First Passage Time Distribution Research Articles

The stretch to stray on time: Resonant length of random walks in a transient.

First-passage times in random walks have a vast number of diverse applications in physics, chemistry, biology, and finance. In general, environmental conditions for a stochastic process are not constant on the time scale of the average first-passage time or control might be applied to reduce noise. We investigate moments of the first-passage time distribution under an exponential transient describing relaxation of environmental conditions. We solve the Laplace-transformed (generalized) master equation analytically using a novel method that is applicable to general state schemes. The first-passage time from one end to the other of a linear chain of states is our application for the solutions. The dependence of its average on the relaxation rate obeys a power law for slow transients. The exponent ν depends on the chain length N like ν=-N/(N+1) to leading order. Slow transients substantially reduce the noise of first-passage times expressed as the coefficient of variation (CV), even if the average first-passage time is much longer than the transient. The CV has a pronounced minimum for some lengths, which we call resonant lengths. These results also suggest a simple and efficient noise control strategy and are closely related to the timing of repetitive excitations, coherence resonance, and information transmission by noisy excitable systems. A resonant number of steps from the inhibited state to the excitation threshold and slow recovery from negative feedback provide optimal timing noise reduction and information transmission.

The one-dimensional asymmetric persistent random walk

Persistent random walks are intermediate transport processes between a uniform rectilinear motion and a Brownian motion. They are formed by successive steps of random finite lengths and directions travelled at a fixed speed. The isotropic and symmetric 1D persistent random walk is governed by the telegrapher’s equation, also called the hyperbolic heat conduction equation. These equations have been designed to resolve the paradox of the infinite speed in the heat and diffusion equations. The finiteness of both the speed and the correlation length leads to several classes of random walks: Persistent random walk in one dimension can display anomalies that cannot arise for Brownian motion such as anisotropy and asymmetries. In this work we focus on the case where the mean free path is anisotropic, the only anomaly leading to a physics that is different from the telegrapher’s case. We derive exact expression of its Green’s function, for its scattering statistics and distribution of first-passage time at the origin. The phenomenology of the latter shows a transition for quantities like the escape probability and the residence time.

Numerically exact full counting statistics of the nonequilibrium Anderson impurity model

The time-dependent full counting statistics of charge transport through an interacting quantum junction is evaluated from its generating function, controllably computed with the inchworm Monte Carlo method. Exact noninteracting results are reproduced; then, we continue to explore the effect of electron-electron interactions on the time-dependent charge cumulants, first-passage time distributions, and $n$-electron transfer distributions. We observe a crossover in the noise from Coulomb blockade to Kondo-dominated physics as the temperature is decreased. In addition, we uncover long-tailed spin distributions in the Kondo regime and analyze queuing behavior caused by correlations between single-electron transfer events.

Extremal behavior of hitting a cone by correlated Brownian motion with drift

This paper derives an exact asymptotic expression for Pxu{∃t≥0X(t)−μt∈U},asu→∞,where X(t)=(X1(t),…,Xd(t))⊤,t≥0 is a correlated d-dimensional Brownian motion starting at the point xu=−αu with α∈Rd, μ∈Rd and U=∏i=1d[0,∞). The derived asymptotics depends on the solution of an underlying multidimensional quadratic optimization problem with constraints, which leads in some cases to dimension-reduction of the considered problem. Complementary, we study asymptotic distribution of the conditional first passage time to U, which depends on the dimension-reduction phenomena.

First-passage times in renewal and nonrenewal systems.

Fluctuations in stochastic systems are usually characterized by full counting statistics, which analyzes the distribution of the number of events taking place in the fixed time interval. In an alternative approach, the distribution of the first-passage times, i.e., the time delays after which the counting variable reaches a certain threshold value, is studied. This paper presents the approach to calculate the first-passage time distribution in systems in which the analyzed current is associated with an arbitrary set of transitions within the Markovian network. Using this approach, it is shown that when the subsequent first-passage times are uncorrelated, there exist strict relations between the cumulants of the full counting statistics and the first-passage time distribution. On the other hand, when the correlations of the first-passage times are present, their distribution may provide additional information about the internal dynamics of the system in comparison to the full counting statistics; for example, it may reveal the switching between different dynamical states of the system. Additionally, I show that breaking of the fluctuation theorem for first-passage times may reveal the multicyclic nature of the Markovian network.

Towards a full quantitative description of single-molecule reaction kinetics in biological cells.

