate model. We subject the data to right censoring, and the censoring mechanism is explainable by observable covariates that could be time dependent. The resulting estimators are valid even if the multistate process is non-Markov. We study the performance of the estimators in two simulation settings. We establish large sample consistency of these estimators. We illustrate our estimators using a data set on bone marrow transplant recipients.

A neural computation model for decision-making times

We introduce two new models for decision-making times for a two-choice decision task with no a priori bias. One of the models is the mean-field Curie–Weiss model of neural computation, and the other is based on dynamics near an unstable equilibrium under a small noise perturbation. As in the existing literature, we interpret exit times as reaction times and show that our models lead to a specific shape of the exit time distributions in the vanishing noise limit. We test the distribution shape against experimental data and show that for almost 90% of the participants, reaction times are described well by the model. Among the features of our model are: the dependence of the exit distribution only on two parameters, the elegance of rigorous mathematical analysis, and the microscopic nature of the noise.

Analysis of inverse stochastic resonance and the long-term firing of Hodgkin–Huxley neurons with Gaussian white noise

In order to explain the occurrence of a minimum in firing rate which occurs for certain mean input levels μ as noise level σ increases (inverse stochastic resonance, ISR) in Hodgkin–Huxley (HH) systems, we analyze the underlying transitions from a stable equilibrium point to limit cycle and vice-versa. For a value of μ at which ISR is pronounced, properties of the corresponding stable equilibrium point are found. A linearized approximation around this point has oscillatory solutions from whose maxima spikes tend to occur. A one dimensional diffusion is also constructed for small noise. Properties of the basin of attraction of the limit cycle (spike) are investigated heuristically. Long term trials of duration 500000 ms are carried out for values of σ from 0 to 2.0. The graph of mean spike count versus σ is divided into 4 regions R1,…,R4, where R3 contains the minimum associated with ISR. In R1 transitions to the basin of attraction of the rest point are not observed until a small critical value of σ=σc1 is reached, at the beginning of R2. The sudden decline in firing rate when σ is just greater than σc1 implies that there is only a small range of noise levels 0<σ<σc1 where repetitive spiking is safe from annihilation by noise. The firing rate remains small throughout R3. At a larger critical value σ=σc2 which signals the beginning of R4, the probability of transitions from the basin of attraction of the equilibrium point to that of the limit cycle apparently becomes greater than zero and the spike rate thereafter increases with increasing σ. The quantitative scheme underlying the ISR curve is outlined in terms of the properties of exit time random variables. In the final subsection, several statistical properties of the main random variables associated with long term spiking activity are given, including distributions of exit times from the two relevant basins of attraction and the interspike interval.