Passage Time Distribution Research Articles

Comments on Earthquake Probabilities Estimated by the Geometrical Mean Likelihood of the Brownian Passage Time Distribution Model for Recurrent Paleoearthquakes

Comments on Earthquake Probabilities Estimated by the Geometrical Mean Likelihood of the Brownian Passage Time Distribution Model for Recurrent Paleoearthquakes

Numerical Modeling of Anisotropic Particle Diffusion through a Cylindrical Channel.

The transport of molecules and particles through single pores is the basis of biological processes, including DNA and protein sequencing. As individual objects pass through a pore, they cause a transient change in the current that can be correlated with the object size, surface charge, and even chemical properties. The majority of experiments and modeling have been performed with spherical objects, while much less is known about the transport characteristics of aspherical particles, which would act as a model system, for example, for proteins and bacteria. The transport kinetics of aspherical objects is an especially important, yet understudied, problem in nanopore analytics. Here, using the Wiener process, we present a simplified model of the diffusion of rod-shaped particles through a cylindrical pore, and apply it to understand the translation and rotation of the particles as they pass through the pore. Specifically, we analyze the influence of the particles' geometrical characteristics on the effective diffusion type, the first passage time distribution, and the particles' orientation in the pore. Our model shows that thicker particles pass through the channel slower than thinner ones, while their lengths do not affect the passage time. We also demonstrate that both spherical and rod-shaped particles undergo normal diffusion, and the first passage time distribution follows an exponential asymptotics. The model provides guidance on how the shape of the particle can be modified to achieve an optimal passage time.

Vibrational Energy Landscapes and Energy Flow in GPCRs.

We construct and analyze disconnectivity graphs to provide the first graphical representation of the vibrational energy landscape of a protein, in this study β2AR, a G-protein coupled receptor (GPCR), in active and inactive states. The graphs, which indicate the relative free energy of each residue and the minimum free energy barriers for energy transfer between them, reveal important composition, structural and dynamic properties that mediate the flow of energy. Prolines and glycines, which contribute to GPCR plasticity and function, are identified as bottlenecks to energy transport along the backbone from which alternative pathways for energy transport via nearby noncovalent contacts emerge, seen also in the analysis of first passage time (FPT) distributions presented here. Striking differences between the disconnectivity graphs and FPT distributions for the inactive and active states of β2AR are found where structural and dynamic changes occur upon activation, contributing to allosteric regulation.

First-passage time distribution of a Brownian motion: two unexpected journeys

The distribution of the first-passage time (FPT) T a for a Brownian particle with drift μ subject to hitting an absorber at a level a > 0 is well-known and given by its density γ ( t ) = a 2 π t 3 e − ( a − μ t ) 2 2 t , t > 0 , which is normalized only if μ≥0. This article demonstrates the existence of two additional diffusion process categories (one with one parameter and the other with two) that have the same first passage-time distributions when μ < 0. For both, we identify the transition densities and thoroughly investigate the processes. A substantial implication is that the first-passage time distribution does not indicate whether the process originates from a drifted Brownian motion or from one of the new processes presented.

Stochastic dynamics of phycocyanin in years of contrasting phosphorus load

AbstractResilience, measured by the distribution of passage times between alternate states, indicates persistence of a state in stochastic dynamic systems such as blooms of cyanobacteria in lakes. We used high‐frequency datasets to compare the resilience of low and high states of phycocyanin, a pigment indicator of cyanobacteria, in Lake Mendota, Wisconsin, USA, for three growing seasons that ranged sevenfold in external phosphorus (P) load. Each year we observed 139–265 passage times across the unstable threshold that separated the low‐ from high‐phycocyanin states. Each sample of passage times is highly skewed with low median, larger mean, much larger SD, and wide tails extending to long lifetimes of a state. About 25% of events, whether low or high phycocyanin, lasted a day or more. Among these 3 years of contrasting external P load, there were no discernible differences in the resilience of either ecosystem state. We attribute this lack of contrast to the sustained recycling of P from sediments and the high stochasticity of phycocyanin in this lake.

Pricing vulnerable lookback options using Laplace transforms

This study develops a closed-form solution for pricing vulnerable lookback options, which combines the Black–Scholes model for the underlying asset and a correlated jump–diffusion model for the issuer’s asset to account for default risks. Our approach relies on applying Laplace transforms to establish a closed-form solution and compute them numerically using Laplace inversion algorithms. To determine the price, we derive the joint distribution of the first passage time of a drifted Brownian motion and the value of the correlated jump–diffusion. Through numerical examples, we demonstrate the accuracy of the numerical solutions obtained through our method and the stability of the algorithm. Subsequently, we conduct a numerical analysis to understand how counterparty risk influences option prices based on our formula.