The first-passage time (FPT), i.e., the moment when a stochastic process reaches a given threshold value for the first time, is a fundamental mathematical concept with immediate applications. In particular, it quantifies the statistics of instances when biomolecules in a biological cell reach their specific binding sites and trigger cellular regulation. Typically, the first-passage properties are given in terms of mean first-passage times. However, modern experiments now monitor single-molecular binding-processes in living cells and thus provide access to the full statistics of the underlying first-passage events, in particular, inherent cell-to-cell fluctuations. We here present a robust explicit approach for obtaining the distribution of FPTs to a small partially reactive target in cylindrical-annulus domains, which represent typical bacterial and neuronal cell shapes. We investigate various asymptotic behaviours of this FPT distribution and show that it is typically very broad in many biological situations, thus, the mean FPT can differ from the most probable FPT by orders of magnitude. The most probable FPT is shown to strongly depend only on the starting position within the geometry and to be almost independent of the target size and reactivity. These findings demonstrate the dramatic relevance of knowing the full distribution of FPTs and thus open new perspectives for a more reliable description of many intracellular processes initiated by the arrival of one or few biomolecules to a small, spatially localised region inside the cell.

Efficient Low-Order Approximation of First-Passage Time Distributions.

We consider the problem of computing first-passage time distributions for reaction processes modeled by master equations. We show that this generally intractable class of problems is equivalent to a sequential Bayesian inference problem for an auxiliary observation process. The solution can be approximated efficiently by solving a closed set of coupled ordinary differential equations (for the low-order moments of the process) whose size scales with the number of species. We apply it to an epidemic model and a trimerization process and show good agreement with stochastic simulations.

On distibutions of first passage times of martingales arising in some gambling problems

Using a martingale technique we derive bounds and asymptotics for tail distributions of first passage times $$\tau _{b}$$ associated with crossing a level b for some gambling strategies. In particilar, for the case of martingale games with so-called “Oscar strategy” we show that $$P\{\tau _{b}>n\}\le C n^{-3/2}\,$$ for any level $$b>0$$ .

First Passage Time Memory Lifetimes for Simple, Multistate Synapses.

Memory models based on synapses with discrete and bounded strengths store new memories by forgetting old ones. Memory lifetimes in such memory systems may be defined in a variety of ways. A mean first passage time (MFPT) definition overcomes much of the arbitrariness and many of the problems associated with the more usual signal-to-noise ratio (SNR) definition. We have previously computed MFPT lifetimes for simple, binary-strength synapses that lack internal, plasticity-related states. In simulation we have also seen that for multistate synapses, optimality conditions based on SNR lifetimes are absent with MFPT lifetimes, suggesting that such conditions may be artifactual. Here we extend our earlier work by computing the entire first passage time (FPT) distribution for simple, multistate synapses, from which all statistics, including the MFPT lifetime, may be extracted. For this, we develop a Fokker-Planck equation using the jump moments for perceptron activation. Two models are considered that satisfy a particular eigenvector condition that this approach requires. In these models, MFPT lifetimes do not exhibit optimality conditions, while in one but not the other, SNR lifetimes do exhibit optimality. Thus, not only are such optimality conditions artifacts of the SNR approach, but they are also strongly model dependent. By examining the variance in the FPT distribution, we may identify regions in which memory storage is subject to high variability, although MFPT lifetimes are nevertheless robustly positive. In such regions, SNR lifetimes are typically (defined to be) zero. FPT-defined memory lifetimes therefore provide an analytically superior approach and also have the virtue of being directly related to a neuron's firing properties.

Generic Schemes for Single-Molecule Kinetics. 2: Information Content of the Poisson Indicator.

Recently, we described a pathway analysis technique (paper 1) for analyzing generic schemes for single-molecule kinetics based upon the first-passage time distribution. Here, we employ this method to derive expressions for the Poisson indicator, a normalized measure of stochastic variation (essentially equivalent to the Fano factor and Mandel's Q parameter), for various renewal (i.e., memoryless) enzymatic reactions. We examine its dependence on substrate concentration, without assuming all steps follow Poissonian kinetics. Based upon fitting to the functional forms of the first two waiting time moments, we show that, to second order, the non-Poissonian kinetics are generally underdetermined but can be specified in certain scenarios. For an enzymatic reaction with an arbitrary intermediate topology, we identify a generic minimum of the Poisson indicator as a function of substrate concentration, which can be used to tune substrate concentration to the stochastic fluctuations and to estimate the largest number of underlying consecutive links in a turnover cycle. We identify a local maximum of the Poisson indicator (with respect to substrate concentration) for a renewal process as a signature of competitive binding, either between a substrate and an inhibitor or between multiple substrates. Our analysis explores the rich connections between Poisson indicator measurements and microscopic kinetic mechanisms.

