The evolution of the largest component has been studied intensely in a variety of random graph processes, starting in 1960 with the Erdős-Rényi process (ER). It is well known that this process undergoes a phase transition at n/2 edges when, asymptotically almost surely, a linear-sized component appears. Moreover, this phase transition is continuous, i.e., in the limit the function f(c) denoting...

To achieve traceability in discontinuous (discrete or batch) processes can be rather straightforward since identification markers, such as bar codes or serial numbers, can be attached to the product/batch or its package and be scanned at predetermined locations in the supply chain. However, continuous processes differ from discontinuous processes in a number of ways and these differences result...

Minimal thinness is a notion that describes the smallness of a set at a boundary point. In this paper, we provide tests for minimal thinness at finite and infinite minimal Martin boundary points for a large class of purely discontinuous symmetric Lévy processes.

We establish an integration by parts formula in an abstract framework in order to study the regularity of the law for processes solution of stochastic differential equations with jumps, including equations with discontinuous coefficients for which the Malliavin calculus developed by Bismut and Bichteler, Gravereaux and Jacod fails. 2000 MSC. Primary: 60H07, Secondary 60G51

The coefficients in the decomposition of the excursions measure as convex combination of excursions measures of reflected processes are computed in order to characterize the discontinuity at one point of the diffusion coefficient. In some sense, this result extends to general diffusions a similar one for the skew Brownian motion, and we advocate it may be used in Monte Carlo methods for discont...

We report the discovery of a discrete hierarchy of microtransitions occurring in models of continuous and discontinuous percolation. The precursory microtransitions allow us to target almost deterministically the location of the transition point to global connectivity. This extends to the class of intrinsically stochastic processes the possibility to use warning signals anticipating phase trans...

1 Abstract This chapter summarizes our recent work on the dynamic resource allocation problem. The question of interest is the performance of simple allocation strategies which can be implemented on-line. The chapter fo-cuses on the least load routing policy. The analysis is based on uid limit equations and the theory of large deviations for Markov processes with discontinuous statistics.

In this paper, we provide a scheme for simulating one-dimensional processes generated by divergence or non-divergence form operators with discontinuous coe cients. We use a space bijection to transform such a process in another one that behaves locally like a Skew Brownian motion. Indeed the behavior of the Skew Brownian motion can easily be approached by an asymmetric random walk.

Feynman-Kac transforms driven by discontinuous additive functionals are studied in this paper for a large class of Markov processes. General gauge and conditional gauge theorems are established for such transforms. Furthermore, the L2-infinitesimal generator of the Schrödinger semigroup given by a non-local Feynman-Kac transform is determined in terms of its associated bilinear form.