Smoothing in linear multicompartment biological processes subject to stochastic input.

Many physical and biological systems rely on the progression of material through multiple independent stages. In viral replication, for example, virions enter a cell to undergo a complex process comprising several disparate stages before the eventual accumulation and release of replicated virions. While such systems may have some control over the internal dynamics that make up this progression, a challenge for many is to regulate behavior under what are often highly variable external environments acting as system inputs. In this work, we study a simple analog of this problem through a linear multicompartment model subject to a stochastic input in the form of a mean-reverting Ornstein-Uhlenbeck process, a type of Gaussian process. By expressing the system as a multidimensional Gaussian process, we derive several closed-form analytical results relating to the covariances and autocorrelations of the system, quantifying the smoothing effect discrete compartments afford multicompartment systems. Semianalytical results demonstrate that feedback and feedforward loops can enhance system robustness, and simulation results probe the intractable problem of the first passage time distribution, which has specific relevance to eventual cell lysis in the viral replication cycle. Finally, we demonstrate that the smoothing seen in the process is a consequence of the discreteness of the system, and does not manifest in systems with continuous transport. While we make progress through analysis of a simple linear problem, many of our insights are applicable more generally, and our work enables future analysis into multicompartment processes subject to stochastic inputs.

First passage time and peak size probability distributions for a complex epidemic model

First passage time and peak size probability distributions for a complex epidemic model

Analytical expressions for the first passage time distribution and hit distribution in two and three dimensions

The distribution of the time elapsed before a random variable reaches a threshold value for the first time, called the first passage time (FPT) distribution, is a fundamental characteristic of stochastic processes. Here, by solving the standard macroscopic diffusion equation, we derive analytical expressions for the FPT distribution of a diffusing particle hitting a spherical object in two dimensions (2D) and three dimensions (3D) in the course of unrestricted diffusion in open space. In addition, we calculate, analytically, the angular dependence of the FPT, known as the hit distribution. The analytical results are also compared to simulations of the motions of a random walker on a discrete lattice. This topic could be of wide pedagogical interest because the FPT is important not only in physics but also in chemistry, biology, medicine, agriculture, engineering, and finance. Additionally, the central equations often appear in physics and engineering with only trivial variations, making the solution techniques widely applicable.

Discounted densities of overshoot and undershoot for Lévy processes with applications in finance

Abstract This paper considers the first passage times to constant boundaries and the two-sided exit problem for Lévy process with a characteristic exponent in which at least one of the two jumps having rational Laplace transforms. The joint distribution of the first passage times and undershoot/overshoot are obtained. The processes recover many models that have appeared in the literature such as the compound Poisson risk models, the perturbed compound Poisson risk models, and their dual ones. As applications, we obtain the solutions for popular path-dependent options such as lookback and barrier options in terms of Laplace transforms.

Analysis and interpretation of first passage time distributions featuring rare events.

In this contribution we consider theory and associated computational tools to treat the kinetics associated with competing pathways on multifunnel energy landscapes. Multifunnel landscapes are associated with molecular switches and multifunctional materials, and are expected to exhibit multiple relaxation time scales and associated thermodynamic signatures in the heat capacity. Our focus here is on the first passage time distribution, which is encoded in a kinetic transition network containing all the locally stable states and the pathways between them. This network can be renormalised to reduce the dimensionality, while exactly conserving the mean first passage time and approximately conserving the full distribution. The structure of the reduced network can be visualised using disconnectivity graphs. We show how features in the first passage time distribution can be associated with specific kinetic traps, and how the appearance of competing relaxation time scales depends on the starting conditions. The theory is tested for two model landscapes and applied to an atomic cluster and a disordered peptide. Our most important contribution is probably the reconstruction of the full distribution for long time scales, where numerical problems prevent direct calculations. Here we combine accurate treatment of the mean first passage time with the reliable part of the distribution corresponding to faster time scales. Hence we now have a fundamental understanding of both thermodynamic and kinetic signatures of multifunnel landscapes.

Time-Dependent Probabilistic Seismic Hazard Analysis for Seismic Sequences Based on Hybrid Renewal Process Models