Performability evaluation of the ERTMS/ETCS – Level 3

Level 3 of the ERTMS/ETCS improves the capacity of railways by replacing fixed-block signalling, which prevents a train to enter a block occupied by another train, with moving block signalling, which allows a train to proceed as long as it receives radio messages ensuring that the track ahead is clear of other trains. If messages are lost, a train must stop for safety reasons within a given deadline, even though the track ahead is clear, making the availability of the communication link crucial for successful operation.We combine analytic evaluation of failures due to burst noise and connection losses with numerical solution of a non-Markovian model representing also failures due to handovers between radio stations. In so doing, we show that handovers experienced by a pair of chasing trains periodically affect the availability of the radio link, making behavior of the overall communication system recurrent over the hyper-period of periodic message releases and periodic arrivals at cell borders. As a notable aspect, non-Markovian transient analysis within two hyper-periods is sufficient to derive an upper bound on the first-passage time distribution to an emergency brake, permitting to achieve a trade-off between railway throughput and stop probability. A sensitivity analysis is performed with respect to train speed and headway distance, permitting to gain insight into the consequences of system-level design choices.

A novel measure of liquidity premium: application to the Korean stock market

ABSTRACTThis study proposes a novel measure for an asset’s liquidity premium. Applying Brownian first-passage time distribution properties, we derive an explicit form of liquidity premium embedded in the asset price. Our liquidity premium measure is intuitive because it assesses the extent to which the value of the asset should be increased from the current market price if investors were allowed to retain the asset until they achieve an investment goal. This measure is readily available for assessing an asset’s liquidity because it does not require information on the asset’s transactional characteristics. Our empirical experiment using Korean stock market data suggests that the liquidity premium in this study is inversely related to Amihud’s (2002) illiquidity ratio, which is commonly used to measure stocks’ illiquidity.

Kramers problem for a dimer: Effect of noise correlations.

The Kramers problem for a dimer in a bistable piecewise linear potential is studied in the presence of correlated noise processes. The effect of such a correlation is to redistribute the thermal power between the dynamical degrees of freedom, and this leads to significant deviations in the dynamics of the system from the case of independent noise processes. The distribution of first passage times from one minima to the basin of attraction of the other minima is found to have exponentially decaying tails with the parameter dependent on the amount of correlation and the coupling between the particles. The strong coupling limit of the problem is analyzed using adiabatic elimination, where it is found that the initial probability density relaxes towards a stationary value on the same time scale as the mean escape time when the noise intensity of the system is low. For higher noise fluctuations, the relaxation towards the stationary state is slower in comparison to escape times. In the extreme limit of perfect anticorrelation, the random dynamical system behaves as a deterministic system in a steady state in which the center of mass starting from the unstable maxima moves down the hill and gets trapped in the potential minima. The implications for polymer dynamics in a potential are discussed.

On the Joint Distribution of First-passage Time and First-passage Area of Drifted Brownian Motion

For drifted Brownian motion $X(t)= x - \mu t + B_t \ (\mu >0)$ starting from $x>0,$ we study the joint distribution of the first-passage time below zero, $\tau(x),$ and the first-passage area, $A(x),$ swept out by $X$ till the time $\tau(x).$ In particular, we establish differential equations with boundary conditions for the joint moments $E[\tau(x)^m A(x)^n],$ and we present an algorithm to find recursively them, for any $m$ and $n.$ Finally, the expected value of the time average of $X$ till the time $\tau(x)$ is obtained.

First-passage time approach to controlling noise in the timing of intracellular events

In the noisy cellular environment, gene products are subject to inherent random fluctuations in copy numbers over time. How cells ensure precision in the timing of key intracellular events despite such stochasticity is an intriguing fundamental problem. We formulate event timing as a first-passage time problem, where an event is triggered when the level of a protein crosses a critical threshold for the first time. Analytical calculations are performed for the first-passage time distribution in stochastic models of gene expression. Derivation of these formulas motivates an interesting question: Is there an optimal feedback strategy to regulate the synthesis of a protein to ensure that an event will occur at a precise time, while minimizing deviations or noise about the mean? Counterintuitively, results show that for a stable long-lived protein, the optimal strategy is to express the protein at a constant rate without any feedback regulation, and any form of feedback (positive, negative, or any combination of them) will always amplify noise in event timing. In contrast, a positive feedback mechanism provides the highest precision in timing for an unstable protein. These theoretical results explain recent experimental observations of single-cell lysis times in bacteriophage [Formula: see text] Here, lysis of an infected bacterial cell is orchestrated by the expression and accumulation of a stable [Formula: see text] protein up to a threshold, and precision in timing is achieved via feedforward rather than feedback control. Our results have broad implications for diverse cellular processes that rely on precise temporal triggering of events.

