Given a survival distribution on the positive half-axis and a Brownian motion, a solution of the inverse first-passage problem consists of a boundary so that the first passage time over the boundary has the given distribution. We show that the solution of the inverse firstpassage problem coincides with the solution of a related optimal stopping problem. Consequently, methods from optimal stoppi...

We study a first passage time problem for a class of spectrally positive Lévy processes. By considering the special case where the Lévy process is a compound Poisson process with negative drift, we obtain the Laplace-Stieltjes transform of the steady-state waiting time distribution of low-priority customers in a twoclass M/GI/1 queue operating under a dynamic non-preemptive priority discipline....

Previous work showed how moving particles that rest along their trajectory lead to timenonlocal advection–dispersion equations. If the waiting times have infinite mean, the model equation contains a fractional time derivative of order between 0 and 1. In this article, we develop a new advection–dispersion equation with an additional fractional time derivative of order between 1 and 2. Solutions...

This work considers weak deterministic Büchi automata reading encodings of non-negative reals in a fixed base. A Real Number Automaton is an automaton which recognizes all encoding of elements of a set of reals. It is explained how to decide in linear time whether a set of reals recognized by a given minimal weak deterministic RNA is FO [R; +, <, 1]definable. Furthermore, it is explained how to...

We derive an explicit expression for the probability density of the first passage time to state 0 for the Ehrenfest diffusion model in continuous time.

It is shown that for any class C closed under linear-time reductions, the complete sets for C under sublogarithmic reductions are also complete under 2DFA reductions, and thus are isomorphic under first-order reductions.

The first-passage time of a Markov process to a moving barrier is considered as a first-exit time for a vector whose components include the process and the barrier. Thus when the barrier is itself a solution of a differential equation, the theory of first-exit times for multidimensional processes may be used to obtain differential equations for the moments and density of the first-passage time ...

We deene a class of modal logics LF by uniformly extending a class of modal logics L. Each logic L is characterised by a class of rst-order deenable frames, but the corresponding logic LF is sometimes characterised by classes of modal frames that are not rst-order deenable. The class LF includes provability logics with deep arithmeti-cal interpretations. Using Belnap's proof-theoretical framewo...

Assume that the problem P0 is not solvable in polynomial time. Let T be a first-order theory containing a sufficiently rich part of true arithmetic. We characterize T∪{ConT } as the minimal extension of T proving for some algorithm that it decides P0 as fast as any algorithm B with the property that T proves that B decides P0. Here, ConT claims the consistency of T . As a byproduct, we obtain a...

We consider a simple random walk on a tree. Exact expressions are obtained for the expectation and the variance of the first passage time, thereby recovering the known result that these are integers. A relationship of the mean first passage matrix with the distance matrix is established and used to derive a formula for the inverse of the mean first passage matrix.