Hitting probability for anomalous diffusion processes

Evidence and quantification of memory effects in competitive first-passage events

Splitting probabilities quantify the likelihood of a given outcome out of competitive events. This key observable of random walk theory, historically introduced as the gambler's ruin problem, is well understood for memoryless (Markovian) processes. However, in complex systems such as polymer fluids, the motion of a particle should typically be described as a process with memory, for which splitting probabilities are much less characterized analytically. Here, we introduce an analytical approach that provides the splitting probabilities for one-dimensional isotropic non-Markovian Gaussian processes with stationary increments, in the case of two targets. This analysis shows that splitting probabilities are controlled by the out-of-equilibrium trajectories observed after the first passage. This is directly evidenced in a prototypical experimental reaction scheme in viscoelastic fluids. These results are extended to d-dimensional processes in large confining volumes, opening a path toward the study of competitive events in complex media.

Nonergodicity of confined superdiffusive fractional Brownian motion

Using stochastic simulations supported by analytics we determine the degree of nonergodicity of box-confined fractional Brownian motion for both sub- and superdiffusive Hurst exponents H. At H>1/2 the nonequivalence of the ensemble- and time-averaged mean-squared displacements (TAMSDs) is found to be most pronounced (with a giant spread of individual TAMSDs at H→1), with two distinct short-lag-time TAMSD exponents.

First-passage functionals of Brownian motion in logarithmic potentials and heterogeneous diffusion

We study the statistics of random functionals Z=∫_{0}^{T}[x(t)]^{γ-2}dt, where x(t) is the trajectory of a one-dimensional Brownian motion with diffusion constant D under the effect of a logarithmic potential V(x)=V_{0}ln(x). The trajectory starts from a point x_{0} inside an interval entirely contained in the positive real axis, and the motion is evolved up to the first-exit time T from the interval. We compute explicitly the PDF of Z for γ=0, and its Laplace transform for γ≠0, which can be inverted for particular combinations of γ and V_{0}. Then we consider the dynamics in (0,∞) up to the first-passage time to the origin and obtain the exact distribution for γ>0 and V_{0}>-D. By using a mapping between Brownian motion in logarithmic potentials and heterogeneous diffusion, we extend this result to functionals measured over trajectories generated by x[over ̇](t)=sqrt[2D][x(t)]^{θ}η(t), where θ<1 and η(t) is a Gaussian white noise. We also emphasize how the different interpretations that can be given to the Langevin equation affect the results. Our findings are illustrated by numerical simulations, with good agreement between data and theory.

Leftward, rightward, and complete exit-time distributions of jump processes

First-passage properties of continuous stochastic processes confined in a one-dimensional interval are well described. However, for jump processes (discrete random walks), the characterization of the corresponding observables remains elusive, despite their relevance in various contexts. Here we derive exact asymptotic expressions for the leftward, rightward, and complete exit-time distributions from the interval [0,x] for symmetric jump processes starting from x_{0}=0, in the large x and large time limit. We show that both the leftward probability F_{[under 0]̲,x}(n) to exit through 0 at step n and rightward probability F_{0,[under x]̲}(n) to exit through x at step n exhibit a universal behavior dictated by the large-distance decay of the jump distribution parametrized by the Levy exponent μ. In particular, we exhaustively describe the n≪(x/a_{μ})^{μ} and n≫(x/a_{μ})^{μ} limits and obtain explicit results in both regimes. Our results finally provide exact asymptotics for exit-time distributions of jump processes in regimes where continuous limits do not apply.

Stochastic kinetics under combined action of two noise sources

We are exploring two archetypal noise-induced escape scenarios: Escape from a finite interval and from the positive half-line under the action of the mixture of Lévy and Gaussian white noises in the overdamped regime, for the random acceleration process and higher-order processes. In the case of escape from finite intervals, the mixture of noises can result in the change of value of the mean first passage time in comparison to the action of each noise separately. At the same time, for the random acceleration process on the (positive) half-line, over the wide range of parameters, the exponent characterizing the power-law decay of the survival probability is equal to the one characterizing the decay of the survival probability under action of the (pure) Lévy noise. There is a transient region, the width of which increases with stability index α, when the exponent decreases from the one for Lévy noise to the one corresponding to the Gaussian white noise driving.

