LAUSR

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 Schrodinger 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 Schrodinger 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 $$N \rightarrow \infty $$ and $$L \rightarrow \infty $$ , 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.