Last-Passage Time for Linear Diffusions and Application to the Emptying Time of a Box – Archive ouverte HAL

Alain Comtet 1 Françoise Cornu 1 Grégory Schehr 1

Alain Comtet, Françoise Cornu, Grégory Schehr. Last-Passage Time for Linear Diffusions and Application to the Emptying Time of a Box. Journal of Statistical Physics, Springer Verlag, 2020, ⟨10.1007/s10955-020-02637-6⟩. ⟨hal-02988500⟩

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 Schr\"{o}dinger 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 Schr\"odinger 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 $N \to \infty$ and $L \to \infty$, keeping the density $\rho = N/L$ 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.

• 1. LPTMS - Laboratoire de Physique Théorique et Modèles Statistiques