Splitting Probabilities of Symmetric Jump Processes

We derive a universal, exact asymptotic form of the splitting probability for symmetric continuous jump processes, which quantifies the probability π_{0,[under x]_}(x_{0}) that the process crosses x before 0 starting from a given position x_{0}∈[0,x] in the regime x_{0}≪x. This analysis provides in particular a fully explicit determination of the transmission probability (x_{0}=0), in striking contrast with the trivial prediction π_{0,[under x]_}(0)=0 obtained by taking the continuous limit of the process, which reveals the importance of the microscopic properties of the dynamics. These results are illustrated with paradigmatic models of jump processes with applications to light scattering in heterogeneous media in realistic 3D slab geometries. In this context, our explicit predictions of the transmission probability, which can be directly measured experimentally, provide a quantitative characterization of the effective random process describing light scattering in the medium.

Joint statistics of space and time exploration of one-dimensional random walks

The statistics of first-passage times of random walks to target sites has proved to play a key role in determining the kinetics of space exploration in various contexts. In parallel, the number of distinct sites visited by a random walker and related observables has been introduced to characterize the geometry of space exploration. Here, we address the question of the joint distribution of the first-passage time to a target and the number of distinct sites visited when the target is reached, which fully quantifies the coupling between the kinetics and geometry of search trajectories. Focusing on one-dimensional systems, we present a general method and derive explicit expressions of this joint distribution for several representative examples of Markovian search processes. In addition, we obtain a general scaling form, which holds also for non-Markovian processes and captures the general dependence of the joint distribution on its space and time variables. We argue that the joint distribution has important applications to various problems, such as a conditional form of the Rosenstock trapping model, and the persistence properties of self-interacting random walks.

Extreme value statistics and arcsine laws for heterogeneous diffusion processes

Heterogeneous diffusion with a spatially changing diffusion coefficient arises in many experimental systems such as protein dynamics in the cell cytoplasm, mobility of cajal bodies, and confined hard-sphere fluids. Here, we showcase a simple model of heterogeneous diffusion where the diffusion coefficient D(x) varies in a power-law way, i.e., D(x)∼|x|^{-α} with the exponent α>-1. This model is known to exhibit anomalous scaling of the mean-squared displacement (MSD) of the form ∼t^{2/2+α} and weak ergodicity breaking in the sense that ensemble averaged and time averaged MSDs do not converge. In this paper, we look at the extreme value statistics of this model and derive, for all α, the exact probability distributions of the maximum spatial displacement M(t) and arg-maximum t_{m}(t) (i.e., the time at which this maximum is reached) till duration t. In the second part of our paper, we analyze the statistical properties of the residence time t_{r}(t) and the last-passage time t_{ℓ}(t) and compute their distributions exactly for all values of α. Our study unravels that the heterogeneous version (α≠0) displays many rich and contrasting features compared to that of the standard Brownian motion (BM). For example, while for BM (α=0), the distributions of t_{m}(t),t_{r}(t), and t_{ℓ}(t) are all identical (á la "arcsine laws" due to Lévy), they turn out to be significantly different for nonzero α. Another interesting property of t_{r}(t) is the existence of a critical α (which we denote by α_{c}=-0.3182) such that the distribution exhibits a local maximum at t_{r}=t/2 for α<α_{c} whereas it has minima at t_{r}=t/2 for α≥α_{c}. The underlying reasoning for this difference hints at the very contrasting natures of the process for α≥α_{c} and α<α_{c} which we thoroughly examine in our paper. All our analytical results are backed by extensive numerical simulations.

Extremal statistics for stochastic resetting systems

While averages and typical fluctuations often play a major role in understanding the behavior of a nonequilibrium system, this nonetheless is not always true. Rare events and large fluctuations are also pivotal when a thorough analysis of the system is being done. In this context, the statistics of extreme fluctuations in contrast to the average plays an important role, as has been discussed in fields ranging from statistical and mathematical physics to climate, finance, and ecology. Herein, we study extreme value statistics (EVS) of stochastic resetting systems, which have recently gained significant interest due to its ubiquitous and enriching applications in physics, chemistry, queuing theory, search processes, and computer science. We present a detailed analysis for the finite and large time statistics of extremals (maximum and arg-maximum, i.e., the time when the maximum is reached) of the spatial displacement in such system. In particular, we derive an exact renewal formula that relates the joint distribution of maximum and arg-maximum of the reset process to the statistical measures of the underlying process. Benchmarking our results for the maximum of a reset trajectory that pertain to the Gumbel class for large sample size, we show that the arg-maximum density attains a uniform distribution independent of the underlying process at a large observation time. This emerges as a manifestation of the renewal property of the resetting mechanism. The results are augmented with a wide spectrum of Markov and non-Markov stochastic processes under resetting, namely, simple diffusion, diffusion with drift, Ornstein-Uhlenbeck process, and random acceleration process in one dimension. Rigorous results are presented for the first two setups, while the latter two are supported with heuristic and numerical analysis.