ABSTRACT Probabilistic seismic hazard analysis (PSHA) is a methodology with a long history and has been widely implemented. However, in the conventional PSHA and sequence-based probabilistic seismic hazard analysis (SPSHA) approaches, the occurrence of mainshocks is modeled as the homogeneous Poisson process, which is unsuitable for large earthquakes. To account for the stationary occurrence of small-to-moderate (STM) mainshocks and the nonstationary behavior of large mainshocks, we propose a time-dependent sequence-based probabilistic seismic hazard analysis (TD-SPSHA) approach by combining the time-dependent mainshock probabilistic seismic hazard analysis (TD-PSHA) and aftershock probabilistic seismic hazard analysis, consisting of four components: (1) STM mainshocks, (2) aftershocks associated with STM mainshocks, (3) large mainshocks, and (4) aftershocks associated with large mainshocks. The approach incorporates an exponential-magnitude, exponential-time model for STM mainshocks, and a renewal-time, characteristic-magnitude model for large mainshocks to assess the time-dependent hazard for mainshocks. Then nonhomogeneous Poisson process is used to model the occurrence of associated aftershocks, in which the aftershock sequences can be modeled using the Reasenberg and Jones (RJ) model or the epidemic-type aftershock sequence (ETAS) model. To demonstrate the proposed TD-SPSHA approach, a representative site of the San Andreas fault is selected as a benchmark case, for which five time-dependent recurrence models, including normal, lognormal, gamma, Weibull, and Brownian passage time (BPT) distributions, are chosen to determine the occurrence of large mainshocks. Then sensitivity tests are presented to show the effects on TD-SPSHA, including (1) time-dependent recurrence models, (2) mainshock magnitude, (3) rupture distance, (4) aftershock duration, (5) escaped time since the last event, and (6) future time interval. Furthermore, the bimodal hybrid renewal model is utilized by TD-SPSHA for another case site. The comparison results illustrate that the sequence hazard analysis approach ignoring time-varying properties of large earthquakes for long periods and the effects of associated aftershocks will result in a significantly underestimated hazard. The TD-SPSHA-based hazard curves using the ETAS model are larger than those of the RJ model. The proposed TD-SPSHA approach may be of significant interest to the field of earthquake engineering, particularly in the context of structural design or seismic risk analysis for the long term.

Optimal resetting strategies for search processes in heterogeneous environments

In many physical situations, there appears the problem of reaching a single target that is spatially distributed. Here we analyse how stochastic resetting, also spatially distributed, can be used to improve the search process when the target location is quenched, i.e. it does not evolve in time. More specifically, we consider a model with minimal but sufficient ingredients that allows us to derive analytical results for the relevant physical quantities, such as the first passage time distribution. We focus on the minimisation of the mean first passage time (MFPT) and its fluctuations (standard deviation), which proves to be non-trivial. Our analysis shows that the no-disorder case is singular: for small disorder, the resetting rate distribution that minimises the MFPT leads to diverging fluctuations—which impinge on the practicality of this minimisation. Interestingly, this issue is healed by minimising the fluctuations: the associated resetting rate distribution gives first passage times that are very close to the optimal ones.

First-passage times under frequent stochastic resetting.

We determine the full distribution and moments of the first passage time for a wide class of stochastic search processes in the limit of frequent stochastic resetting. Our results apply to any system whose short-time behavior of the search process without resetting can be estimated. In addition to the typical case of exponentially distributed resetting times, we prove our results for a variety of resetting time distributions. We illustrate our results in several examples and show that the errors of our approximations vanish exponentially fast in typical scenarios of diffusive search.

The winner takes it all but one

AbstractWe study competing first passage percolation on graphs generated by the configuration model with infinite-mean degrees. Initially, two uniformly chosen vertices are infected with a type 1 and type 2 infection, respectively, and the infection then spreads via nearest neighbors in the graph. The time it takes for the type 1 (resp. 2) infection to traverse an edge e is given by a random variable $X_1(e)$ (resp. $X_2(e)$ ) and, if the vertex at the other end of the edge is still uninfected, it then becomes type 1 (resp. 2) infected and immune to the other type. Assuming that the degrees follow a power-law distribution with exponent $\tau \in (1,2)$ , we show that with high probability as the number of vertices tends to infinity, one of the infection types occupies all vertices except for the starting point of the other type. Moreover, both infections have a positive probability of winning regardless of the passage-time distribution. The result is also shown to hold for the erased configuration model, where self-loops are erased and multiple edges are merged, and when the degrees are conditioned to be smaller than $n^\alpha$ for some $\alpha\gt 0$ .

Numerical approximation of the first-passage time distribution of time-varying diffusion decision models: A mesh-free approach

Sequential sampling theory is the dominant theoretical framework for explaining human decision making behavior. In this theory, the decision process is determined by a stochastic process in a bounded domain, with the likelihood function of the model being the first passage-time distribution of this process. However, many sequential sampling models have no analytic formulation for the first passage-time distribution, making these models difficult and computationally taxing to fit. Thus, developing an efficient and general approximation procedure for the first-passage time distribution of sequential sampling models is crucial for the field of mathematical psychology. Here, we have developed a numerical algorithm based on the mesh-free techniques for numerically solving the Fokker–Planck differential equation, which allows us to approximate the first passage-time distribution of some important classes of sequential sampling models, including the urgency signal model, Ornstein–Uhlenbeck model, and collapsing threshold model. Since the considered equation is a singular partial differential equation in a moving boundary domain, the proposed numerical method first transforms the equation to a regular domain, and then the solution is approximated based on the collocation method. However, since satisfying the boundary conditions in the collocation algorithms can be very tricky, we have imposed the boundary conditions in the basis functions. Furthermore, for the time-discrete and the full-discrete schemes error estimate has been presented to show the unconditional stability and convergence of the developed numerical method. The performance of the proposed method is illustrated by solving several examples.

