Passage Time Density Research Articles

Boundary crossing problems and functional transformations for Ornstein–Uhlenbeck processes

ABSTRACT We are interested in the law of the first passage time of an Ornstein–Uhlenbeck process to time-varying thresholds. We show that this problem is connected to the laws of the first passage time of the process to members of a two-parameter family of functional transformations of a time-varying boundary. For specific values of the parameters, these transformations appear in a realisation of a standard Ornstein–Uhlenbeck bridge. We provide three different proofs of this connection. The first is based on a similar result for Brownian motion, the second uses a generalisation of the so-called Gauss–Markov processes, and the third relies on the Lie group symmetry method. We investigate the properties of these transformations and study the algebraic and analytical properties of an involution operator which is used in constructing them. We also show that these transformations map the space of solutions of Sturm–Liouville equations into the space of solutions of the associated nonlinear ordinary differential equations. Lastly, we interpret our results through the method of images and give new examples of curves with explicit first passage time densities.

Encounter-based reaction-subdiffusion model I: surface adsorption and the local time propagator

In this paper, we develop an encounter-based model of partial surface adsorption for fractional diffusion in a bounded domain. We take the probability of adsorption to depend on the amount of particle-surface contact time, as specified by a Brownian functional known as the boundary local time . If the rate of adsorption is state dependent, then the adsorption process is non-Markovian, reflecting the fact that surface activation/deactivation proceeds progressively by repeated particle encounters. The generalized adsorption event is identified as the first time that the local time crosses a randomly generated threshold. Different models of adsorption (Markovian and non-Markovian) then correspond to different choices for the random threshold probability density . The marginal probability density for particle position prior to absorption depends on ψ and the joint probability density for the pair , also known as the local time propagator. In the case of normal diffusion one can use a Feynman–Kac formula to derive an evolution equation for the propagator. Here we derive the local time propagator equation for fractional diffusion by taking a continuum limit of a heavy-tailed continuous-time random walk (CTRW). We begin by considering a CTRW on a one-dimensional lattice with a reflecting boundary at n = 0. We derive an evolution equation for the joint probability density of the particle location and the amount of time spent at the origin. The continuum limit involves rescaling by a factor , where is the lattice spacing. In the limit , the rescaled functional becomes the Brownian local time at x = 0. We use our encounter-based model to investigate the effects of subdiffusion and non-Markovian adsorption on the long-time behavior of the first passage time (FPT) density in a finite interval with a reflecting boundary at x = L. In particular, we determine how the choice of function ψ affects the large-t power law decay of the FPT density. Finally, we indicate how to extend the model to higher spatial dimensions.

Impact of Evanescence Process on Three-Dimensional Sub-Diffusion based Molecular Communication Channel.

In most of the existing works of molecular communication (MC), the standard diffusion environment is taken into account where the mean square displacement (MSD) of an information molecule (IM) scales linearly with time. On the contrary, this work considers the sub-diffusion motion that appears in crowded and complex (porous or fractal) environments (movement of the particles in the living cells) where the particle's MSD scales as a fractional order power law in time. Moreover, we examine an additional evanescence process resulting from which the molecules can degrade before hitting the boundary of the receiver (RX). Thus, in this work, we present a 3D MC system with a point transmitter (TX) and the spherical RX with the sub-diffusive behavior of an IM along with its evanescence. Furthermore, an IM's closed-form expressions for the arrival probability and the first passage time density (FPTD) are emulated in the above context. Additionally, we investigate the performance of MC by using the concentration-based modulation technique in a sub-diffusion channel. Finally, the considered MC channel is exploited in terms of the probability of detection, probability of false alarm, and probability of error for different parameters such as the reaction rate, fractional power, and radius of the RX.

Approximating the First Passage Time Density of Diffusion Processes with State-Dependent Jumps

We study the problem of the first passage time through a constant boundary for a jump diffusion process whose infinitesimal generator is a nonlocal Jacobi operator. Due to the lack of analytical results, we address the problem using a discretization scheme for simulating the trajectories of jump diffusion processes with state-dependent jumps in both frequency and amplitude. We obtain numerical approximations on their first passage time probability density functions and results for the qualitative behavior of other statistics of this random variable. Finally, we provide two examples of application of the method for different choices of the distribution involved in the mechanism of generation of the jumps.

Some new infinite series expansions for the first passage time densities in a jump diffusion model with phase-type jumps

We propose for the first time some explicit closed-form Laguerre series expansion formulas for the first passage time density functions of a jump diffusion model. Suppose that the jumps in the model are phase-type distributed. In contrast to existing methods in the literature based on numerical Laplace transform inversion, the proposed formulas are in analytical closed-form and simple to implement. Two different methods are proposed to compute the Laguerre coefficients. Various numerical examples are also given to show the effectiveness of our method.