Distribution of the time between maximum and minimum of random walks

We consider a one-dimensional Brownian motion of fixed duration T. Using a path-integral technique, we compute exactly the probability distribution of the difference τ=t_{min}-t_{max} between the time t_{min} of the global minimum and the time t_{max} of the global maximum. We extend this result to a Brownian bridge, i.e., a periodic Brownian motion of period T. In both cases, we compute analytically the first few moments of τ, as well as the covariance of t_{max} and t_{min}, showing that these times are anticorrelated. We demonstrate that the distribution of τ for Brownian motion is valid for discrete-time random walks with n steps and with a finite jump variance, in the limit n→∞. In the case of Lévy flights, which have a divergent jump variance, we numerically verify that the distribution of τ differs from the Brownian case. For random walks with continuous and symmetric jumps we numerically verify that the probability of the event "τ=n" is exactly 1/(2n) for any finite n, independently of the jump distribution. Our results can be also applied to describe the distance between the maximal and minimal height of (1+1)-dimensional stationary-state Kardar-Parisi-Zhang interfaces growing over a substrate of finite size L. Our findings are confirmed by numerical simulations. Some of these results have been announced in a recent Letter [Phys. Rev. Lett. 123, 200201 (2019)PRLTAO0031-900710.1103/PhysRevLett.123.200201].

Time Between the Maximum and the Minimum of a Stochastic Process

We present an exact solution for the probability density function P(τ=t_{min}-t_{max}|T) of the time difference between the minimum and the maximum of a one-dimensional Brownian motion of duration T. We then generalize our results to a Brownian bridge, i.e., a periodic Brownian motion of period T. We demonstrate that these results can be directly applied to study the position difference between the minimal and the maximal heights of a fluctuating (1+1)-dimensional Kardar-Parisi-Zhang interface on a substrate of size L, in its stationary state. We show that the Brownian motion result is universal and, asymptotically, holds for any discrete-time random walk with a finite jump variance. We also compute this distribution numerically for Lévy flights and find that it differs from the Brownian motion result.

First passage in an interval for fractional Brownian motion

Let X_{t} be a random process starting at x∈[0,1] with absorbing boundary conditions at both ends of the interval. Denote by P_{1}(x) the probability to first exit at the upper boundary. For Brownian motion, P_{1}(x)=x, which is equivalent to P_{1}^{'}(x)=1. For fractional Brownian motion with Hurst exponent H, we establish that P_{1}^{'}(x)=N[x(1-x)]^{1/H-2}e^{εF(x)+O(ε^{2})}, where ε=H-1/2. The function F(x) is analytic and well approximated by its Taylor expansion F(x)≃16(C-1)(x-1/2)^{2}+O(x-1/2)^{4}, where C=0.915... is the Catalan constant. A similar result holds for moments of the exit time starting at x. We then consider the span of X_{t}, i.e., the size of the (compact) domain visited up to time t. For Brownian motion, we derive an analytic expression for the probability that the span reaches 1 for the first time and then generalize it to fractional Brownian motion. Using large-scale numerical simulations with system sizes up to N=2^{24} and a broad range of H, we confirm our analytic results. There are important finite-discretization corrections which we quantify. They are most severe for small H, necessitating going to the large systems mentioned above.

Universal first-passage statistics in aging media

It has been known for a long time that the kinetics of diffusion-limited reactions can be quantified by the time needed for a diffusing molecule to reach a target: the first-passage time (FPT). So far the general determination of the mean first-passage time to a target in confinement has left aside aging media, such as glassy materials, cellular media, or cold atoms in optical lattices. Here we consider general non-Markovian scale-invariant diffusion processes, which model a broad class of transport processes of molecules in aging media, and demonstrate that all the moments of the FPT obey universal scalings with the confining volume with nontrivial exponents. Our analysis shows that a nonlinear scaling with the volume of the mean FPT, which quantities the mean reaction time, is the hallmark of aging and provides a general tool to quantify its impact on reaction kinetics in confinement.

