N G van Kampen 1981 Amsterdam: North-Holland xiv + 419 pp price Dfl 180 This is a book which, at a lower price, could be expected to become an essential part of the library of every physical scientist concerned with problems involving fluctuations and stochastic processes, as well as those who just enjoy a beautifully written book. It provides an extensive graduate-level introduction which is clear, cautious, interesting and readable.

https://iopscience.iop.org/article/10.1088/0031-9112/34/4/040/pdf

Stochastic Processes in Physics and Chemistry

Proceedings of the American Mathematical Society

Abstract The remaining travel time of a plane shortens with every minute that passes from its departure, and a flame diminishes a candle with every second it burns. Such everyday occurrences bias us to think that processes which have already begun will end before those which have just started. Yet, the inspection paradox teaches us that the converse can also happen when randomness is at play. The paradox comes from probability theory, where it is often illustrated by measuring how long passengers wait upon arriving at a bus stop at a random time. Interestingly, such passengers may on average wait longer than the mean time between bus arrivals—a counter-intuitive result, since one expects to wait less when coming some time after the previous bus departed. In this viewpoint , we review the inspection paradox and its origins. The insight gained is then used to explain why, and under which conditions, stochastic resetting expedites the completion of random processes. Importantly, this is done with elementary mathematical tools which help develop a probabilistic intuition for stochastic resetting and how it works. This viewpoint can thus be used as an accessible introduction to the subject. 

https://iopscience.iop.org/article/10.1088/1751-8121/ac3cdf/pdf

The inspection paradox in stochastic resetting

Stochastic processes offer a fundamentally different paradigm of dynamics than deterministic processes, the most prominent example of the latter being Newton's laws of motion. Here, we discuss in a pedagogical manner a simple and illustrative example of stochastic processes in the form of a particle undergoing standard Brownian diffusion, with the additional feature of the particle resetting repeatedly and at random times to its initial condition. Over the years, many different variants of this simple setting have been studied, all of which serve as illustrations of non-trivial and interesting static and dynamic features that characterize stochastic dynamics at long times. We will provide in this work a brief overview of this active and rapidly evolving field by considering the arguably simplest example of Brownian diffusion in one dimension. Along the way, we will learn about some of the general techniques that a physicist employs to study stochastic processes. Relevant to the special issue, we will discuss in detail how introducing resetting in an otherwise diffusive dynamics provides an explicit optimization of the time to locate a target through a special choice of the resetting protocol. We also discuss thermodynamics of resetting, and provide a bird's eye view of some of the recent work in the field of resetting. 

/pdf/stochastic-resetting-a-very-brief-review-1971jqub.pdf

Stochastic Resetting: A (Very) Brief Review

We study the effects of stochastic resetting on geometric Brownian motion with drift (GBM), a canonical stochastic multiplicative process for nonstationary and nonergodic dynamics. Resetting is a sudden interruption of a process, which consecutively renews its dynamics. We show that, although resetting renders GBM stationary, the resulting process remains nonergodic. Quite surprisingly, the effect of resetting is pivotal in manifesting the nonergodic behavior. In particular, we observe three different long-time regimes: a quenched state, an unstable state, and a stable annealed state depending on the resetting strength. Notably, in the last regime, the system is self-averaging and thus the sample average will always mimic ergodic behavior establishing a stand-alone feature for GBM under resetting. Crucially, the above-mentioned regimes are well separated by a self-averaging time period which can be minimized by an optimal resetting rate. Our results can be useful to interpret data emanating from stock market collapse or reconstitution of investment portfolios.

Geometric Brownian motion under stochastic resetting: A stationary yet nonergodic process

We investigate classic diffusion with the added feature that a diffusing particle is reset to its starting point each time the particle reaches a specified threshold. In an infinite domain, this process is nonstationary and its probability distribution exhibits rich features. In a finite domain, we define a nontrivial optimization in which a cost is incurred whenever the particle is reset and a reward is obtained while the particle stays near the reset point. We derive the condition to optimize the net gain in this system, namely, the reward minus the cost.

Optimization in First-Passage Resetting.

We introduce a resetting Brownian bridge as a simple model to study search processes where the total search time t_{f} is finite and the searcher returns to its starting point at t_{f}. This is simply a Brownian motion with a Poissonian resetting rate r to the origin which is constrained to start and end at the origin at time t_{f}. We unveil a surprising general mechanism that enhances fluctuations of a Brownian bridge, by introducing a small amount of resetting. This is verified for different observables, such as the mean-square displacement, the hitting probability of a fixed target and the expected maximum. This mechanism, valid for a Brownian bridge in arbitrary dimensions, leads to a finite optimal resetting rate that minimizes the time to search a fixed target. The physical reason behind an optimal resetting rate in this case is entirely different from that of resetting Brownian motions without the bridge constraint. We also derive an exact effective Langevin equation that generates numerically the trajectories of a resetting Brownian bridge in all dimensions via a completely rejection-free algorithm.

Optimal Resetting Brownian Bridges via Enhanced Fluctuations.

We combine the processes of resetting and first-passage to define \emph{first-passage resetting}, where the resetting of a random walk to a fixed position is triggered by a first-passage event of the walk itself. In an infinite domain, first-passage resetting of isotropic diffusion is non-stationary, with the number of resetting events growing with time as $\sqrt{t}$. We calculate the resulting spatial probability distribution of the particle analytically, and also obtain this distribution by a geometric path decomposition. In a finite interval, we define an optimization problem that is controlled by first-passage resetting; this scenario is motivated by reliability theory. The goal is to operate a system close to its maximum capacity without experiencing too many breakdowns. However, when a breakdown occurs the system is reset to its minimal operating point. We define and optimize an objective function that maximizes the reward (being close to maximum operation) minus a penalty for each breakdown. We also investigate extensions of this basic model to include delay after each reset and to two dimensions. Finally, we study the growth dynamics of a domain in which the domain boundary recedes by a specified amount whenever the diffusing particle reaches the boundary after which a resetting event occurs. We determine the growth rate of the domain for the semi-infinite line and the finite interval and find a wide range of behaviors that depend on how much the recession occurs when the particle hits the boundary.

Optimization and Growth in First-Passage Resetting

Universal Order Statistics for Random Walks & Lévy Flights

We investigate classic diffusion with the added feature that a diffusing particle is reset to its starting point each time the particle reaches a specified threshold. In an infinite domain, this process is non-stationary and its probability distribution exhibits rich features. In a finite domain, we define a non-trivial optimization in which a cost is incurred whenever the particle is reset and a reward is obtained while the particle stays near the reset point. We derive the condition to optimize the net gain in this system, namely, the reward minus the cost.

First-Passage Resetting and Optimization

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