Last-Passage Time for Linear Diffusions and Application to the Emptying Time of a Box
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Last-Passage Time for Linear Diffusions and Application to the Emptying Time of a Box Alain Comtet1 · Françoise Cornu1 · Grégory Schehr1 Received: 21 June 2020 / Accepted: 12 September 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract We study the statistics of last-passage time for linear diffusions. First we present an elementary derivation of the Laplace transform of the probability density of the last-passage time, thus recovering known results from the mathematical literature. We then illustrate them on several explicit examples. In a second step we study the spectral properties of the Schrödinger operator associated to such diffusions in an even potential U (x) = U (−x), unveiling the role played by the so-called Weyl coefficient. Indeed, in this case, our approach allows us to relate the last-passage times for dual diffusions (i.e., diffusions driven by opposite force fields) and to obtain new explicit formulae for the mean last-passage time. We further show that, for such even potentials, the small time t expansion of the mean last-passage time on the interval [0, t] involves the Korteveg–de Vries invariants, which are well known in the theory of Schrödinger operators. Finally, we apply these results to study the emptying time of a one-dimensional box, of size L, containing N independent Brownian particles subjected to a constant drift. In the scaling limit where both N → ∞ and L → ∞, keeping the density ρ = N /L fixed, we show that the limiting density of the emptying time is given by a Gumbel distribution. Our analysis provides a new example of the applications of extreme value statistics to out-of-equilibrium systems. Keywords Stochastic processes · Last-passage times · First-passage times · Extreme value statistics
1 Introduction First-passage times of random walks or diffusion processes have been extensively studied in the physics literature [1–6]. Several physical or chemical properties are controlled by first-passage events [1,2], a key example being reaction rates. In the Kramers approach they are given in terms of the inverse mean first-passage time [3]. First-passage times also play a crucial role in the study of transport properties, in particular in the context of biochemical
Communicated by Sidney Redner.
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Grégory Schehr [email protected] Université Paris-Saclay, CNRS, LPTMS, 91405 Orsay, France
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reactions [6–8] and in the description of persistence properties of spatially extended systems [5]. However, there are several cases of physical interest where the mean first-passage does not give the proper time scale. In particular, in nuclear physics, it is known that the fission rates of heavy nuclei are best described in terms of last-passage time events [9]. The simpler case of the overdamped motion of a particle in a potential well already allows for a better understanding of the problem. In this case, it has been shown [10] that the mean last-passage time provides a more accurate formula for the escape rate at a saddle poin
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