Log-correlated random-energy models with extensive free-energy fluctuations: Pathologies caused by rare events as signatures of phase transitions

We address systematically an apparent nonphysical behavior of the free-energy moment generating function for several instances of the logarithmically correlated models: the fractional Brownian motion with Hurst index H=0 (fBm0) (and its bridge version), a one-dimensional model appearing in decaying Burgers turbulence with log-correlated initial conditions and, finally, the two-dimensional log-correlated random-energy model (logREM) introduced in Cao et al. [Phys. Rev. Lett. 118, 090601 (2017)PRLTAO0031-900710.1103/PhysRevLett.118.090601] based on the two-dimensional Gaussian free field with background charges and directly related to the Liouville field theory. All these models share anomalously large fluctuations of the associated free energy, with a variance proportional to the log of the system size. We argue that a seemingly nonphysical vanishing of the moment generating function for some values of parameters is related to the termination point transition (i.e., prefreezing). We study the associated universal log corrections in the frozen phase, both for logREMs and for the standard REM, filling a gap in the literature. For the above mentioned integrable instances of logREMs, we predict the nontrivial free-energy cumulants describing non-Gaussian fluctuations on the top of the Gaussian with extensive variance. Some of the predictions are tested numerically.

Perturbative expansion for the maximum of fractional Brownian motion

Brownian motion is the only random process which is Gaussian, scale invariant, and Markovian. Dropping the Markovian property, i.e., allowing for memory, one obtains a class of processes called fractional Brownian motion, indexed by the Hurst exponent H. For H=1/2, Brownian motion is recovered. We develop a perturbative approach to treat the nonlocality in time in an expansion in ɛ=H-1/2. This allows us to derive analytic results beyond scaling exponents for various observables related to extreme value statistics: the maximum m of the process and the time t_{max} at which this maximum is reached, as well as their joint distribution. We test our analytical predictions with extensive numerical simulations for different values of H. They show excellent agreement, even for H far from 1/2.

Maximum of a Fractional Brownian Motion: Analytic Results from Perturbation Theory

Fractional Brownian motion is a non-Markovian Gaussian process X_{t}, indexed by the Hurst exponent H. It generalizes standard Brownian motion (corresponding to H=1/2). We study the probability distribution of the maximum m of the process and the time t_{max} at which the maximum is reached. They are encoded in a path integral, which we evaluate perturbatively around a Brownian, setting H=1/2+ϵ. This allows us to derive analytic results beyond the scaling exponents. Extensive numerical simulations for different values of H test these analytical predictions and show excellent agreement, even for large ϵ.

Stationary states in two-dimensional systems driven by bivariate Lévy noises

Systems driven by α-stable noises could be very different from their Gaussian counterparts. Stationary states in single-well potentials can be multimodal. Moreover, a potential well needs to be steep enough in order to produce stationary states. Here it is demonstrated that two-dimensional (2D) systems driven by bivariate α-stable noises are even more surprising than their 1D analogs. In 2D systems, intriguing properties of stationary states originate not only due to heavy tails of noise pulses, which are distributed according to α-stable densities, but also because of properties of spectral measures. Consequently, 2D systems are described by a whole family of Langevin and fractional diffusion equations. Solutions of these equations bear some common properties, but also can be very different. It is demonstrated that also for 2D systems potential wells need to be steep enough in order to produce bounded states. Moreover, stationary states can have local minima at the origin. The shape of stationary states reflects symmetries of the underlying noise, i.e., its spectral measure. Finally, marginal densities in power-law potentials also have power-law asymptotics.