Efficient Computation of First Passage Times in Kou’s Jump-diffusion Model

S. G. Kou and H. Wang [First passage times of a jump diffusion process, Adv. Appl. Probab.35 (2003) 504–531] give expressions of both the real Laplace transform of the distribution of first passage time and the real Laplace transform of the joint distribution of the first passage time and the running maxima of a jump-diffusion model called Kou model. These authors invert the former Laplace transform by using Gaver-Stehfest algorithm, and for the latter they need a large computing time with an algebra computer system. In the present paper, we give a much simpler expression of the Laplace transform of the joint distribution, and we also show, using Complex Analysis techniques, that both Laplace transforms can be extended to the complex plane. Hence, we can use inversion methods based on the complex inversion formula or Bromwich integral which are very efficent. The improvement in the computing times and accuracy is remarkable.

Random Walk Analysis of the Effect of Mechanical Degradation on All-Solid-State Battery Power

Mechanical and electrochemical phenomena are coupled in defining the battery reliability, particularly for solid-state batteries. Micro-cracks act as barriers to Li-ion diffusion in the electrolyte, increasing the average electrode’s tortuosity. In our previous work, we showed that solid electrolytes are likely to suffer from mechanical degradation if their fracture energy is lower than 4 J m− 2 [G. Bucci, T. Swamy, Y.-M. Chiang, and W. C. Carter, J. Mater. Chem. A (2017)]. Here we study the effect of electrolyte micro-cracking on the effective conductivity of composite electrodes. Via random analyzes, we predict the average diffusivity of lithium in a solid-state electrode to decrease linearly with the extension of mechanical degradation. Furthermore, the statistical distribution of first passage times indicates that the microstructure becomes more and more heterogeneous as damage progresses. In addition to power and capacity loss, a non-uniform increase of the electrode tortuosity can lead to heterogeneous lithiation and further stress localization. The understanding of these phenomena at the mesoscale is essential to the implementation of safe high-energy solid-state batteries.

Extremes ofγ-reflected Gaussian processes with stationary increments

For a given centered Gaussian process with stationary increments $\{X(t), t\geq 0\}$ and $c>0$, let $$ W_\gamma(t)=X(t)-ct-\gamma\inf_{0\leq s\leq t}\left(X(s)-cs\right), \quad t\geq 0$$ denote the $\gamma$-reflected process, where $\gamma\in (0,1)$. This process is introduced in the context of risk theory to model surplus process that include tax payments of loss-carry forward type.In this contribution we derive asymptotic approximations of both the ruin probability and the joint distribution of first and last passage times given that ruin occurs. We apply our findings to the cases with $X$ being the multiplex fractional Brownian motion and the integrated Gaussian processes. As a by-product we derive an extension of Piterbarg inequality \KD{for} threshold-dependent random fields.

The first-passage time distribution for the diffusion model with variable drift

The Ratcliff diffusion model is now arguably the most widely applied model for response time data. Its major advantage is its description of both response times and the probabilities for correct as well as incorrect responses. The model assumes a Wiener process with drift between two constant absorbing barriers. The first-passage times at the upper and lower boundary describe the responses in simple two-choice decision tasks, for example, in experiments with perceptual discrimination or memory search. In applications of the model, a usual assumption is a varying drift of the Wiener process across trials. This extra flexibility allows accounting for slow errors that often occur in response time experiments. So far, the predicted response time distributions were obtained by numerical evaluation as analytical solutions were not available. Here, we present an analytical expression for the cumulative first-passage time distribution in the diffusion model with normally distributed trial-to-trial variability in the drift. The solution is obtained with predefined precision, and its evaluation turns out to be extremely fast.

Waiting for rare entropic fluctuations

Nonequilibrium fluctuations of various stochastic variables, such as work and entropy production, have been widely discussed recently in the context of large deviations, cumulants and fluctuation relations. Typically one looks at the probability distributions for entropic fluctuations of various sizes to occur in a fixed time interval. An important and natural question is to ask for the time one has to wait to see fluctuations of a desired size. We address this question by studying the first-passage time distribution (FPTD). We derive the general basic equation to get the FPTD for entropic variables. Based on this, the FPTD on entropy production in a driven colloidal particle in the ring geometry is illustrated. A general asymptotic form of the FPTD and integral fluctuation relation symmetry in terms of the first passages are found.