Anomalous diffusion and long-range memory in the scaled voter model.

We analyze the scaled voter model, which is a generalization of the noisy voter model with time-dependent herding behavior. We consider the case when the intensity of herding behavior grows as a power-law function of time. In this case, the scaled voter model reduces to the usual noisy voter model, but it is driven by the scaled Brownian motion. We derive analytical expressions for the time evolution of the first and second moments of the scaled voter model. In addition, we have derived an analytical approximation of the first passage time distribution. By numerical simulation, we confirm our analytical results as well as showing that the model exhibits long-range memory indicators despite being a Markov model. The proposed model has steady-state distribution consistent with the bounded fractional Brownian motion, thus we expect it to be a good substitute model for the bounded fractional Brownian motion.

Temporal distribution characteristics of earthquakes in Taiwan, China

The characteristics of seismic temporal distribution represent an important basis for earthquake prediction and seismic hazard analysis. In this paper, based on the seismic catalogs in Taiwan, and using Poisson (exponential distribution), Gamma, Lognormal, Weibull, and Brownian passage time distributions as target models, we adopt the maximum likelihood method for estimating model parameters. The optimal model for describing the temporal distribution of earthquakes in Taiwan is determined according to the Akaike Information Criterion (AIC), Bayesian Information Criterion (BIC), K-S test, Chi-square test, and coefficient of determination R2 results. The results show that for moderate-strong earthquakes events (MW &amp;lt; 7.0), the Gamma distribution model can well describe the temporal distribution characteristics of earthquakes, while large earthquakes (MW ≥ 7.0) can be described entirely by exponential distribution. In addition, the temporal correlation between earthquakes is also examined through diffusion entropy analysis. The results show that seismic activity features temporal correlation, and earthquakes with relatively small magnitude (MW &amp;lt; 7) are affected by larger events (MW ≥ 7.0), thus suggesting long-term memory in time. In this study, the probability of the occurrence of a major earthquake in Taiwan is also calculated. The results show that the probability of an MW ≥ 7.0 earthquake in Taiwan in the next 10 years reaches 91.3%. The results may be used to inform the selection of seismic time distribution models and the calculation of seismic activity parameters in earthquake prediction and seismic hazard calculation, and hold scientific significance for understanding the mechanism of earthquake genesis.

Degradation modeling based on the gamma process with random initial degradation level and random threshold

ABSTRACT The stochastic modeling of performance characteristics is an important approach for the quality assessment of products. As part of the stochastic modeling, it is possible to consider different variation sources such as temporal, unit-to-unit heterogeneity and measurement error. These sources have been incorporated into different stochastic processes in the literature. Considering the unit-to-unit heterogeneity, two sources of variation can be found for certain products, particularly when the initial level and the critical threshold vary from unit-to-unit. Although, these sources have been studied separately, they have not been studied simultaneously. In this paper, a modeling approach is proposed which considers the gamma process to take into account the temporal uncertainty and the initial degradation level and the critical threshold level as two additional sources of variation. The modeling consists in first obtaining the distribution of the subtraction of the critical threshold and the initial level via a deconvolution operation. Then, the cumulative distribution function of the first passage time distribution is obtained through numerical integration. A simulation study is conducted to evaluate the performance of the proposed method, where different scenarios are analyzed. Furthermore, this approach is applied to a fatigue-crack propagation case study, from which a reliability assessment is performed.

Theoretical Tools to Quantify Stochastic Fluctuations in Single-Molecule Catalysis by Enzymes and Nanoparticles.

Single-molecule microscopic techniques allow the counting of successive turnover events and the study of the time-dependent fluctuations of the catalytic activities of individual enzymes and different sites on a single heterogeneous nanocatalyst. It is important to establish theoretical methods to obtain the statistical measurements of such stochastic fluctuations that provide insight into the catalytic mechanism. In this review, we discuss a few theoretical frameworks for evaluating the first passage time distribution functions using a self-consistent pathway approach and chemical master equations, to establish a connection with experimental observables. The measurable probability distribution functions and their moments depend on the molecular details of the reaction and provide a way to quantify the molecular mechanisms of the reaction process. The statistical measurements of these fluctuations should provide insight into the enzymatic mechanism.