First-passage statistics for aging diffusion in systems with annealed and quenched disorder

Aging, the dependence of the dynamics of a physical process on the time ta since its original preparation, is observed in systems ranging from the motion of charge carriers in amorphous semiconductors over the blinking dynamics of quantum dots to the tracer dispersion in living biological cells. Here we study the effects of aging on one of the most fundamental properties of a stochastic process, the first-passage dynamics. We find that for an aging continuous time random walk process, the scaling exponent of the density of first-passage times changes twice as the aging progresses and reveals an intermediate scaling regime. The first-passage dynamics depends on ta differently for intermediate and strong aging. Similar crossovers are obtained for the first-passage dynamics for a confined and driven particle. Comparison to the motion of an aged particle in the quenched trap model with a bias shows excellent agreement with our analytical findings. Our results demonstrate how first-passage measurements can be used to unravel the age ta of a physical system.

Anomalous fluctuations of currents in Sinai-type random chains with strongly correlated disorder

We study properties of a random walk in a generalized Sinai model, in which a quenched random potential is a trajectory of a fractional Brownian motion with arbitrary Hurst parameter H, 0<H<1, so that the random force field displays strong spatial correlations. In this case, the disorder-average mean-square displacement grows in proportion to log(2/H)(n), n being time. We prove that moments of arbitrary order k of the steady-state current J(L) through a finite segment of length L of such a chain decay as L(-(1-H)), independently of k, which suggests that despite a logarithmic confinement the average current is much higher than its Fickian counterpart in homogeneous systems. Our results reveal a paradoxical behavior such that, for fixed n and L, the mean-square displacement decreases when one varies H from 0 to 1, while the average current increases. This counterintuitive behavior is explained via an analysis of representative realizations of disorder.

Counting statistics: a Feynman-Kac perspective

By building upon a Feynman-Kac formalism, we assess the distribution of the number of collisions in a given region for a broad class of discrete-time random walks in absorbing and nonabsorbing media. We derive the evolution equation for the generating function of the number of collisions, and we complete our analysis by examining the moments of the distribution and their relation to the walker equilibrium density. Some significant applications are discussed in detail: in particular, we revisit the gambler's ruin problem and generalize to random walks with absorption the arcsine law for the number of collisions on the half-line.

Perturbation theory for fractional Brownian motion in presence of absorbing boundaries

Fractional Brownian motion is a Gaussian process x(t) with zero mean and two-time correlations (x(t(1))x(t(2)))=D(t(1)(2H)+t(2)(2H)-|t(1)-t(2)|(2H)), where H, with 0<H<1, is called the Hurst exponent. For H=1/2, x(t) is a Brownian motion, while for H≠1/2, x(t) is a non-Markovian process. Here we study x(t) in presence of an absorbing boundary at the origin and focus on the probability density P(+)(x,t) for the process to arrive at x at time t, starting near the origin at time 0, given that it has never crossed the origin. It has a scaling form P(+)(x,t)~t(-H)R(+)(x/t(H)). Our objective is to compute the scaling function R(+)(y), which up to now was only known for the Markov case H=1/2. We develop a systematic perturbation theory around this limit, setting H=1/2+ε, to calculate the scaling function R(+)(y) to first order in ε. We find that R(+)(y) behaves as R(+)(y)~y(ϕ) as y→0 (near the absorbing boundary), while R(+)(y)~y(γ)exp(-y(2)/2) as y→∞, with ϕ=1-4ε+O(ε(2)) and γ=1-2ε+O(ε(2)). Our ε-expansion result confirms the scaling relation ϕ=(1-H)/H proposed in Zoia, Rosso, and Majumdar [Phys. Rev. Lett. 102, 120602 (2009)]. We verify our findings via numerical simulations for H=2/3. The tools developed here are versatile, powerful, and adaptable to different situations.

Fractional Lévy stable motion can model subdiffusive dynamics

We show in this paper that the sample (time average) mean-squared displacement (MSD) of the fractional Lévy α -stable motion behaves very differently from the corresponding ensemble average (second moment). While the ensemble average MSD diverges for α<2 , the sample MSD may exhibit either subdiffusion, normal diffusion, or superdiffusion. Thus, H -self-similar Lévy stable processes can model either a subdiffusive, diffusive or superdiffusive dynamics in the sense of sample MSD. We show that the character of the process is controlled by a sign of the memory parameter d=H-1/α . We also introduce a sample p -variation dynamics test which allows to distinguish between two models of subdiffusive dynamics. Finally, we illustrate a subdiffusive behavior of the fractional Lévy stable motion on biological data describing the motion of individual fluorescently labeled mRNA molecules inside live E. coli cells, but it may concern many other fields of contemporary experimental physics.