Effect of Time-Dependent Drift and Diffusion in Molecular Communication Systems

This letter presents analysis of a molecular communication (MC) system considering time-dependent drift and diffusion of molecules. The proportionality in diffusion and drift is considered, and by applying the method of images, the general solution for Fokker-Planck (FP) equation is provided. Two different functional forms of the time-dependent drift and diffusion, viz., power-law time dependence and periodic driver, are used to analyze the MC system. The expressions for cumulative distribution function (CDF), also known as first passage probability (FPP), corresponding to first passage time density function (FPTDF) are derived. Moreover, the expressions for pulse-peak time and pulse peak of FPTDF are obtained. Further, the impacts of time-dependent drift and diffusion are examined in terms of average bit-error-rate (BER). Monte-Carlo and particle-based simulations (PBS) have also been performed to validate the feasibility of the used analytical models.

First passage time density of an Ornstein–Uhlenbeck process with broken drift

We consider an Ornstein–Uhlenbeck process with different drift rates below and above zero. We derive an analytic expression for the density of the first time, where the process hits a given level. The passage time density is linked to the joint law of the process and its running supremum, and we also provide an analytic formula of the joint density/distribution function. Results from a numerical experiment reveal that our formulas allow to numerically evaluate the joint law and the density of the first passage time faster than a simulation-based method.

First passage times for some classes of fractional time-changed diffusions

We consider some time-changed diffusion processes obtained by applying the Doob transformation rule to a time-changed Brownian motion. The time-change is obtained via the inverse of an α-stable subordinator. These processes are specified in terms of time-changed Gauss-Markov processes and fractional time-changed diffusions. A fractional pseudo-Fokker-Planck equation for such processes is given. We investigate their first passage time densities providing a generalized integral equation they satisfy and some transformation rules. First passage time densities for time-changed Brownian motion and Ornstein-Uhlenbeck processes are provided in several forms. Connections with closed form results and numerical evaluations through the level zero are given.

On the Simulation of a Special Class of Time-Inhomogeneous Diffusion Processes

General methods to simulate probability density functions and first passage time densities are provided for time-inhomogeneous stochastic diffusion processes obtained via a composition of two Gauss–Markov processes conditioned on the same initial state. Many diffusion processes with time-dependent infinitesimal drift and infinitesimal variance are included in the considered class. For these processes, the transition probability density function is explicitly determined. Moreover, simulation procedures are applied to the diffusion processes obtained starting from Wiener and Ornstein–Uhlenbeck processes. Specific examples in which the infinitesimal moments include periodic functions are discussed.

The First Passage Time Density of Brownian Motion and the Heat Equation with Dirichlet Boundary Condition in Time Dependent Domains

The First Passage Time Density of Brownian Motion and the Heat Equation with Dirichlet Boundary Condition in Time Dependent Domains

The first passage time density of Brownian motion and the heat equation with Dirichlet boundary condition in time dependent domains

В книге [Jimyeong Lee, "First passage time densities through Hölder curves", ALEA Lat. Am. J. Probab. Math. Stat., 15:2 (2018), 837-849] доказано, что плотность момента первого пересечения границы одномерным стандартным броуновским движением будет непрерывной, когда граница непрерывна по Гeльдеру с показателем больше $1/2$. С целью распространить результат [Jimyeong Lee, "First passage time densities through Hölder curves", ALEA Lat. Am. J. Probab. Math. Stat., 15:2 (2018), 837-849] на многомерные области мы показываем, что существует непрерывная функция плотности момента первого пересечения подвижных границ в $\mathbf R^d$, $d \ge 2$, стандартным $d$-мерным броуновским движением при $C^3$-диффеоморфизме. Как и в [Jimyeong Lee, "First passage time densities through Hölder curves", ALEA Lat. Am. J. Probab. Math. Stat., 15:2 (2018), 837-849], используя свойство локального времени стандартного $d$-мерного броуновского движения и уравнение теплопроводности с граничным условием Дирихле, мы находим достаточное условие существования непрерывной функции плотности.

Molecular Communication in Fractional Diffusive Channel

The molecular communication system with anomalous/fractional diffusion inside a one-dimensional (1-D) environment is considered. The time-dependent diffusivity is incorporated in terms of the power-law diffusivity, and the expression of first passage time density (FPTD) is derived. Further, the peak pulse time and peak concentration corresponding to the derived FPTD function are obtained. Moreover, the analysis is extended in terms of the average probability of error and throughput for the anomalous diffusion channel. The analytical results are validated through simulations.

First passage leapovers of Lévy flights and the proper formulation of absorbing boundary conditions

An important open problem in the theory of Lévy flights concerns the analytically tractable formulation of absorbing boundary conditions. Although the nonlocal approach, where the dynamical operators are given restrictions on their domain properties, has yielded substantial insights regarding the statistics of first passage, the resultant modifications to the dynamical equations hinder the detailed analysis possible in the absence of these nonlocal restrictions. In this study it is demonstrated that using the first-hit distribution, related to the first passage leapover, as the absorbing sink preserves the tractability of the dynamical equations for a particle undergoing Lévy flight. In particular, knowledge of the first-hit distribution is sufficient to fully determine the first passage time and position density of the particle, without requiring operator domain restrictions or numerical simulations. In addition, we report on the first-hit and leapover properties of first passages and arrivals for Lévy flights of arbitrary skew parameter, and extend these results to Lévy flights in a certain ubiquitous class of potentials satisfying an integral condition.

