Longest excursion of stochastic processes in nonequilibrium systems

Controlling Uncertainty of Empirical First-Passage Times in the Small-Sample Regime

We derive general bounds on the probability that the empirical first-passage time τ[over ¯]_{n}≡∑_{i=1}^{n}τ_{i}/n of a reversible ergodic Markov process inferred from a sample of n independent realizations deviates from the true mean first-passage time by more than any given amount in either direction. We construct nonasymptotic confidence intervals that hold in the elusive small-sample regime and thus fill the gap between asymptotic methods and the Bayesian approach that is known to be sensitive to prior belief and tends to underestimate uncertainty in the small-sample setting. We prove sharp bounds on extreme first-passage times that control uncertainty even in cases where the mean alone does not sufficiently characterize the statistics. Our concentration-of-measure-based results allow for model-free error control and reliable error estimation in kinetic inference, and are thus important for the analysis of experimental and simulation data in the presence of limited sampling.

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.

Extreme value theory for constrained physical systems

We investigate extreme value theory for physical systems with a global conservation law which describes renewal processes, mass transport models, and long-range interacting spin models. As shown previously, a special feature is that the distribution of the extreme value exhibits a nonanalytical point in the middle of the support. We expose exact relationships between constrained extreme value theory and well-known quantities of the underlying stochastic dynamics, all valid beyond the midpoint in general, i.e., even far from the thermodynamic limit. For example, for renewal processes the distribution of the maximum time between two renewal events is exactly related to the mean number of these events. In the thermodynamic limit, we show how our theory is suitable to describe typical and rare events which deviate from classical extreme value theory. For example, for the renewal process we unravel dual scaling of the extreme value distribution, pointing out two types of limiting laws: a normalizable scaling function for the typical statistics and a non-normalized state describing the rare events.

Exact extreme-value statistics at mixed-order transitions

We study extreme-value statistics for spatially extended models exhibiting mixed-order phase transitions (MOT). These are phase transitions that exhibit features common to both first-order (discontinuity of the order parameter) and second-order (diverging correlation length) transitions. We consider here the truncated inverse distance squared Ising model, which is a prototypical model exhibiting MOT, and study analytically the extreme-value statistics of the domain lengths The lengths of the domains are identically distributed random variables except for the global constraint that their sum equals the total system size L. In addition, the number of such domains is also a fluctuating variable, and not fixed. In the paramagnetic phase, we show that the distribution of the largest domain length l_{max} converges, in the large L limit, to a Gumbel distribution. However, at the critical point (for a certain range of parameters) and in the ferromagnetic phase, we show that the fluctuations of l_{max} are governed by novel distributions, which we compute exactly. Our main analytical results are verified by numerical simulations.

Memory effects in fractional Brownian motion with Hurst exponent H<1/3

We study the regression to the origin of a walker driven by dynamically generated fractional Brownian motion (FBM) and we prove that when the FBM scaling, i.e., the Hurst exponent H<1/3 , the emerging inverse power law is characterized by a power index that is a compelling signature of the infinitely extended memory of the system. Strong memory effects leads to the relation H=θ/2 between the Hurst exponent and the persistent exponent θ , which is different from the widely used relation H=1-θ . The latter is valid for 1/3<H<1 and is known to be compatible with the renewal assumption.

Longest excursion of fractional Brownian motion: numerical evidence of non-Markovian effects

We study, using exact numerical simulations, the statistics of the longest excursion l(max)(t) up to time t for the fractional Brownian motion with Hurst exponent 0<H<1. We show that in the large t limit, <l(max)(t)> proportional to variantQ(infinity)t, where Q(infinity) identical with Q(infinity)(H) depends continuously on H. These results are compared with exact analytical results for a renewal process with an associated persistence exponent theta=1-H. This comparison shows that Q(infinity)(H) carries the clear signature of non-Markovian effects for H not equal 1/2. The preasymptotic behavior of <l(max)(t)> is also discussed.

Universality of persistence exponents in two-dimensional Ostwald ripening

We measured persistence exponents theta(phi) of Ostwald ripening in two dimensions, as a function of the area fraction phi occupied by coarsening domains. The values of theta(phi) in two systems, succinonitrile and brine, quenched to their liquid-solid coexistence region, compare well with one another, providing compelling evidence for the universality of the one-parameter family of exponents. For small phi, theta(phi) approximately = 0.39phi, as predicted by a model that assumes no correlations between evolving domains. These constitute the first measurements of persistence exponents in the case of phase transitions with a conserved order parameter.