The problem of computing the moment generating function of the first passage time T to a > 0 or −b < 0 for a one-dimensional Wiener process {X(t), t ≥ 0} is generalized by assuming that the infinitesimal parameters of the process may depend on the sign of X(t). The probability that the process is absorbed at a is also computed explicitly, as is the expected value of T .

The discrete Green’s function (without boundary) G is a pseudo-inverse of the combinatorial Laplace operator of a graph G = (V,E). We reveal the intimate connection between Green’s function and the theory of exact stopping rules for random walks on graphs. We give an elementary formula for Green’s function in terms of state-to-state hitting times of the underlying graph. Namely, G(i, j) = πj (∑...

The hitting time, huv , of a random walk on a finite graph G, is the expected time for the walk to reach vertex v given that it started at vertex u. We present two methods of calculating the hitting time between vertices of finite graphs, along with applications to specific classes of graphs, including grids, trees, and the ’tadpole’ graphs. keywords: random walks, hitting time

We show that the hitting time of the discrete time quantum random walk on the n-bit hypercube from one corner to its opposite is polynomial in n. This gives the first exponential quantum-classical gap in the hitting time of discrete quantum random walks. We provide the framework for quantum hitting time and give two alternative definitions to set the ground for its study on general graphs. We t...

We study the complexity of the Hitting Set problem in set systems (hypergraphs) that avoid certain sub-structures. In particular, we characterize the classical and parameterized complexity of the problem when the Vapnik-Chervonenkis dimension (VC-dimension) of the input is small. VC-dimension is a natural measure of complexity of set systems. Several tractable instances of Hitting Set with a ge...

We define encodings of a Process Hitting into a Petri Net and conversely. It results notably that • the Process Hitting is a sub-class of Petri Nets; • any safe Petri Net can be encoded into a weakly-bisimilar Process Hittings with 3 priority classes. We finally discuss the use of the encoding of a Petri Net into a Process Hitting for providing efficient static analysis of dynamical properties.

A hitting-set generator is a deterministic algorithm that generates a set of strings such that this set intersects every dense set that is recognizable by a small circuit. A polynomial time hitting-set generator readily implies RP = P , but it is not apparent what this implies for BPP. Nevertheless, Andreev et al. (ICALP’96, and JACM 1998) showed that a polynomial-time hitting-set generator imp...

A hitting-set generator is a deterministic algorithm which generates a set of strings that intersects every dense set recognizable by a small circuit. A polynomial time hitting-set generator readily implies $RP=P$. Andreev et. Al (ICALP’96 and JACM 1998) showed that if polynomial-time hitting-set generator in fact implies the much stronger conclusion $BPP=P$. We simplify and improve their (and ...