Lie Symmetries Methods in Boundary Crossing Problems for Diffusion Processes

This paper uses Lie symmetry methods to analyze boundary crossing probabilities for a large class of diffusion processes. We show that if the Fokker–Planck–Kolmogorov equation has non-trivial Lie symmetry, then the boundary crossing identity exists and depends only on parameters of process and symmetry. For time-homogeneous diffusion processes we found the necessary and sufficient conditions of the symmetries’ existence. This paper shows that if a drift function satisfies one of a family of Riccati equations, then the problem has nontrivial Lie symmetries. For each case we present symmetries in explicit form. Based on obtained results, we derive two-parametric boundary crossing identities and prove its uniqueness. Further, we present boundary crossing identities between different process. We show, that if the problem has 6 or 4 group of symmetries then the first passage time density to any boundary can be explicitly represented in terms of the first passage time by a Brownian motion or a Bessel process. Many examples are presented to illustrate the method.

Explicit asymptotics on first passage times of diffusion processes

AbstractWe introduce a unified framework for solving first passage times of time-homogeneous diffusion processes. Using potential theory and perturbation theory, we are able to deduce closed-form truncated probability densities, as asymptotics or approximations to the original first passage time densities, for single-side level crossing problems. The framework is applicable to diffusion processes with continuous drift functions; in particular, for bounded drift functions, we show that the perturbation series converges. In the present paper, we demonstrate examples of applying our framework to the Ornstein–Uhlenbeck, Bessel, exponential-Shiryaev, and hypergeometric diffusion processes (the latter two being previously studied by Dassios and Li (2018) and Borodin (2009), respectively). The purpose of this paper is to provide a fast and accurate approach to estimating first passage time densities of various diffusion processes.

A Note on First Passage Time Densities for Stochastic Simulation of Diffusion

This brief note extends our working paper [1]. A central claim in [1] is that the first passage time densities in 1-3 dimensions have upper bounds that allow the implementation of efficient von Neumann rejection schemes for generating random first passage times. This approach was adopted, among other methods, in [4]. The claim mentioned above was not proved in [1]. This note attempts to close this gap using the arguments provided in [2].

A Symmetry-Based Approach for First-Passage-Times of Gauss-Markov Processes through Daniels-Type Boundaries

Symmetry properties of the Brownian motion and of some diffusion processes are useful to specify the probability density functions and the first passage time density through specific boundaries. Here, we consider the class of Gauss-Markov processes and their symmetry properties. In particular, we study probability densities of such processes in presence of a couple of Daniels-type boundaries, for which closed form results exit. The main results of this paper are the alternative proofs to characterize the transition probability density between the two boundaries and the first passage time density exploiting exclusively symmetry properties. Explicit expressions are provided for Wiener and Ornstein-Uhlenbeck processes.

On an algorithm for solving Fredholm integrals of the first kind

In this paper we use an iterative algorithm for solving Fredholm equations of the first kind. The basic algorithm is known and is based on an EM algorithm when involved functions are non-negative and integrable. With this algorithm we demonstrate two examples involving the estimation of a mixing density and a first passage time density function involving Brownian motion. We also develop the basic algorithm to include functions which are not necessarily non-negative and again present illustrations under this scenario. A self contained proof of convergence of all the algorithms employed is presented.

First Passage Time Densities through Hölder curves

We prove that for a standard Brownian motion, there exists a first-passage-time density function through a locally H\"older continuous curve with exponent greater than 1/2. By using a property of local time of a standard Brownian motion and the theories of partial differential equations in Cannon [2], we find a sufficient condition for existence of the density function. We also show that this density function is proportional to the space derivative of the Green function of the heat equation with Dirichlet boundary condition at the moving boundary.

Linked Gauss-Diffusion processes for modeling a finite-size neuronal network

A Leaky Integrate-and-Fire (LIF) model with stochastic current-based linkages is considered to describe the firing activity of neurons interacting in a (2×2)-size feed-forward network. In the subthreshold regime and under the assumption that no more than one spike is exchanged between coupled neurons, the stochastic evolution of the neuronal membrane voltage is subject to random jumps due to interactions in the network. Linked Gauss-Diffusion processes are proposed to describe this dynamics and to provide estimates of the firing probability density of each neuron. To this end, an iterated integral equation-based approach is applied to evaluate numerically the first passage time density of such processes through the firing threshold. Asymptotic approximations of the firing densities of surrounding neurons are used to obtain closed-form expressions for the mean of the involved processes and to simplify the numerical procedure. An extension of the model to an (N×N)-size network is also given. Histograms of firing times obtained by simulations of the LIF dynamics and numerical firings estimates are compared.