Cramér asymptotics for finite time first passage probabilities of general Lévy processes

Asymptotics of rare events in birth–death processes bypassing the exact solutions

We investigate the near-continuum asymptotics of mean first passage times in some one-variable birth–death processes. The particular problem we address is how to extract mean first passage times in the near-continuum limit from their defining finite-difference equations alone. For the simple class of processes we consider here, exact closed-form solutions for the mean first passage time between...

Asymptotic Behaviour of First Passage Time Distributions for Lévy Processes

Overshoots and undershoots of Lévy Processes

We obtain a new fluctuation identity for a general Lévy process giving a quintuple law describing the time of first passage, the time of the last maximum before first passage, the overshoot, the undershoot and the undershoot of the last maximum. With the help of this identity, we revisit the results of Klüppelberg, Kyprianou and Maller [Ann. Appl. Probab. 14 (2004) 1766–1801] concerning asympto...

Universal proximity effect in target search kinetics in the few-encounter limit

When does a diffusing particle reach its target for the first time? This first-passage time (FPT) problem is central to the kinetics of molecular reactions in chemistry and molecular biology. Here, we explain the behavior of smooth FPT densities, for which all moments are finite, and demonstrate universal yet generally non-Poissonian long-time asymptotics for a broad variety of transport proces...

An Analogue of the Lévy-cramér Theorem for Multi-dimensional Rayleigh Distributions

In the present paper we prove that every k-dimensional Cartesian product of Kingman convolutions can be embedded into a k-dimensional symmetric convolution (k=1, 2, . . . ) and obtain an analogue of the Cramér-Lévy theorem for multi-dimensional Rayleigh distributions. A new and more general class of multi-dimensional Rayleygh distributions and associated higher dimensional Bessel processes are ...